Research Article On Newton-Kantorovich Method for Solving the...

Research ArticleOn Newton-Kantorovich Method for Solving the NonlinearOperator Equation

Hameed Husam Hameed12 Z K Eshkuvatov34

Anvarjon Ahmedov45 and N M A Nik Long14

1Department of Mathematics Faculty of Science Universiti Putra Malaysia (UPM) Selangor Malaysia2Technical Institute of Alsuwerah The Middle Technical University Baghdad Iraq3Faculty of Science and Technology Universiti Sains Islam Malaysia (USIM) Negeri Sembilan Malaysia4Institute for Mathematical Research Universiti Putra Malaysia (UPM) Selangor Malaysia5Department of Process and Food Engineering Faculty of Engineering Universiti Putra Malaysia (UPM) Selangor Malaysia

Correspondence should be addressed to Z K Eshkuvatov zainidinupmedumy

Received 19 July 2014 Accepted 6 October 2014

Academic Editor Gaohang Yu

Copyright copy 2015 Hameed Husam Hameed et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We develop the Newton-Kantorovich method to solve the system of 2times2 nonlinear Volterra integral equations where the unknownfunction is in logarithmic form A new majorant function is introduced which leads to the increment of the convergence intervalThe existence and uniqueness of approximate solution are proved and a numerical example is provided to show the validation ofthe method

1 Introduction

Nonlinear phenomenon appears in many scientific areassuch as physics fluid mechanics population models chem-ical kinetics economic systems and medicine and can bemodeled by system of nonlinear integral equations Thedifficulty lies in finding the exact solution for such systemAlternatively the approximate or numerical solutions canbe sought One of the well known approximate method isNewton-Kantorovich method which reduces the nonlinearinto sequence of linear integral equations The the approxi-mate solution is then obtained by processing the convergentsequence In 1939 Kantorovich [1] presented an iterativemethod for functional equation in Banach space and derivedthe convergence theorem for Newton method In 1948Kantorovich [2] proved a semilocal convergence theoremfor Newton method in Banach space later known as theNewton-Kantorovich method Uko and Argyros [3] proveda weak Kantorovich-type theorem which gives the sameconclusion under theweaker conditions Shen and Li [4] have

established the Kantorovich-type convergence criterion forinexact Newton methods assuming that the first derivativeof an operator satisfies the Lipschitz condition Argyros [5]provided a sufficient condition for the semilocal convergenceofNewtonrsquosmethod to a locally unique solution of a nonlinearoperator equation Saberi-Nadjafi andHeidari [6] introduceda combination of the Newton-Kantorovich and quadraturemethods to solve the nonlinear integral equation of Urysohntype in the systematic procedure Ezquerro et al [7] studiedthe nonlinear integral equation of mixed Hammerstein typeusing Newton-Kantorovich method with majorant principleEzquerro et al [8] provided the semilocal convergence ofNewton method in Banach space under a modification of theclassic conditions of Kantorovich There are many methodsof solving the system of nonlinear integral equations forexample product integration method [9] Adomian method[10] RBF network method [11] biorthogonal system method[12] Chebyshevwaveletsmethod [13] analyticalmethod [14]reproducing kernel method [15] step method [16] and singleterm Wlash series [17] In 2003 Boikov and Tynda [18]

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015 Article ID 219616 12 pageshttpdxdoiorg1011552015219616

2 Abstract and Applied Analysis

implemented theNewton-Kantorovichmethod to the follow-ing system

119909 (119905) minus int

119905

119910(119905)

ℎ (119905 120591) 119892 (120591) 119909 (120591) 119889120591 = 0

int

119905

119910(119905)

119896 (119905 120591) [1 minus 119892 (120591)] 119909 (120591) 119889120591 = 119891 (119905)

(1)

where 0 lt 1199050le 119905 le 119879 119910(119905) lt 119905 and the functions ℎ(119905 120591)

119896(119905 120591) isin 119862[1199050 119879]times[1199050 119879]

119891(119905) 119892(119905) isin 119862[1199050 119879]

and (0 lt 119892(119905) lt 1)In 2010 Eshkuvatov et al [19] used the Newton-Kantorovichhypothesis to solve the system of nonlinear Volterra integralequation of the form

119909 (119905) minus int

119905

119910(119905)

ℎ (119905 120591) 1199092(120591) 119889120591 = 0

int

119905

119910(119905)

119896 (119905 120591) 1199092(120591) 119889120591 = 119891 (119905)

(2)

where 119909(119905) and 119910(119905) are unknown functions defined on[1199050infin) 119905

0gt 0 and ℎ(119905 120591) 119896(119905 120591) isin 119862

[1199050 infin]times[1199050 infin] 119891(119905) isin

119862[1199050 infin]

In 2010 Eshkuvatov et al [20] developed themodifiedNewton-Kantorovich to obtain an approximate solution ofsystem with the form

119909 (119905) minus int

119905

119910(119905)

119867(119905 120591) 119909119899(120591) 119889120591 = 0

int

119905

119910(119905)

119870 (119905 120591) 119909119899(120591) 119889120591 = 119891 (119905)

(3)

where 0 lt 1199050le 119905 le 119879 119910(119905) lt 119905 and the functions 119867(119905

120591) 119870(119905 120591) isin 119862[1199050 infin]times[1199050 infin]

119891(119905) isin 119862[1199050 infin]

and the unknownfunctions 119909(119905) isin 119862

[1199050 infin] 119910(119905) isin 119862

1

[1199050 infin] 119910(119905) lt 119905

In this paper we consider the systems of nonlinearintegral equation of the form

119909 (119905) minus int

119905

119910(119905)

ℎ (119905 120591) log |119909 (120591)| 119889120591 = 119892 (119905)

int

119905

119910(119905)

119896 (119905 120591) log |119909 (120591)| 119889120591 = 119891 (119905)

(4)

where 0 lt 1199050le 119905 le 119879 119910(119905) lt 119905 119909(119905) = 0 ℎ(119905 120591) ℎ

120591(119905 120591)

119896(119905 120591) 119896120591(119905 120591) isin 119862(119863) and the unknown functions 119909(119905) isin

119862[1199050 119879] 119910(119905) isin 119862

1[1199050 119879] to be determined and119863 = [119905

0 119879] times

[1199050 119879]The paper is organized as follows in Section 2 Newton-

Kantorovich method for the system of integral equations (4)is presented Section 3 deals with mixed method followedby discretizations In Section 4 the rate of convergence ofthemethod is investigated Lastly Section 5 demonstrates thenumerical example to verify the validity and accuracy of theproposed method followed by the conclusion in Section 6

2 Newton-Kantorovich Method for the System

Let us rewrite the system of nonlinear Volterra integral equa-tion (4) in the operator form

119875 (119883) = (1198751(119883) 119875

2(119883)) = 0 (5)

where119883 = (119909(119905) 119910(119905)) and

1198751(119883) = 119909 (119905) minus int

119905

119910(119905)

ℎ (119905 120591) log |119909 (120591)| 119889120591 minus 119892 (119905)

1198752(119883) = int

119905

119910(119905)

119896 (119905 120591) log |119909 (120591)| 119889120591 minus 119891 (119905)

(6)

To solve (5) we use initial iteration of Newton-Kantorovichmethod which is of the form

1198751015840(1198830) (119883 minus 119883

0) + 119875 (119883

0) = 0 (7)

where 1198830= (1199090(119905) 1199100(119905)) is the initial guess and 119909

0(119905) and

1199100(119905) can be any continuous functions provided that 119905

0lt

119910(119905) lt 119905 and 119909(119905) = 0The Frechet derivative of 119875(119883) at the point 119883

0is defined

as

1198751015840(1198830)119883

= ( lim119904rarr0

1

119904[1198751(1198830+ 119904119883) minus 119875

1(119883)]

lim119904rarr0

1

119904[1198752(1198830+ 119904119883) minus 119875

2(119883)])

= ( lim119904rarr0

1

119904[1198751(1199090+ 119904119909 119910

0+ 119904119910) minus 119875

1(1199090 1199100)]

lim119904rarr0

1

119904[1198752(1199090+ 119904119909 119910

0+ 119904119910) minus 119875

2(1199090 1199100)])

= ( lim119904rarr0

[1205971198751(1199090 1199100)

120597119909119904119909 +

1205971198751(1199090 1199100)

120597119910119904119910

+1

2(12059721198751

1205971199092(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059721198751

120597119909120597119910(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059721198751

1205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904119910

2)]

Abstract and Applied Analysis 3

lim119904rarr0

1

119904[1205971198752

120597119909(1199090 1199100) 119904119909 +

1205971198752

120597119910(1199090 1199100) 119904119910

+1

2(12059721198752

1205971199092(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059721198752

120597119909120597119910(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059721198752

1205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904119910

2)])

= (1205971198751(1199090 1199100)

120597119909119909 +

1205971198751(1199090 1199100)

120597119910119910

1205971198752(1199090 1199100)

120597119909119909 +

1205971198752(1199090 1199100)

120597119910119910)

(8)

Hence

1198751015840(1198830)119883 = (

1205971198751

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

1205971198751

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

1205971198752

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

1205971198752

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

)(119909

119910) (9)

From (7) and (9) it follows that

1205971198751

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

(Δ119909 (119905)) +1205971198751

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

(Δ119910 (119905))

= minus1198751(1199090(119905) 1199100(119905))

1205971198752

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

(Δ119909 (119905)) +1205971198752

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

(Δ119910 (119905))

= minus1198752(1199090(119905) 1199100(119905))

(10)

where Δ119909(119905) = 1199091(119905)minus119909

0(119905) Δ119910(119905) = 119910

1(119905)minus119910

0(119905) and (119909

0(119905)

1199100(119905)) is the initial given functions To solve (10) with respect

to Δ119909 and Δ119910 we need to compute all partial derivatives

1205971198751

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

= lim119904rarr0

1

119904(1198751(1199090+ 119904119909 119910

0) minus 1198751(1199090 1199100))

= lim119904rarr0

1

119904[119904119909 (119905)

minus int

119905

1199100(119905)

ℎ (119905 120591) (log 10038161003816100381610038161199090 (120591) + 119904119909 (120591)1003816100381610038161003816

minus log 10038161003816100381610038161199090 (120591)1003816100381610038161003816) 119889120591]

= 119909 (119905) minus int

119905

1199100(119905)

ℎ (119905 120591)119909 (120591)

1199090(120591)

119889120591

1205971198751

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

= lim119904rarr0

1

119904(1198751(1199090 1199100+ 119904119910) minus 119875

1(1199090 1199100))

= lim119904rarr0

1

119904[int

1199100(119905)+119904119910(119905)

1199100(119905)

ℎ (119905 120591) log 1003816100381610038161003816(1199090 (120591))1003816100381610038161003816 119889120591]

= ℎ (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 119910 (119905)

(11)

and in the same manner we obtain

1205971198752

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

= int

119905

1199100(119905)

119896 (119905 120591)119909 (120591)

1199090(120591)

119889120591

1205971198752

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

= minus119896 (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 119910 (119905)

(12)

So that from (10)ndash(12) it follows that

Δ119909 (119905) minus int

119905

1199100(119905)

ℎ (119905 120591)Δ119909 (120591)

1199090(120591)

119889120591

+ ℎ (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 Δ119910 (119905)

= int

119905

1199100(119905)

ℎ (119905 120591) log 10038161003816100381610038161199090 (120591)1003816100381610038161003816 119889120591 minus 119909

0(119905) + 119892 (119905)

int

119905

1199100(119905)

119896 (119905 120591)Δ119909 (120591)

1199090(120591)

119889120591

minus 119896 (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 Δ119910 (119905)

= minusint

119905

1199100(119905)

119896 (119905 120591) log 10038161003816100381610038161199090 (120591)1003816100381610038161003816 119889120591 + 119891 (119905)

(13)

Equation (13) is a linear and by solving it for Δ119909 and Δ119910 weobtain (119909

1(119905) 1199101(119905)) By continuing this process a sequence

of approximate solution (119909119898(119905) 119910119898(119905)) can be evaluated from

1198751015840(1198830) Δ119883119898+ 119875 (119883

119898) = 0 (14)

which is equivalent to the system

Δ119909119898(119905) minus int

119905

1199100(119905)

ℎ (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

+ ℎ (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 Δ119910119898 (119905)

= int

119905

1199100(119905)

ℎ (119905 120591) log 10038161003816100381610038161199090 (120591)1003816100381610038161003816 119889120591 minus 119909

0(119905) + 119892 (119905)

int

119905

1199100(119905)

119896 (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

minus 119896 (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 Δ119910119898 (119905)

= minusint

119905

1199100(119905)

119896 (119905 120591) log 10038161003816100381610038161199090 (120591)1003816100381610038161003816 119889120591 + 119891 (119905)

(15)

whereΔ119909119898(119905) = 119909

119898(119905)minus119909119898minus1

(119905) andΔ119910119898(119905) = 119910

119898(119905)minus119910119898minus1

(119905)119898 = 1 2 3


Thus one should solve a system of two linear Volterraintegral equations to find each successive approximationLet us eliminate Δ119910(119905) from the system (13) by finding theexpression of Δ119910(119905) from the first equation of this system andsubstitute it in the second equation to yield

Δ119910 (119905) =1

119867 (119905)[int

119905

1199100(119905)

ℎ (119905 120591) [Δ119909 (120591)

1199090(120591)

+ log 10038161003816100381610038161199090 (120591)1003816100381610038161003816] 119889120591

minus [Δ119909 (119905) + 1199090(119905) minus 119892 (119905)]]

119866 (119905) [int

119905

1199100(119905)

ℎ (119905 120591) [Δ119909 (120591)

1199090(120591)

+ log 10038161003816100381610038161199090 (120591)1003816100381610038161003816] 119889120591

minus [Δ119909 (119905) + 1199090(119905) minus 119892 (119905)]]

= int

119905

1199100(119905)

119896 (119905 120591)Δ119909 (120591)

1199090(120591)

119889120591

minus int

119905

1199100(119905)

119896 (119905 120591) log 10038161003816100381610038161199090 (120591)1003816100381610038161003816 119889120591 + 119891 (119905)

(16)

where 119866(119905) = 119896(119905 1199100(119905))ℎ(119905 119910

0(119905)) and 119867(119905) = 1[ℎ(119905

1199100(119905)) log |119909

0(1199100(119905))|] and the second equation of (16) yields

Δ119909 (119905) minus int

119905

1199100(119905)

1198961(119905 120591)

Δ119909 (120591)

1199090(120591)

120591 = 1198650(119905) (17)

where

1198961(119905 120591) = ℎ (119905 120591) minus

119896 (119905 120591)

119866 (119905)

119866 (119905) =119896 (119905 119910

0(119905))

ℎ (119905 1199100(119905))

119896 (119905 1199100(119905)) = 0 forall119905 isin [119905

0 119879]

1198650(119905) = int

119905

1199100(119905)

1198961(119905 120591) log 10038161003816100381610038161199090 (120591)

1003816100381610038161003816 119889120591 minus 1199090(119905) + 119892 (119905) +

119891 (119905)

119866 (119905)

(18)

In an analogous wayΔ119910119898(119905) andΔ119909

119898(119905) can be written in the

form

Δ119910119898(119905)

=1

119867 (119905)[int

119905

1199100(119905)

ℎ (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

+ int

119905

119910119898minus1(119905)

ℎ (119905 120591) log 1003816100381610038161003816119909119898minus1 (120591)1003816100381610038161003816 119889120591

minusΔ119909119898(119905) minus 119909

119898minus1(119905) + 119892 (119905)]

(19)

Δ119909119898(119905) minus int

119905

1199100(119905)

1198961(119905 120591)

Δ119909119898(120591)

1199090(120591)

119889120591 = 119865119898minus1

(119905) (20)

where

119865119898minus1

(119905) = int

119905

119910119898minus1(119905)

1198961(119905 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591 minus 119909119898minus1

(119905)

+ 119892 (119905) +119891 (119905)

119866 (119905)

(21)

3 The Mixed Method (Simpsonand Trapezoidal) for Approximate Solution

At each step of the iterative process we have to find thesolution of (18) and (20) on the closed interval [119905

0 119879] To do

this the grid (120596) of points 119905119894= 1199050+ 119894ℎ 119894 = 1 2 3 2119873

ℎ = (119879minus1199050)2119873 is introduced and by the collocationmethod

with mixed rule we require that the approximate solutionsatisfies (18) and (20) Hence

Δ119909119898(1199050) = minus119909

119898minus1(1199050) + 119892 (119905

0) +

119891 (1199050)

119866 (1199050) (22)

Δ119909119898(1199052119894) minus int

1199052119894

1199100(1199052119894)

1198961(1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

= 119865119898minus1

(1199052119894) 119894 = 1 2 119873

(23)

On the grid (120596) we set V2119894= 1199100(1199052119894) suct that

119905V2119894 = 119905V2119894 119905

0le 1199100(1199052119894) lt 1199052119894minus2

1199052119894 1199052119894minus2

le 1199100(1199052119894) lt 1199052119894

(24)

Consequently the system (23) can be written in the form

Δ119909119898(1199052119894) minus int

119905V2119894

1199100(1199052119894)

1198961(1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

minus

119894minus1

sum

119895=V2119894

int

1199052119895+2

1199052119895

1198961(1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

= 119865119898minus1

(1199052119894) 119894 = 1 2 119873

(25)

By computing the integral in (26) using tapezoidal formulaon the first integrals and Simpson formula on the secondintegral we consider two cases

Case 1When V2119894

= 2119894 119894 = 1 2 119873 then

Δ119909119898(1199052119894) =

119865119898minus1

(1199052119894) + 119860 (119894) + 119861 (119894) + 119862 (119894)

1 minus ((1199052119894minus 1199052119894minus2

) 6 sdot 1199090(1199052119894)) 1198961(1199052119894 1199052119894)

(26)


where

119860 (119894) = 05 (119905V2119894 minus 1199100(1199052119894))

times [1198961(1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ 1198961(1199052119894 1199100(1199052119894))

times

Δ119909119898(119905V2119894) (119905V2119894 minus 119910

0(1199052119894))

(119905V2119894 minus 119905V2119894minus2) (1199090 (1199100 (1199052119894)))

+ 1198961(1199052119894 1199100(1199052119894))

times

Δ119909119898(119905V2119894minus2) (1199100 (1199052119894) minus 119905V2119894minus2)

(119905V2119894 minus 119905V2119894minus2) (1199090 (1199100 (1199052119894)))]

119861 (119894) =

119894minus2

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times [1198961(1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 41198961(1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ 1198961(1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

]

119862 (119894) =(1199052119894minus 1199052119894minus2

)

6[1198961(1199052119894 1199052119894minus2

)Δ119909119898(1199052119894minus2

)

1199090(1199052119894minus2

)

+ 41198961(1199052119894 1199052119894minus1

)Δ119909119898(1199052119894minus1

)

1199090(1199052119894minus1

)]

(27)

Case 2When V2119894= 2119894 119894 = 1 2 119873 then

Δ119909119898(1199052119894) =

1198631(119894)

1198632(119894)

(28)

where

1198631(119894) = 119865

119898minus1(1199052119894) + 05119896

1(1199052119894 1199100(1199052119894))

times [Δ119909119898(1199052119894minus2

)

1199090(1199100(1199052119894))

(1199052119894minus 1199100(1199052119894)) (1199100(1199052119894) minus 1199052119894minus2

)

1199052119894minus 1199052119894minus2

]

1198632(119894) = [1 minus 05 (119905

2119894minus 1199100(1199052))

1198961(1199052119894 1199052119894)

1199090(1199052119894)

minus 051198961(1199052119894 1199100(1199052119894))

(1199052119894minus 1199100(1199052119894))2

1199090(1199100(1199052119894)) (1199052119894minus 1199052119894minus2

)]

(29)

Also to compute Δ119910119898(119905) on the grid (120596) (18) can be re-

presented in the form

Δ119910119898(1199052119894) =

1

119867 (1199052119894)

times [int

1199052119894

1199100(1199052119894)

ℎ (1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

+ int

1199052119894

119910119898minus1(1199052119894)

ℎ (1199052119894 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591

minusΔ119909119898(1199052119894) minus 119909119898minus1

(1199052119894) + 119892 (119905

2119894) ]

(30)

Let us set V2119894= 1199100(1199052119894) and 119906

2119894= 119910119898minus1

(1199052119894) and

119905V2119894 =

1199052119894 1199052119894minus2

le 1199100(1199052119894) lt 1199052119894

119905V2119894 1199050le 1199100(1199052119894) lt 1199052119894minus2

1199051199062119894

=

1199052119894 1199052119894minus2

le 119910119898minus1

(1199052119894) lt 1199052119894

1199051199062119894 1199050le 119910119898minus1

(1199052119894) lt 1199052119894minus2

(31)

Then (30) can be written as

Δ119910119898(1199052119894) =

1

119867 (1199052119894)

times [int

119905V2119894

1199100(1199052119894)

ℎ (1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

+

119894minus1

sum

119895=V2119894

int

1199052119895+2

1199052119895

ℎ (1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

+ int

1199051199062119894

119910119898minus1(1199052119894)

ℎ (1199052119894 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591

+

119894minus1

sum

119895=1199062119894

int

1199052119895+2

1199052119895

ℎ (1199052119894 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591


(1199052119894) + 119892 (119905

2119894) ]

(32)

and by applying mixed formula for (32) we obtain the follow-ing four cases


Case 1When V2119894

= 2119894 and 1199062119894

= 2119894 we have

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (119905V2119894 minus 1199100(1199052119894))

times (ℎ (1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+

119894minus1

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 4ℎ (1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ ℎ (1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

)

+ 05 (1199051199062119894

minus 119910119898minus1

(1199052119894))

times (ℎ (1199052119894 1199051199062119894) log 10038161003816100381610038161003816119909119898minus1 (1199051199062119894)

10038161003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816(119909119898minus1 (119910119898minus1 (1199052119894)))

1003816100381610038161003816 )

+

119894minus1

sum

119895=1199062119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895) log (119909

119898minus1(1199052119895))

+ 4ℎ (1199052119894 1199052119895+1

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+1)10038161003816100381610038161003816

+ ℎ (1199052119894 1199052119895+2

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+2)10038161003816100381610038161003816)

minus Δ119909119898(1199052119894) minus 119909119898minus1

(1199052119894) + 119892 (119905

2119894) ]

]

(33)

Case 2 If V2119894= 2119894 and 119906

2119894= 2119894 then

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (1199052119894minus 1199100(1199052119894))

times (ℎ (1199052119894 1199052119894)

Δ119909119898(119905V2119894)

1199090(1199052119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+ 05 (1199051199062119894


(1199052119894))

times (ℎ (1199052119894 1199051199062119894) log 10038161003816100381610038161003816119909119898minus1 (1199051199062119894)

10038161003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )

+

119894minus1

sum

119895=1199062119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895) log 10038161003816100381610038161003816119909119898minus1 (1199052119895)

10038161003816100381610038161003816

+ 4ℎ (1199052119894 1199052119895+1

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+1)10038161003816100381610038161003816

+ ℎ (1199052119894 1199052119895+2

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+2)10038161003816100381610038161003816)


(1199052119894) + 119892 (119905

2119894) ]

]

(34)

Case 3When V2119894

= 2119894 and 1199062119894= 2119894 we get

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (119905V2119894 minus 1199100(1199052119894))

times (ℎ (1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)


+

119894minus1

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 4ℎ (1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ ℎ (1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

)

+ 05 (1199052119894minus 119910119898minus1

(1199052119894))

times (ℎ (1199052119894 1199052119894) log 1003816100381610038161003816119909119898minus1 (1199052119894)

1003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )


(1199052119894) + 119892 (119905

2119894) ]

]

(35)

Case 4 If V2119894= 2119894 and 119906

2119894= 2119894 then

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (1199052119894minus 1199100(1199052119894))

times (ℎ (1199052119894 1199052119894)Δ119909119898(1199052119894)

1199090(1199052119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+ 05 (1199052119894minus 119910119898minus1

(1199052119894))

times (ℎ (1199052119894 1199052119894) log 1003816100381610038161003816119909119898minus1 (1199052119894)

1003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )


(1199052119894) + 119892 (119905

2119894) ]

]

(36)

Thus (32) can be computed by one of (33)ndash(36) according tothe cases

4 The Convergence Analysis of the Method

On the basis of general theorems of Newton-Kantorovichmethod [21 Chapter XVIII] for the convergence we state thefollowing theorem regarding the successive approximationsdescribed by (18)ndash(20)

First consider the following classes of functions

(i) 119862[1199050 119879]

the set of all continuous functions 119891(119905) definedon the interval [119905

0 119879]

(ii) 119862[1199050 119879]times[1199050 119879]

the set of all continuous functions 120595(119905 120591)defined on the region [119905

0 119879] times [119905

0 119879]

(iii) 119862 = 119883 119883 = (119909(119905) 119910(119905)) 119909(119905) 119910(119905) isin 119862[1199050 119879]

(iv) 119862lt[1199050 119879]

= 119910(119905) isin 1198621

[1199050 119879] 119910(119905) lt 119905

And define the following norms

119909 = max119905isin[1199050 119879]

|119909 (119905)|

Δ119883119862= max Δ119909

119862[1199050119879]1003817100381710038171003817Δ119910

1003817100381710038171003817119862[1199050119879]

1198831198621 = max 119909

119862[1199050119879]10038171003817100381710038171003817119909101584010038171003817100381710038171003817119862[1199050 119879]

1003817100381710038171003817100381711988310038171003817100381710038171003817119862

= max 119909119862[1199050119879]

1003817100381710038171003817119910

1003817100381710038171003817119862[1199050 119879]

ℎ (119905 120591) = 1198671

10038171003817100381710038171003817ℎ1015840

120591(119905 120591)

10038171003817100381710038171003817= 1198671015840

1

119896 (119905 120591) = 1198672

100381710038171003817100381710038171198961015840

120591(119905 120591)

10038171003817100381710038171003817= 1198671015840

2

10038171003817100381710038171003817100381710038171003817

1

1199090

10038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

10038161003816100381610038161003816100381610038161003816

1

1199090(119905)

10038161003816100381610038161003816100381610038161003816

= 1198881

100381710038171003817100381710038171003817100381710038171003817

1

1199092

0

100381710038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

100381610038161003816100381610038161003816100381610038161003816

1

1199092

0(119905)

100381610038161003816100381610038161003816100381610038161003816

= 1198882

10038171003817100381710038171003817100381710038171003817

1

119866 (119905)

10038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

10038161003816100381610038161003816100381610038161003816

1

119866 (119905)

10038161003816100381610038161003816100381610038161003816

= 1198883

100381710038171003817100381711990901003817100381710038171003817 = max119905isin[1199050119879]

10038161003816100381610038161199090 (119905)1003816100381610038161003816 = 1198673

100381710038171003817100381710038171199091015840

0

10038171003817100381710038171003817= max119905isin[1199050119879]

100381610038161003816100381610038161199091015840

0(119905)

10038161003816100381610038161003816= 1198671015840

3

min119905isin[1199050119879]

10038161003816100381610038161199100 (119905)1003816100381610038161003816 = 1198674

1003817100381710038171003817log1003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816log (119909 (119905))1003816100381610038161003816 = 1198675

10038171003817100381710038171198921003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816119892 (119905)1003816100381610038161003816 = 1198676

10038171003817100381710038171198911003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816119891 (119905)1003816100381610038161003816 = 1198677

(37)

Let

1205781= max 119867

11198882(119879 minus 119867

4) 11986711198881 1198671015840

11198675+ 11986711198671015840

31198881

11986721198882(119879 minus 119867

4) 11986721198881 1198671015840

21198675+ 11986721198671015840

31198881

(38)


Let us consider real valued function

120595 (119905) = 119870 (119905 minus 1199050)2

minus (1 + 119870120578) (119905 minus 1199050) + 120578 (39)

where119870 gt 0 and 120578 are nonnegative real coefficients

Theorem 1 Assume that the operator 119875(119883) = 0 in (5) is de-fined in Ω = 119883 isin 119862([119905

0 119879]) 119883 minus 119883

0 le 119877 and has

continuous second derivative in closed ball Ω0= 119883 isin 119862([119905

0

119879]) 119883 minus 1198830 le 119903 where 119879 = 119905

0+ 119903 le 119905

0+ 119877 Suppose the

following conditions are satisfied

(1) Γ0119875(1198830) le 120578(1 + 119870120578)

(2) Γ011987510158401015840(119883) le 2119870(1+119870120578) when 119883minus119883

0 le 119905minus119905

0le 119903

where 119870 and 120578 as in (39) Then the function 120595(119905) defined by(39) majorizes the operator 119875(119883)

Proof Let us rewrite (5) and (39) in the form

119905 = 120601 (119905) 120601 (119905) = 119905 + 1198880120595 (119905) (40)

119883 = 119878 (119883) 119878 (119883) = 119883 minus Γ0119875 (119883) (41)

where 1198880= minus1120595

1015840(1199050) = 1(1 + 119870120578) and Γ

0= [1198751015840(1198830)]minus1

Let us show that (40) and (41) satisfy the majorizingconditions [21 Theorem 1 page 525] In fact

1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817 =1003817100381710038171003817minusΓ0119875 (119883

0)1003817100381710038171003817 le

120578

1 + 119870120578= 120601 (119905

0) minus 1199050 (42)

and for the 119883minus1198830 le 119905 minus 119905

0with the Remark in [21 Remark

1 page 504] we have

100381710038171003817100381710038171198781015840(119883)

10038171003817100381710038171003817=100381710038171003817100381710038171198781015840(119883) minus 119878

1015840(1198830)10038171003817100381710038171003817

le int

119883

1198830

1003817100381710038171003817100381711987810158401015840(119883)

10038171003817100381710038171003817119889119883 = int

119883

1198830

10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817119889119883

le int

119905

1199050

119888012059510158401015840(120591) 119889120591 = int

119905

1199050

2119870

1 + 119870120578119889120591

=2119870

1 + 119870120578(119905 minus 1199050) = 1206011015840(119905)

(43)

Hence 120595(119905) = 0 is a majorant function of 119875(119883) = 0

Theorem2 Let the functions119891(119905) 119892(119905) isin 119862[1199050 119879]

1199090(119905) isin 119862

1[1199050

119879] 1199090(1199100(119905)) = 0 1199092

0(119905) = 0 and the kernels ℎ(119905 120591) 119896(119905 120591) isin

1198621

[1199050 119879]times[1199050 119879]and (119909

0(119905) 1199100(119905)) isin Ω

0 then

(1) the system (7) has unique solution in the interval [1199050

119879] that is there exists Γ0 and Γ

0 le sum

infin

119895=1(11988811198671+

119888111988831198672)119895((119879 minus 119867

4)119895minus1

(119895 minus 1)) = 1205782

(2) Δ119883 le 120578(1 + 119870120578)

(3) 11987510158401015840(119883) le 1205781

(4) 120578 gt 1119870 and 119903 lt 120578 + 1199050

where 119870 and 120578 as in (39) Then the system (4) has uniquesolution 119883

lowast in the closed ball Ω0and the sequence 119883

119898(119905) =

(119909119898(119905) 119910119898(119905))119898 ge 0 of successive approximations

Δ119910119898(119905) =

1

119867 (119905)[int

119905

1199100(119905)

ℎ (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

+ int

119905

119910119898minus1(119905)

ℎ (119905 120591) log 1003816100381610038161003816119909119898minus1 (120591)1003816100381610038161003816 119889120591

minusΔ119909119898(119905) minus 119909

119898minus1(119905) + 119892 (119905) ]

Δ119909119898(119905) minus int

119905

1199100(119905)

1198961(119905 120591)

Δ119909119898(120591)

1199090(120591)

119889120591 = 119865119898minus1

(119905)

(44)

whereΔ119909119898(119905) = 119909

119898(119905)minus119909119898minus1

(119905) andΔ119910119898(119905) = 119910

119898(119905)minus119910119898minus1

(119905)119898 = 2 3 and 119883

119898converge to the solution 119883

lowast The rate ofconvergence is given by

1003817100381710038171003817119883lowastminus 119883119898

1003817100381710038171003817 le (2

1 + 119870120578)

119898

(1

119870) (45)

Proof It is shown that (7) is reduced to (17) Since (17) is alinear Volterra integral equation of 2nd kind with respect toΔ119909(119905) and since 119896(119905 119910

0(119905)) = 0 forall119905 isin [119905

0 119879] which implies

that the kernel 1198961(119905 120591) defined by (18) is continues it follows

that (17) has a unique solution which can be obtained bythe method of successive approximations Then the functionΔ119910(119905) is uniquely determined from (16) Hence the existenceof Γ0is archived

To verify that Γ0is boundedwe need to establish the resol-

vent kernel Γ0(119905 120591) of (17) so we assume the integral operator

119880 from 119862[1199050 119879] rarr 119862[119905

0 119879] is given by

119885 = 119880 (Δ119909) 119885 (119905) = int

119905

1199100(119905)

1198962(119905 120591) Δ119909 (120591) 119889120591 (46)

where 1198962(119905 120591) = 119896

1(119905 120591)119909

0(120591) and 119896

1(119905 120591) is defined in (18)

Due to (46) (17) can be written as

Δ119909 minus 119880 (Δ119909) = 1198650 (47)

The solution Δ119909lowast of (47) is expressed in terms of 119865

0by means

of the formula

Δ119909lowast= 1198650+ 119861 (119865

0) (48)

where 119861 is an integral operator and can be expanded as aseries in powers of 119880 [21 Theorem 1 page 378]

119861 (1198650) = 119880 (119865

0) + 1198802(1198650) + sdot sdot sdot + 119880

119899(1198650) + sdot sdot sdot (49)

and it is known that the powers of 119880 are also integral opera-tors In fact

119885119899= 119880119899 119885119899(119905) = int

119905

1199100(119905)

119896(119899)

2(119905 120591) Δ119909 (120591) 119889120591

(119899 = 1 2 )

(50)

where 119896(119899)2

is the iterated kernel


Substituting (50) into (48) we obtain an expression for thesolution of (47)

Δ119909lowast= 1198650(119905) +

infin

sum

119895=1

int

119905

1199100(119905)

119896(119895)

2(119905 120591) 119865

0(120591) 119889120591 (51)

Next we show that the series in (51) is convergent uniformlyfor all 119905 isin [119905

0 119879] Since

10038161003816100381610038161198962 (119905 120591)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

1198961(119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

ℎ (119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119896 (119905 120591)

1199090(120591) 119866 (119905)

10038161003816100381610038161003816100381610038161003816

le 11988811198671+ 119888111988831198672

(52)

Let 119872 = 11988811198671+ 119888111988831198672 then by mathematical induction we

get

10038161003816100381610038161003816119896(2)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

10038161003816100381610038161198962 (119905 119906) 1198962 (119906 120591)1003816100381610038161003816 119889119906 le

1198722(119905 minus 119867

4)

(1)

10038161003816100381610038161003816119896(3)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(2)

2(119906 120591)

10038161003816100381610038161003816119889119906 le

1198723(119905 minus 119867

4)2

(2)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(119899minus1)

2(119906 120591)

10038161003816100381610038161003816119889119906

le119872119899(119905 minus 119867

4)119899minus1

(119899 minus 1)

(119899 = 1 2 )

(53)

then

10038171003817100381710038171198801198991003817100381710038171003817 = max119905isin[1199050119879]

int

119905

1199100(119905)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816119889120591 le

119872119899(119879 minus 119867

4)(119899minus1)

(119899 minus 1)

(54)

Therefore the 119899th root test of the sequence yields

119899radic119880119899 le

119872(119879 minus 1198674)1minus1119899

119899radic(119899 minus 1)

997888997888997888rarr119899rarrinfin

0 (55)

Hence 120588 = 1lim119899rarrinfin

119899radic119880119899 = infin and a Volterra integral

equations (17) has no characteristic values Since the seriesin (51) converges uniformly (48) can be written in terms ofresolvent kernel of (17)

Δ119909lowast= 1198650+ int

119905

1199100(119905)

Γ0(119905 120591) 119865

0(120591) 119889120591 (56)

where

Γ0(119905 120591) =

infin

sum

119895=1

119896(119895)

2(119905 120591) (57)

Since the series in (57) is convergent we obtain

1003817100381710038171003817Γ01003817100381710038171003817 =

1003817100381710038171003817119861 (1198650)1003817100381710038171003817 le

infin

sum

119895=1

1003817100381710038171003817100381711988011989510038171003817100381710038171003817

le

infin

sum

119895=1

119872119895 (119879 minus 119867)

119895minus1

(119895 minus 1)le 1205782 (58)

To establish the validity of second condition let us representoperator equation

119875 (119883) = 0 (59)

as in (41) and its the successive approximations is

119883119899+1

= 119878 (119883119899) (119899 = 0 1 2 ) (60)

For initial guess1198830we have

119878 (1198830) = 119883

0minus Γ0119875 (1198830) (61)

From second condition of (Theorem 1) we have

1003817100381710038171003817Γ0119875 (1198830)1003817100381710038171003817 =

1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817

=10038171003817100381710038171198831 minus 119883

0

1003817100381710038171003817 = Δ119883 le120578

1 + 119870120578

(62)

In addition we need to show that 11987510158401015840(119883) le 1205781for all

119883 isin Ω0where 120578

1is defined in (38) It is known that the

second derivative 11987510158401015840(1198830)(119883119883) of the nonlinear operator

119875(119883) is described by 3-dimensional array 11987510158401015840(1198830)119883119883 =

(1198631 1198632)(119883119883) which is called bilinear operator that is

11987510158401015840(1198830)(119883119883) = 119861(119883

0 119883119883) where

11987510158401015840(1198830) (119883119883)

= lim119904rarr0

1

119904[1198751015840(1199090+ 119904119883) minus 119875

1015840(1198830)]

= lim119904rarr0

1

119904[(

1205971198751

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119909(1199090 1199100)) 119909

+ (1205971198751

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119910(1199090 1199100)) 119910]

lim119904rarr0

1

119904[(

1205971198752

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119909(1199090 1199100)) 119909

+ (1205971198752

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119910(1199090 1199100)) 119910]


= lim119904rarr0

1

119904[(

12059721198751

1205971199092(1199090 1199100) 119904119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198751

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198751

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198751

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

lim119904rarr0

1

119904[(

12059721198752

1205971199092(1199090 1199100) 119904119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198752

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198752

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198752

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

= (12059721198751

1205971199092(1199090 1199100) 119909119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119910119909

+12059721198751

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198751

1205971199102(1199090 1199100) 119910119909

12059721198752

1205971199092(1199090 1199100) 119909119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119910119909

+12059721198752

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198752

1205971199102(1199090 1199100) 119910119909)

(63)

where 120579 120575 isin (0 1) so we have

11987510158401015840(1198830) (119883119883) = (119863

11198632) (

119909

119910)(

119909

119910) (64)

where

1198631= (

12059721198751

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

1198632= (

12059721198752

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

(65)

Then the norms of every components of 1198631and 119863

2have the

estimate100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

ℎ (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986711198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[ℎ1015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ ℎ (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

11198675+ 11986711198671015840

31198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

119896 (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986721198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881


100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[1198961015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ 119896 (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

21198675+ 11986721198671015840

31198881

(66)

Therefore all the second derivatives exist and are bounded1003817100381710038171003817100381711987510158401015840(119883)

10038171003817100381710038171003817le 1205781 (67)

Since 120595(119905) majorizes operator 119875(119883) and utilizing the secondcondition of (Theorem 1) we get

10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817le

2119870

1 + 119870120578 (68)

Let us consider the discriminant of equation 120595(119905) = 0

119863 = 11987021205782minus 2119870120578 + 1 = (119896120578 minus 1)

2

(69)

and the two roots of120595(119905) = 0 are 1199031= 1119870+119905

0and 1199032= 120578+119905

0

therefore when 1199031lt 119903 lt 119903

2implies

120595 (119903) le 0 (70)

then under the assumption of the fourth condition that is1119870 + 119905

0is the unique solution of 120595(119905) = 0 in [119905

0 119879] and the

condition in (70) [21Theorem 4 page 530] implies that119883lowast isthe unique solution of operator equation (5) [21 Theorem 6page 532] and

1003817100381710038171003817119883lowastminus 1198830

1003817100381710038171003817 le 119905lowastminus 1199050 (71)

where 119905lowast is the unique solution of 120595(119905) = 0 in [1199050 119903]

To show the rate of convergence let us write the equation120595(119905) = 0 in a same form as in (40) then its successiveapproximation is

119905119898+1

= 120601 (119905119898) 119898 = 0 1 2 (72)

To estimate the difference between 119905lowast and successive approx-

imation 119905119898

119905lowastminus 119905119898= 120601 (119905

lowast) minus 120601 (119905

119898minus1) = 1206011015840(119905119898) (119905lowastminus 119905119898minus1

) (73)

where 119905119898isin (119905119898minus1

119905lowast) and

1206011015840(119905) = 1 + 119888

01205951015840(119905) =

2119870

1 + 119870120578(119905 minus 1199050) (74)

therefore

1206011015840(119905119898) =

2119870

1 + 119870120578(119905119898minus 1199050)

le2119870

1 + 119870120578(119905lowastminus 1199050) =

2

1 + 119870120578

(75)

then

119905lowastminus 119905119898le

2

1 + 119870120578(119905lowastminus 119905119898minus1

)

119905lowastminus 119905119898minus1

le2


)


2

1 + 119870120578(119905lowastminus 1199050)

(76)

consequently

119905lowastminus 119905119898le (

2

1 + 119870120578)

1198981

119870 (77)

it implies

1003817100381710038171003817119883lowastminus 119883119898

1003817100381710038171003817 le (119905lowastminus 119905119898) le (

2

1 + 119870120578)

1198981

119870 (78)

5 Numerical Example

Consider the system of nonlinear equation

119909 (119905) minus int

119905

119910(119905)

119905120591 log (|119909 (120591)|) 119889120591 = 119890119905minus1199052

3

int

119905

119910(119905)

120591 log (|119909 (120591)|) 119889120591 =119905

3 119905 isin [10 15]

(79)

The exact solution is119909lowast(119905) = 119890

119905

119910lowast(119905) =

3radic1199053minus 119905

(80)

and the initial guesses are

1199090(119905) = 119890

10(119905 minus 9)

1199100(119905) = 06119905 + 4

(81)

Table 1 shows that 119909119898(119905) coincides with the exact 119909lowast(119905)

from the first iteration whereas only six iterations are neededfor 119910119898(119905) to be very close to 119910

lowast(119905) Notations used here are

as follows 119873 is the number of nodes 119898 is the number ofiterations and 120598

119909= max

119905isin[1015]|119909119898(119905) minus 119909

lowast(119905)| and 120598

119910=

max119905isin[1015]

|119910119898(119905) minus 119910

lowast(119905)|

6 Conclusion

In this paper the Newton-Kantorovich method is developedto solve the system of nonlinear Volterra integral equationswhich contains logarithmic function We have introduceda new majorant function that leads to the increment ofrange of convergence of successive approximation processA new theorem is stated based on the general theoremsof Kantorovich Numerical example is given to show thevalidation of the method Table 1 shows that the proposedmethod is in good agreement with the theoretical findings


Table 1 Numerical results for (79)

119873 = 20 ℎ = 025

119898 120598119909

120598119910

1 000 000292 000 43597119864 minus 006

3 000 31061119864 minus 008

4 000 10140119864 minus 009

5 000 12541119864 minus 010

6 000 39968119864 minus 011

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by University Putra Malaysia underFundamental Research Grant Scheme (FRGS) Project codeis 01-12-10-989FR

References

[1] L V Kantorovich ldquoThemethod of successive approximation forfunctional equationsrdquo Acta Mathematica vol 71 no 1 pp 63ndash97 1939

[2] L V Kantorovich ldquoOn Newtonrsquos method for functional equa-tionsrdquo Doklady Akademii Nauk SSSR vol 59 pp 1237ndash12401948 (Russian)

[3] L U Uko and I K Argyros ldquoA weak Kantorovich existencetheorem for the solution of nonlinear equationsrdquo Journal ofMathematical Analysis andApplications vol 342 no 2 pp 909ndash914 2008

[4] W Shen and C Li ldquoKantorovich-type convergence criterion forinexact NewtonmethodsrdquoApplied Numerical Mathematics vol59 no 7 pp 1599ndash1611 2009

[5] I K Argyros ldquoOn Newtonrsquos method for solving equationscontaining Frechet-differentiable operators of order at leasttwordquo Applied Mathematics and Computation vol 215 no 4 pp1553ndash1560 2009

[6] J Saberi-Nadjafi and M Heidari ldquoSolving nonlinear inte-gral equations in the Urysohn form by Newton-Kantorovich-quadrature methodrdquo Computers amp Mathematics with Applica-tions vol 60 no 7 pp 2058ndash2065 2010

[7] J A Ezquerro D Gonzalez and M A Hernandez ldquoA variantof the Newton-Kantorovich theorem for nonlinear integralequations of mixed Hammerstein typerdquo Applied Mathematicsand Computation vol 218 no 18 pp 9536ndash9546 2012

[8] J A Ezquerro D Gonzalez and M A Hernandez ldquoA mod-ification of the classic conditions of Newton-Kantorovich forNewtonrsquos methodrdquoMathematical and Computer Modelling vol57 no 3-4 pp 584ndash594 2013

[9] B Jumarhon and S McKee ldquoProduct integration methodsfor solving a system of nonlinear Volterra integral equationsrdquoJournal of Computational and Applied Mathematics vol 69 no2 pp 285ndash301 1996

[10] H Sadeghi Goghary S Javadi and E Babolian ldquoRestartedAdomian method for system of nonlinear Volterra integral

equationsrdquo Applied Mathematics and Computation vol 161 no3 pp 745ndash751 2005

[11] AGolbabaiMMammadov and S Seifollahi ldquoSolving a systemof nonlinear integral equations by an RBF networkrdquo Computersamp Mathematics with Applications vol 57 no 10 pp 1651ndash16582009

[12] M I Berenguer D Gamez A I Garralda-Guillem M RuizGalan and M C Serrano Perez ldquoBiorthogonal systems forsolving Volterra integral equation systems of the second kindrdquoJournal of Computational and AppliedMathematics vol 235 no7 pp 1875ndash1883 2011

[13] J Biazar and H Ebrahimi ldquoChebyshev wavelets approach fornonlinear systems of Volterra integral equationsrdquo Computers ampMathematics with Applications vol 63 no 3 pp 608ndash616 2012

[14] M Ghasemi M Fardi and R Khoshsiar Ghaziani ldquoSolution ofsystem of the mixed Volterra-Fredholm integral equations byan analytical methodrdquo Mathematical and Computer Modellingvol 58 no 7-8 pp 1522ndash1530 2013

[15] L-H Yang H-Y Li and J-RWang ldquoSolving a system of linearVolterra integral equations using the modified reproducingkernel methodrdquoAbstract and Applied Analysis vol 2013 ArticleID 196308 5 pages 2013

[16] M Dobritoiu and M-A Serban ldquoStep method for a system ofintegral equations from biomathematicsrdquo Applied Mathematicsand Computation vol 227 pp 412ndash421 2014

[17] V Balakumar and K Murugesan ldquoSingle-term Walsh seriesmethod for systems of linear Volterra integral equations of thesecond kindrdquo Applied Mathematics and Computation vol 228pp 371ndash376 2014

[18] I V Boikov and A N Tynda ldquoApproximate solution of non-linear integral equations of the theory of developing systemsrdquoDifferential Equations vol 39 no 9 pp 1277ndash1288 2003

[19] Z K Eshkuvatov A Ahmedov N M A Nik Long andO Shafiq ldquoApproximate solution of the system of nonlinearintegral equation by Newton-Kantorovich methodrdquo AppliedMathematics and Computation vol 217 no 8 pp 3717ndash37252010

[20] Z K Eshkuvatov A Ahmedov N M A Nik Long and OShafiq ldquoApproximate solution of nonlinear system of integralequation usingmodifiedNewton-Kantorovichmethodrdquo in Pro-ceeding of the Reggional Annual Fumdamental Science Seminarpp 595ndash607 2010

[21] L V Kantorovich and G P Akilov functional Analysis Perga-mon Press Nauka 1982

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


implemented theNewton-Kantorovichmethod to the follow-ing system

119909 (119905) minus int

119905

119910(119905)

ℎ (119905 120591) 119892 (120591) 119909 (120591) 119889120591 = 0

int

119905

119910(119905)

119896 (119905 120591) [1 minus 119892 (120591)] 119909 (120591) 119889120591 = 119891 (119905)

(1)

where 0 lt 1199050le 119905 le 119879 119910(119905) lt 119905 and the functions ℎ(119905 120591)

119896(119905 120591) isin 119862[1199050 119879]times[1199050 119879]

119891(119905) 119892(119905) isin 119862[1199050 119879]

and (0 lt 119892(119905) lt 1)In 2010 Eshkuvatov et al [19] used the Newton-Kantorovichhypothesis to solve the system of nonlinear Volterra integralequation of the form

119909 (119905) minus int

119905

119910(119905)

ℎ (119905 120591) 1199092(120591) 119889120591 = 0

int

119905

119910(119905)

119896 (119905 120591) 1199092(120591) 119889120591 = 119891 (119905)

(2)

where 119909(119905) and 119910(119905) are unknown functions defined on[1199050infin) 119905

0gt 0 and ℎ(119905 120591) 119896(119905 120591) isin 119862

[1199050 infin]times[1199050 infin] 119891(119905) isin

119862[1199050 infin]

In 2010 Eshkuvatov et al [20] developed themodifiedNewton-Kantorovich to obtain an approximate solution ofsystem with the form

119909 (119905) minus int

119905

119910(119905)

119867(119905 120591) 119909119899(120591) 119889120591 = 0

int

119905

119910(119905)

119870 (119905 120591) 119909119899(120591) 119889120591 = 119891 (119905)

(3)

where 0 lt 1199050le 119905 le 119879 119910(119905) lt 119905 and the functions 119867(119905

120591) 119870(119905 120591) isin 119862[1199050 infin]times[1199050 infin]

119891(119905) isin 119862[1199050 infin]

and the unknownfunctions 119909(119905) isin 119862

[1199050 infin] 119910(119905) isin 119862

1

[1199050 infin] 119910(119905) lt 119905

In this paper we consider the systems of nonlinearintegral equation of the form

119909 (119905) minus int

119905

119910(119905)

ℎ (119905 120591) log |119909 (120591)| 119889120591 = 119892 (119905)

int

119905

119910(119905)

119896 (119905 120591) log |119909 (120591)| 119889120591 = 119891 (119905)

(4)

where 0 lt 1199050le 119905 le 119879 119910(119905) lt 119905 119909(119905) = 0 ℎ(119905 120591) ℎ

120591(119905 120591)

119896(119905 120591) 119896120591(119905 120591) isin 119862(119863) and the unknown functions 119909(119905) isin

119862[1199050 119879] 119910(119905) isin 119862

1[1199050 119879] to be determined and119863 = [119905

0 119879] times

[1199050 119879]The paper is organized as follows in Section 2 Newton-

Kantorovich method for the system of integral equations (4)is presented Section 3 deals with mixed method followedby discretizations In Section 4 the rate of convergence ofthemethod is investigated Lastly Section 5 demonstrates thenumerical example to verify the validity and accuracy of theproposed method followed by the conclusion in Section 6

2 Newton-Kantorovich Method for the System

Let us rewrite the system of nonlinear Volterra integral equa-tion (4) in the operator form

119875 (119883) = (1198751(119883) 119875

2(119883)) = 0 (5)

where119883 = (119909(119905) 119910(119905)) and

1198751(119883) = 119909 (119905) minus int

119905

119910(119905)

ℎ (119905 120591) log |119909 (120591)| 119889120591 minus 119892 (119905)

1198752(119883) = int

119905

119910(119905)

119896 (119905 120591) log |119909 (120591)| 119889120591 minus 119891 (119905)

(6)

To solve (5) we use initial iteration of Newton-Kantorovichmethod which is of the form

1198751015840(1198830) (119883 minus 119883

0) + 119875 (119883

0) = 0 (7)

where 1198830= (1199090(119905) 1199100(119905)) is the initial guess and 119909

0(119905) and

1199100(119905) can be any continuous functions provided that 119905

0lt

119910(119905) lt 119905 and 119909(119905) = 0The Frechet derivative of 119875(119883) at the point 119883

0is defined

as

1198751015840(1198830)119883

= ( lim119904rarr0

1

119904[1198751(1198830+ 119904119883) minus 119875

1(119883)]

lim119904rarr0

1

119904[1198752(1198830+ 119904119883) minus 119875

2(119883)])

= ( lim119904rarr0

1

119904[1198751(1199090+ 119904119909 119910

0+ 119904119910) minus 119875

1(1199090 1199100)]

lim119904rarr0

1

119904[1198752(1199090+ 119904119909 119910

0+ 119904119910) minus 119875

2(1199090 1199100)])

= ( lim119904rarr0

[1205971198751(1199090 1199100)

120597119909119904119909 +

1205971198751(1199090 1199100)

120597119910119904119910

+1

2(12059721198751

1205971199092(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059721198751

120597119909120597119910(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059721198751

1205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904119910

2)]


lim119904rarr0

1

119904[1205971198752

120597119909(1199090 1199100) 119904119909 +

1205971198752

120597119910(1199090 1199100) 119904119910

+1

2(12059721198752

1205971199092(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059721198752

120597119909120597119910(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059721198752

1205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904119910

2)])

= (1205971198751(1199090 1199100)

120597119909119909 +

1205971198751(1199090 1199100)

120597119910119910

1205971198752(1199090 1199100)

120597119909119909 +

1205971198752(1199090 1199100)

120597119910119910)

(8)

Hence

1198751015840(1198830)119883 = (

1205971198751

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

1205971198751

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

1205971198752

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

1205971198752

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

)(119909

119910) (9)


1205971198751

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

(Δ119909 (119905)) +1205971198751

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

(Δ119910 (119905))

= minus1198751(1199090(119905) 1199100(119905))

1205971198752

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

(Δ119909 (119905)) +1205971198752

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

(Δ119910 (119905))

= minus1198752(1199090(119905) 1199100(119905))

(10)

where Δ119909(119905) = 1199091(119905)minus119909

0(119905) Δ119910(119905) = 119910

1(119905)minus119910

0(119905) and (119909

0(119905)



1205971198751

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

= lim119904rarr0

1

119904(1198751(1199090+ 119904119909 119910

0) minus 1198751(1199090 1199100))

= lim119904rarr0

1

119904[119904119909 (119905)

minus int

119905

1199100(119905)

ℎ (119905 120591) (log 10038161003816100381610038161199090 (120591) + 119904119909 (120591)1003816100381610038161003816

minus log 10038161003816100381610038161199090 (120591)1003816100381610038161003816) 119889120591]

= 119909 (119905) minus int

119905

1199100(119905)

ℎ (119905 120591)119909 (120591)

1199090(120591)

119889120591

1205971198751

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

= lim119904rarr0

1

119904(1198751(1199090 1199100+ 119904119910) minus 119875

1(1199090 1199100))

= lim119904rarr0

1

119904[int

1199100(119905)+119904119910(119905)

1199100(119905)

ℎ (119905 120591) log 1003816100381610038161003816(1199090 (120591))1003816100381610038161003816 119889120591]

= ℎ (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 119910 (119905)

(11)


1205971198752

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

= int

119905

1199100(119905)

119896 (119905 120591)119909 (120591)

1199090(120591)

119889120591

1205971198752

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

= minus119896 (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 119910 (119905)

(12)


Δ119909 (119905) minus int

119905

1199100(119905)

ℎ (119905 120591)Δ119909 (120591)

1199090(120591)

119889120591

+ ℎ (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 Δ119910 (119905)

= int

119905

1199100(119905)

ℎ (119905 120591) log 10038161003816100381610038161199090 (120591)1003816100381610038161003816 119889120591 minus 119909

0(119905) + 119892 (119905)

int

119905

1199100(119905)

119896 (119905 120591)Δ119909 (120591)

1199090(120591)

119889120591

minus 119896 (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 Δ119910 (119905)

= minusint

119905

1199100(119905)

119896 (119905 120591) log 10038161003816100381610038161199090 (120591)1003816100381610038161003816 119889120591 + 119891 (119905)

(13)




1198751015840(1198830) Δ119883119898+ 119875 (119883

119898) = 0 (14)


Δ119909119898(119905) minus int

119905

1199100(119905)

ℎ (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

+ ℎ (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 Δ119910119898 (119905)

= int

119905

1199100(119905)

ℎ (119905 120591) log 10038161003816100381610038161199090 (120591)1003816100381610038161003816 119889120591 minus 119909

0(119905) + 119892 (119905)

int

119905

1199100(119905)

119896 (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

minus 119896 (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 Δ119910119898 (119905)

= minusint

119905

1199100(119905)

119896 (119905 120591) log 10038161003816100381610038161199090 (120591)1003816100381610038161003816 119889120591 + 119891 (119905)

(15)

whereΔ119909119898(119905) = 119909

119898(119905)minus119909119898minus1

(119905) andΔ119910119898(119905) = 119910

119898(119905)minus119910119898minus1

(119905)119898 = 1 2 3



Δ119910 (119905) =1

119867 (119905)[int

119905

1199100(119905)

ℎ (119905 120591) [Δ119909 (120591)

1199090(120591)

+ log 10038161003816100381610038161199090 (120591)1003816100381610038161003816] 119889120591

minus [Δ119909 (119905) + 1199090(119905) minus 119892 (119905)]]

119866 (119905) [int

119905

1199100(119905)

ℎ (119905 120591) [Δ119909 (120591)

1199090(120591)

+ log 10038161003816100381610038161199090 (120591)1003816100381610038161003816] 119889120591

minus [Δ119909 (119905) + 1199090(119905) minus 119892 (119905)]]

= int

119905

1199100(119905)

119896 (119905 120591)Δ119909 (120591)

1199090(120591)

119889120591

minus int

119905

1199100(119905)

119896 (119905 120591) log 10038161003816100381610038161199090 (120591)1003816100381610038161003816 119889120591 + 119891 (119905)

(16)

where 119866(119905) = 119896(119905 1199100(119905))ℎ(119905 119910

0(119905)) and 119867(119905) = 1[ℎ(119905

1199100(119905)) log |119909


Δ119909 (119905) minus int

119905

1199100(119905)

1198961(119905 120591)

Δ119909 (120591)

1199090(120591)

120591 = 1198650(119905) (17)

where

1198961(119905 120591) = ℎ (119905 120591) minus

119896 (119905 120591)

119866 (119905)

119866 (119905) =119896 (119905 119910

0(119905))

ℎ (119905 1199100(119905))

119896 (119905 1199100(119905)) = 0 forall119905 isin [119905

0 119879]

1198650(119905) = int

119905

1199100(119905)

1198961(119905 120591) log 10038161003816100381610038161199090 (120591)

1003816100381610038161003816 119889120591 minus 1199090(119905) + 119892 (119905) +

119891 (119905)

119866 (119905)

(18)



form

Δ119910119898(119905)

=1

119867 (119905)[int

119905

1199100(119905)

ℎ (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

+ int

119905

119910119898minus1(119905)

ℎ (119905 120591) log 1003816100381610038161003816119909119898minus1 (120591)1003816100381610038161003816 119889120591

minusΔ119909119898(119905) minus 119909

119898minus1(119905) + 119892 (119905)]

(19)

Δ119909119898(119905) minus int

119905

1199100(119905)

1198961(119905 120591)

Δ119909119898(120591)

1199090(120591)

119889120591 = 119865119898minus1

(119905) (20)

where

119865119898minus1

(119905) = int

119905

119910119898minus1(119905)

1198961(119905 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591 minus 119909119898minus1

(119905)

+ 119892 (119905) +119891 (119905)

119866 (119905)

(21)



0 119879] To do




Δ119909119898(1199050) = minus119909

119898minus1(1199050) + 119892 (119905

0) +

119891 (1199050)

119866 (1199050) (22)

Δ119909119898(1199052119894) minus int

1199052119894

1199100(1199052119894)

1198961(1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

= 119865119898minus1

(1199052119894) 119894 = 1 2 119873

(23)


119905V2119894 = 119905V2119894 119905

0le 1199100(1199052119894) lt 1199052119894minus2

1199052119894 1199052119894minus2

le 1199100(1199052119894) lt 1199052119894

(24)


Δ119909119898(1199052119894) minus int

119905V2119894

1199100(1199052119894)

1198961(1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

minus

119894minus1

sum

119895=V2119894

int

1199052119895+2

1199052119895

1198961(1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

= 119865119898minus1

(1199052119894) 119894 = 1 2 119873

(25)


Case 1When V2119894

= 2119894 119894 = 1 2 119873 then

Δ119909119898(1199052119894) =

119865119898minus1

(1199052119894) + 119860 (119894) + 119861 (119894) + 119862 (119894)


) 6 sdot 1199090(1199052119894)) 1198961(1199052119894 1199052119894)

(26)


where

119860 (119894) = 05 (119905V2119894 minus 1199100(1199052119894))

times [1198961(1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ 1198961(1199052119894 1199100(1199052119894))

times

Δ119909119898(119905V2119894) (119905V2119894 minus 119910

0(1199052119894))

(119905V2119894 minus 119905V2119894minus2) (1199090 (1199100 (1199052119894)))

+ 1198961(1199052119894 1199100(1199052119894))

times


(119905V2119894 minus 119905V2119894minus2) (1199090 (1199100 (1199052119894)))]

119861 (119894) =

119894minus2

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times [1198961(1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 41198961(1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ 1198961(1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

]

119862 (119894) =(1199052119894minus 1199052119894minus2

)

6[1198961(1199052119894 1199052119894minus2

)Δ119909119898(1199052119894minus2

)

1199090(1199052119894minus2

)

+ 41198961(1199052119894 1199052119894minus1

)Δ119909119898(1199052119894minus1

)

1199090(1199052119894minus1

)]

(27)

Case 2When V2119894= 2119894 119894 = 1 2 119873 then

Δ119909119898(1199052119894) =

1198631(119894)

1198632(119894)

(28)

where

1198631(119894) = 119865

119898minus1(1199052119894) + 05119896

1(1199052119894 1199100(1199052119894))

times [Δ119909119898(1199052119894minus2

)

1199090(1199100(1199052119894))


)

1199052119894minus 1199052119894minus2

]

1198632(119894) = [1 minus 05 (119905

2119894minus 1199100(1199052))

1198961(1199052119894 1199052119894)

1199090(1199052119894)

minus 051198961(1199052119894 1199100(1199052119894))

(1199052119894minus 1199100(1199052119894))2

1199090(1199100(1199052119894)) (1199052119894minus 1199052119894minus2

)]

(29)



Δ119910119898(1199052119894) =

1

119867 (1199052119894)

times [int

1199052119894

1199100(1199052119894)

ℎ (1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

+ int

1199052119894

119910119898minus1(1199052119894)

ℎ (1199052119894 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591


(1199052119894) + 119892 (119905

2119894) ]

(30)

Let us set V2119894= 1199100(1199052119894) and 119906

2119894= 119910119898minus1

(1199052119894) and

119905V2119894 =

1199052119894 1199052119894minus2

le 1199100(1199052119894) lt 1199052119894

119905V2119894 1199050le 1199100(1199052119894) lt 1199052119894minus2

1199051199062119894

=

1199052119894 1199052119894minus2

le 119910119898minus1

(1199052119894) lt 1199052119894

1199051199062119894 1199050le 119910119898minus1

(1199052119894) lt 1199052119894minus2

(31)


Δ119910119898(1199052119894) =

1

119867 (1199052119894)

times [int

119905V2119894

1199100(1199052119894)

ℎ (1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

+

119894minus1

sum

119895=V2119894

int

1199052119895+2

1199052119895

ℎ (1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

+ int

1199051199062119894

119910119898minus1(1199052119894)

ℎ (1199052119894 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591

+

119894minus1

sum

119895=1199062119894

int

1199052119895+2

1199052119895

ℎ (1199052119894 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591


(1199052119894) + 119892 (119905

2119894) ]

(32)



Case 1When V2119894

= 2119894 and 1199062119894

= 2119894 we have

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (119905V2119894 minus 1199100(1199052119894))

times (ℎ (1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+

119894minus1

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 4ℎ (1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ ℎ (1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

)

+ 05 (1199051199062119894


(1199052119894))

times (ℎ (1199052119894 1199051199062119894) log 10038161003816100381610038161003816119909119898minus1 (1199051199062119894)

10038161003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816(119909119898minus1 (119910119898minus1 (1199052119894)))

1003816100381610038161003816 )

+

119894minus1

sum

119895=1199062119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895) log (119909

119898minus1(1199052119895))

+ 4ℎ (1199052119894 1199052119895+1

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+1)10038161003816100381610038161003816

+ ℎ (1199052119894 1199052119895+2

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+2)10038161003816100381610038161003816)


(1199052119894) + 119892 (119905

2119894) ]

]

(33)

Case 2 If V2119894= 2119894 and 119906

2119894= 2119894 then

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (1199052119894minus 1199100(1199052119894))

times (ℎ (1199052119894 1199052119894)

Δ119909119898(119905V2119894)

1199090(1199052119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+ 05 (1199051199062119894


(1199052119894))

times (ℎ (1199052119894 1199051199062119894) log 10038161003816100381610038161003816119909119898minus1 (1199051199062119894)

10038161003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )

+

119894minus1

sum

119895=1199062119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895) log 10038161003816100381610038161003816119909119898minus1 (1199052119895)

10038161003816100381610038161003816

+ 4ℎ (1199052119894 1199052119895+1

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+1)10038161003816100381610038161003816

+ ℎ (1199052119894 1199052119895+2

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+2)10038161003816100381610038161003816)


(1199052119894) + 119892 (119905

2119894) ]

]

(34)

Case 3When V2119894

= 2119894 and 1199062119894= 2119894 we get

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (119905V2119894 minus 1199100(1199052119894))

times (ℎ (1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)


+

119894minus1

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 4ℎ (1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ ℎ (1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

)

+ 05 (1199052119894minus 119910119898minus1

(1199052119894))

times (ℎ (1199052119894 1199052119894) log 1003816100381610038161003816119909119898minus1 (1199052119894)

1003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )


(1199052119894) + 119892 (119905

2119894) ]

]

(35)

Case 4 If V2119894= 2119894 and 119906

2119894= 2119894 then

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (1199052119894minus 1199100(1199052119894))

times (ℎ (1199052119894 1199052119894)Δ119909119898(1199052119894)

1199090(1199052119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+ 05 (1199052119894minus 119910119898minus1

(1199052119894))

times (ℎ (1199052119894 1199052119894) log 1003816100381610038161003816119909119898minus1 (1199052119894)

1003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )


(1199052119894) + 119892 (119905

2119894) ]

]

(36)





(i) 119862[1199050 119879]


0 119879]

(ii) 119862[1199050 119879]times[1199050 119879]


0 119879] times [119905

0 119879]

(iii) 119862 = 119883 119883 = (119909(119905) 119910(119905)) 119909(119905) 119910(119905) isin 119862[1199050 119879]

(iv) 119862lt[1199050 119879]

= 119910(119905) isin 1198621

[1199050 119879] 119910(119905) lt 119905


119909 = max119905isin[1199050 119879]

|119909 (119905)|

Δ119883119862= max Δ119909

119862[1199050119879]1003817100381710038171003817Δ119910

1003817100381710038171003817119862[1199050119879]

1198831198621 = max 119909

119862[1199050119879]10038171003817100381710038171003817119909101584010038171003817100381710038171003817119862[1199050 119879]

1003817100381710038171003817100381711988310038171003817100381710038171003817119862

= max 119909119862[1199050119879]

1003817100381710038171003817119910

1003817100381710038171003817119862[1199050 119879]

ℎ (119905 120591) = 1198671

10038171003817100381710038171003817ℎ1015840

120591(119905 120591)

10038171003817100381710038171003817= 1198671015840

1

119896 (119905 120591) = 1198672

100381710038171003817100381710038171198961015840

120591(119905 120591)

10038171003817100381710038171003817= 1198671015840

2

10038171003817100381710038171003817100381710038171003817

1

1199090

10038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

10038161003816100381610038161003816100381610038161003816

1

1199090(119905)

10038161003816100381610038161003816100381610038161003816

= 1198881

100381710038171003817100381710038171003817100381710038171003817

1

1199092

0

100381710038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

100381610038161003816100381610038161003816100381610038161003816

1

1199092

0(119905)

100381610038161003816100381610038161003816100381610038161003816

= 1198882

10038171003817100381710038171003817100381710038171003817

1

119866 (119905)

10038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

10038161003816100381610038161003816100381610038161003816

1

119866 (119905)

10038161003816100381610038161003816100381610038161003816

= 1198883

100381710038171003817100381711990901003817100381710038171003817 = max119905isin[1199050119879]

10038161003816100381610038161199090 (119905)1003816100381610038161003816 = 1198673

100381710038171003817100381710038171199091015840

0

10038171003817100381710038171003817= max119905isin[1199050119879]

100381610038161003816100381610038161199091015840

0(119905)

10038161003816100381610038161003816= 1198671015840

3

min119905isin[1199050119879]

10038161003816100381610038161199100 (119905)1003816100381610038161003816 = 1198674

1003817100381710038171003817log1003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816log (119909 (119905))1003816100381610038161003816 = 1198675

10038171003817100381710038171198921003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816119892 (119905)1003816100381610038161003816 = 1198676

10038171003817100381710038171198911003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816119891 (119905)1003816100381610038161003816 = 1198677

(37)

Let

1205781= max 119867

11198882(119879 minus 119867

4) 11986711198881 1198671015840

11198675+ 11986711198671015840

31198881

11986721198882(119879 minus 119867

4) 11986721198881 1198671015840

21198675+ 11986721198671015840

31198881

(38)



120595 (119905) = 119870 (119905 minus 1199050)2

minus (1 + 119870120578) (119905 minus 1199050) + 120578 (39)



0 119879]) 119883 minus 119883

0 le 119877 and has


0

119879]) 119883 minus 1198830 le 119903 where 119879 = 119905

0+ 119903 le 119905



(1) Γ0119875(1198830) le 120578(1 + 119870120578)

(2) Γ011987510158401015840(119883) le 2119870(1+119870120578) when 119883minus119883

0 le 119905minus119905

0le 119903



119905 = 120601 (119905) 120601 (119905) = 119905 + 1198880120595 (119905) (40)

119883 = 119878 (119883) 119878 (119883) = 119883 minus Γ0119875 (119883) (41)

where 1198880= minus1120595

1015840(1199050) = 1(1 + 119870120578) and Γ

0= [1198751015840(1198830)]minus1


1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817 =1003817100381710038171003817minusΓ0119875 (119883

0)1003817100381710038171003817 le

120578

1 + 119870120578= 120601 (119905

0) minus 1199050 (42)



1 page 504] we have

100381710038171003817100381710038171198781015840(119883)

10038171003817100381710038171003817=100381710038171003817100381710038171198781015840(119883) minus 119878

1015840(1198830)10038171003817100381710038171003817

le int

119883

1198830

1003817100381710038171003817100381711987810158401015840(119883)

10038171003817100381710038171003817119889119883 = int

119883

1198830

10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817119889119883

le int

119905

1199050

119888012059510158401015840(120591) 119889120591 = int

119905

1199050

2119870

1 + 119870120578119889120591

=2119870

1 + 119870120578(119905 minus 1199050) = 1206011015840(119905)

(43)



1199090(119905) isin 119862

1[1199050

119879] 1199090(1199100(119905)) = 0 1199092


1198621

[1199050 119879]times[1199050 119879]and (119909

0(119905) 1199100(119905)) isin Ω

0 then



0 le sum

infin

119895=1(11988811198671+

119888111988831198672)119895((119879 minus 119867

4)119895minus1

(119895 minus 1)) = 1205782

(2) Δ119883 le 120578(1 + 119870120578)

(3) 11987510158401015840(119883) le 1205781

(4) 120578 gt 1119870 and 119903 lt 120578 + 1199050



119898(119905) =


Δ119910119898(119905) =

1

119867 (119905)[int

119905

1199100(119905)

ℎ (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

+ int

119905

119910119898minus1(119905)

ℎ (119905 120591) log 1003816100381610038161003816119909119898minus1 (120591)1003816100381610038161003816 119889120591

minusΔ119909119898(119905) minus 119909

119898minus1(119905) + 119892 (119905) ]

Δ119909119898(119905) minus int

119905

1199100(119905)

1198961(119905 120591)

Δ119909119898(120591)

1199090(120591)

119889120591 = 119865119898minus1

(119905)

(44)

whereΔ119909119898(119905) = 119909

119898(119905)minus119909119898minus1

(119905) andΔ119910119898(119905) = 119910

119898(119905)minus119910119898minus1

(119905)119898 = 2 3 and 119883



1003817100381710038171003817119883lowastminus 119883119898

1003817100381710038171003817 le (2

1 + 119870120578)

119898

(1

119870) (45)


0(119905)) = 0 forall119905 isin [119905






119880 from 119862[1199050 119879] rarr 119862[119905


119885 = 119880 (Δ119909) 119885 (119905) = int

119905

1199100(119905)

1198962(119905 120591) Δ119909 (120591) 119889120591 (46)

where 1198962(119905 120591) = 119896

1(119905 120591)119909

0(120591) and 119896

1(119905 120591) is defined in (18)


Δ119909 minus 119880 (Δ119909) = 1198650 (47)


0by means

of the formula

Δ119909lowast= 1198650+ 119861 (119865

0) (48)


119861 (1198650) = 119880 (119865

0) + 1198802(1198650) + sdot sdot sdot + 119880

119899(1198650) + sdot sdot sdot (49)


119885119899= 119880119899 119885119899(119905) = int

119905

1199100(119905)

119896(119899)

2(119905 120591) Δ119909 (120591) 119889120591

(119899 = 1 2 )

(50)

where 119896(119899)2




Δ119909lowast= 1198650(119905) +

infin

sum

119895=1

int

119905

1199100(119905)

119896(119895)

2(119905 120591) 119865

0(120591) 119889120591 (51)


0 119879] Since

10038161003816100381610038161198962 (119905 120591)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

1198961(119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

ℎ (119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119896 (119905 120591)

1199090(120591) 119866 (119905)

10038161003816100381610038161003816100381610038161003816

le 11988811198671+ 119888111988831198672

(52)


get

10038161003816100381610038161003816119896(2)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

10038161003816100381610038161198962 (119905 119906) 1198962 (119906 120591)1003816100381610038161003816 119889119906 le

1198722(119905 minus 119867

4)

(1)

10038161003816100381610038161003816119896(3)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(2)

2(119906 120591)

10038161003816100381610038161003816119889119906 le

1198723(119905 minus 119867

4)2

(2)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(119899minus1)

2(119906 120591)

10038161003816100381610038161003816119889119906

le119872119899(119905 minus 119867

4)119899minus1

(119899 minus 1)

(119899 = 1 2 )

(53)

then

10038171003817100381710038171198801198991003817100381710038171003817 = max119905isin[1199050119879]

int

119905

1199100(119905)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816119889120591 le

119872119899(119879 minus 119867

4)(119899minus1)

(119899 minus 1)

(54)


119899radic119880119899 le

119872(119879 minus 1198674)1minus1119899

119899radic(119899 minus 1)

997888997888997888rarr119899rarrinfin

0 (55)




Δ119909lowast= 1198650+ int

119905

1199100(119905)

Γ0(119905 120591) 119865

0(120591) 119889120591 (56)

where

Γ0(119905 120591) =

infin

sum

119895=1

119896(119895)

2(119905 120591) (57)


1003817100381710038171003817Γ01003817100381710038171003817 =

1003817100381710038171003817119861 (1198650)1003817100381710038171003817 le

infin

sum

119895=1

1003817100381710038171003817100381711988011989510038171003817100381710038171003817

le

infin

sum

119895=1

119872119895 (119879 minus 119867)

119895minus1

(119895 minus 1)le 1205782 (58)


119875 (119883) = 0 (59)


119883119899+1

= 119878 (119883119899) (119899 = 0 1 2 ) (60)


119878 (1198830) = 119883

0minus Γ0119875 (1198830) (61)


1003817100381710038171003817Γ0119875 (1198830)1003817100381710038171003817 =

1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817

=10038171003817100381710038171198831 minus 119883

0

1003817100381710038171003817 = Δ119883 le120578

1 + 119870120578

(62)


119883 isin Ω0where 120578





11987510158401015840(1198830)(119883119883) = 119861(119883

0 119883119883) where

11987510158401015840(1198830) (119883119883)

= lim119904rarr0

1

119904[1198751015840(1199090+ 119904119883) minus 119875

1015840(1198830)]

= lim119904rarr0

1

119904[(

1205971198751

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119909(1199090 1199100)) 119909

+ (1205971198751

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119910(1199090 1199100)) 119910]

lim119904rarr0

1

119904[(

1205971198752

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119909(1199090 1199100)) 119909

+ (1205971198752

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119910(1199090 1199100)) 119910]


= lim119904rarr0

1

119904[(

12059721198751

1205971199092(1199090 1199100) 119904119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198751

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198751

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198751

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

lim119904rarr0

1

119904[(

12059721198752

1205971199092(1199090 1199100) 119904119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198752

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198752

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198752

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

= (12059721198751

1205971199092(1199090 1199100) 119909119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119910119909

+12059721198751

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198751

1205971199102(1199090 1199100) 119910119909

12059721198752

1205971199092(1199090 1199100) 119909119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119910119909

+12059721198752

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198752

1205971199102(1199090 1199100) 119910119909)

(63)


11987510158401015840(1198830) (119883119883) = (119863

11198632) (

119909

119910)(

119909

119910) (64)

where

1198631= (

12059721198751

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

1198632= (

12059721198752

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

(65)


2have the

estimate100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

ℎ (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986711198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[ℎ1015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ ℎ (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

11198675+ 11986711198671015840

31198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

119896 (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986721198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881


100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[1198961015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ 119896 (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

21198675+ 11986721198671015840

31198881

(66)


10038171003817100381710038171003817le 1205781 (67)


10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817le

2119870

1 + 119870120578 (68)


119863 = 11987021205782minus 2119870120578 + 1 = (119896120578 minus 1)

2

(69)


0and 1199032= 120578+119905

0


2implies

120595 (119903) le 0 (70)



0 119879] and the


1003817100381710038171003817119883lowastminus 1198830

1003817100381710038171003817 le 119905lowastminus 1199050 (71)



119905119898+1

= 120601 (119905119898) 119898 = 0 1 2 (72)


imation 119905119898

119905lowastminus 119905119898= 120601 (119905

lowast) minus 120601 (119905


) (73)

where 119905119898isin (119905119898minus1

119905lowast) and

1206011015840(119905) = 1 + 119888

01205951015840(119905) =

2119870

1 + 119870120578(119905 minus 1199050) (74)

therefore

1206011015840(119905119898) =

2119870

1 + 119870120578(119905119898minus 1199050)

le2119870

1 + 119870120578(119905lowastminus 1199050) =

2

1 + 119870120578

(75)

then


2


)


le2


)


2

1 + 119870120578(119905lowastminus 1199050)

(76)

consequently

119905lowastminus 119905119898le (

2

1 + 119870120578)

1198981

119870 (77)

it implies

1003817100381710038171003817119883lowastminus 119883119898


2

1 + 119870120578)

1198981

119870 (78)

5 Numerical Example


119909 (119905) minus int

119905

119910(119905)

119905120591 log (|119909 (120591)|) 119889120591 = 119890119905minus1199052

3

int

119905

119910(119905)

120591 log (|119909 (120591)|) 119889120591 =119905

3 119905 isin [10 15]

(79)


119905

119910lowast(119905) =


(80)


1199090(119905) = 119890

10(119905 minus 9)

1199100(119905) = 06119905 + 4

(81)





119909= max

119905isin[1015]|119909119898(119905) minus 119909

lowast(119905)| and 120598

119910=

max119905isin[1015]

|119910119898(119905) minus 119910

lowast(119905)|

6 Conclusion




119873 = 20 ℎ = 025

119898 120598119909

120598119910

1 000 000292 000 43597119864 minus 006

3 000 31061119864 minus 008

4 000 10140119864 minus 009

5 000 12541119864 minus 010

6 000 39968119864 minus 011



Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










lim119904rarr0

1

119904[1205971198752

120597119909(1199090 1199100) 119904119909 +

1205971198752

120597119910(1199090 1199100) 119904119910

+1

2(12059721198752

1205971199092(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059721198752

120597119909120597119910(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059721198752

1205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904119910

2)])

= (1205971198751(1199090 1199100)

120597119909119909 +

1205971198751(1199090 1199100)

120597119910119910

1205971198752(1199090 1199100)

120597119909119909 +

1205971198752(1199090 1199100)

120597119910119910)

(8)

Hence

1198751015840(1198830)119883 = (

1205971198751

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

1205971198751

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

1205971198752

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

1205971198752

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

)(119909

119910) (9)


1205971198751

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

(Δ119909 (119905)) +1205971198751

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

(Δ119910 (119905))

= minus1198751(1199090(119905) 1199100(119905))

1205971198752

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

(Δ119909 (119905)) +1205971198752

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

(Δ119910 (119905))

= minus1198752(1199090(119905) 1199100(119905))

(10)

where Δ119909(119905) = 1199091(119905)minus119909

0(119905) Δ119910(119905) = 119910

1(119905)minus119910

0(119905) and (119909

0(119905)



1205971198751

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

= lim119904rarr0

1

119904(1198751(1199090+ 119904119909 119910

0) minus 1198751(1199090 1199100))

= lim119904rarr0

1

119904[119904119909 (119905)

minus int

119905

1199100(119905)

ℎ (119905 120591) (log 10038161003816100381610038161199090 (120591) + 119904119909 (120591)1003816100381610038161003816

minus log 10038161003816100381610038161199090 (120591)1003816100381610038161003816) 119889120591]

= 119909 (119905) minus int

119905

1199100(119905)

ℎ (119905 120591)119909 (120591)

1199090(120591)

119889120591

1205971198751

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

= lim119904rarr0

1

119904(1198751(1199090 1199100+ 119904119910) minus 119875

1(1199090 1199100))

= lim119904rarr0

1

119904[int

1199100(119905)+119904119910(119905)

1199100(119905)

ℎ (119905 120591) log 1003816100381610038161003816(1199090 (120591))1003816100381610038161003816 119889120591]

= ℎ (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 119910 (119905)

(11)


1205971198752

120597119909

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

= int

119905

1199100(119905)

119896 (119905 120591)119909 (120591)

1199090(120591)

119889120591

1205971198752

120597119910

10038161003816100381610038161003816100381610038161003816(1199090 1199100)

= minus119896 (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 119910 (119905)

(12)


Δ119909 (119905) minus int

119905

1199100(119905)

ℎ (119905 120591)Δ119909 (120591)

1199090(120591)

119889120591

+ ℎ (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 Δ119910 (119905)

= int

119905

1199100(119905)

ℎ (119905 120591) log 10038161003816100381610038161199090 (120591)1003816100381610038161003816 119889120591 minus 119909

0(119905) + 119892 (119905)

int

119905

1199100(119905)

119896 (119905 120591)Δ119909 (120591)

1199090(120591)

119889120591

minus 119896 (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 Δ119910 (119905)

= minusint

119905

1199100(119905)

119896 (119905 120591) log 10038161003816100381610038161199090 (120591)1003816100381610038161003816 119889120591 + 119891 (119905)

(13)




1198751015840(1198830) Δ119883119898+ 119875 (119883

119898) = 0 (14)


Δ119909119898(119905) minus int

119905

1199100(119905)

ℎ (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

+ ℎ (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 Δ119910119898 (119905)

= int

119905

1199100(119905)

ℎ (119905 120591) log 10038161003816100381610038161199090 (120591)1003816100381610038161003816 119889120591 minus 119909

0(119905) + 119892 (119905)

int

119905

1199100(119905)

119896 (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

minus 119896 (119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816 Δ119910119898 (119905)

= minusint

119905

1199100(119905)

119896 (119905 120591) log 10038161003816100381610038161199090 (120591)1003816100381610038161003816 119889120591 + 119891 (119905)

(15)

whereΔ119909119898(119905) = 119909

119898(119905)minus119909119898minus1

(119905) andΔ119910119898(119905) = 119910

119898(119905)minus119910119898minus1

(119905)119898 = 1 2 3



Δ119910 (119905) =1

119867 (119905)[int

119905

1199100(119905)

ℎ (119905 120591) [Δ119909 (120591)

1199090(120591)

+ log 10038161003816100381610038161199090 (120591)1003816100381610038161003816] 119889120591

minus [Δ119909 (119905) + 1199090(119905) minus 119892 (119905)]]

119866 (119905) [int

119905

1199100(119905)

ℎ (119905 120591) [Δ119909 (120591)

1199090(120591)

+ log 10038161003816100381610038161199090 (120591)1003816100381610038161003816] 119889120591

minus [Δ119909 (119905) + 1199090(119905) minus 119892 (119905)]]

= int

119905

1199100(119905)

119896 (119905 120591)Δ119909 (120591)

1199090(120591)

119889120591

minus int

119905

1199100(119905)

119896 (119905 120591) log 10038161003816100381610038161199090 (120591)1003816100381610038161003816 119889120591 + 119891 (119905)

(16)

where 119866(119905) = 119896(119905 1199100(119905))ℎ(119905 119910

0(119905)) and 119867(119905) = 1[ℎ(119905

1199100(119905)) log |119909


Δ119909 (119905) minus int

119905

1199100(119905)

1198961(119905 120591)

Δ119909 (120591)

1199090(120591)

120591 = 1198650(119905) (17)

where

1198961(119905 120591) = ℎ (119905 120591) minus

119896 (119905 120591)

119866 (119905)

119866 (119905) =119896 (119905 119910

0(119905))

ℎ (119905 1199100(119905))

119896 (119905 1199100(119905)) = 0 forall119905 isin [119905

0 119879]

1198650(119905) = int

119905

1199100(119905)

1198961(119905 120591) log 10038161003816100381610038161199090 (120591)

1003816100381610038161003816 119889120591 minus 1199090(119905) + 119892 (119905) +

119891 (119905)

119866 (119905)

(18)



form

Δ119910119898(119905)

=1

119867 (119905)[int

119905

1199100(119905)

ℎ (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

+ int

119905

119910119898minus1(119905)

ℎ (119905 120591) log 1003816100381610038161003816119909119898minus1 (120591)1003816100381610038161003816 119889120591

minusΔ119909119898(119905) minus 119909

119898minus1(119905) + 119892 (119905)]

(19)

Δ119909119898(119905) minus int

119905

1199100(119905)

1198961(119905 120591)

Δ119909119898(120591)

1199090(120591)

119889120591 = 119865119898minus1

(119905) (20)

where

119865119898minus1

(119905) = int

119905

119910119898minus1(119905)

1198961(119905 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591 minus 119909119898minus1

(119905)

+ 119892 (119905) +119891 (119905)

119866 (119905)

(21)



0 119879] To do




Δ119909119898(1199050) = minus119909

119898minus1(1199050) + 119892 (119905

0) +

119891 (1199050)

119866 (1199050) (22)

Δ119909119898(1199052119894) minus int

1199052119894

1199100(1199052119894)

1198961(1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

= 119865119898minus1

(1199052119894) 119894 = 1 2 119873

(23)


119905V2119894 = 119905V2119894 119905

0le 1199100(1199052119894) lt 1199052119894minus2

1199052119894 1199052119894minus2

le 1199100(1199052119894) lt 1199052119894

(24)


Δ119909119898(1199052119894) minus int

119905V2119894

1199100(1199052119894)

1198961(1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

minus

119894minus1

sum

119895=V2119894

int

1199052119895+2

1199052119895

1198961(1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

= 119865119898minus1

(1199052119894) 119894 = 1 2 119873

(25)


Case 1When V2119894

= 2119894 119894 = 1 2 119873 then

Δ119909119898(1199052119894) =

119865119898minus1

(1199052119894) + 119860 (119894) + 119861 (119894) + 119862 (119894)


) 6 sdot 1199090(1199052119894)) 1198961(1199052119894 1199052119894)

(26)


where

119860 (119894) = 05 (119905V2119894 minus 1199100(1199052119894))

times [1198961(1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ 1198961(1199052119894 1199100(1199052119894))

times

Δ119909119898(119905V2119894) (119905V2119894 minus 119910

0(1199052119894))

(119905V2119894 minus 119905V2119894minus2) (1199090 (1199100 (1199052119894)))

+ 1198961(1199052119894 1199100(1199052119894))

times


(119905V2119894 minus 119905V2119894minus2) (1199090 (1199100 (1199052119894)))]

119861 (119894) =

119894minus2

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times [1198961(1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 41198961(1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ 1198961(1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

]

119862 (119894) =(1199052119894minus 1199052119894minus2

)

6[1198961(1199052119894 1199052119894minus2

)Δ119909119898(1199052119894minus2

)

1199090(1199052119894minus2

)

+ 41198961(1199052119894 1199052119894minus1

)Δ119909119898(1199052119894minus1

)

1199090(1199052119894minus1

)]

(27)

Case 2When V2119894= 2119894 119894 = 1 2 119873 then

Δ119909119898(1199052119894) =

1198631(119894)

1198632(119894)

(28)

where

1198631(119894) = 119865

119898minus1(1199052119894) + 05119896

1(1199052119894 1199100(1199052119894))

times [Δ119909119898(1199052119894minus2

)

1199090(1199100(1199052119894))


)

1199052119894minus 1199052119894minus2

]

1198632(119894) = [1 minus 05 (119905

2119894minus 1199100(1199052))

1198961(1199052119894 1199052119894)

1199090(1199052119894)

minus 051198961(1199052119894 1199100(1199052119894))

(1199052119894minus 1199100(1199052119894))2

1199090(1199100(1199052119894)) (1199052119894minus 1199052119894minus2

)]

(29)



Δ119910119898(1199052119894) =

1

119867 (1199052119894)

times [int

1199052119894

1199100(1199052119894)

ℎ (1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

+ int

1199052119894

119910119898minus1(1199052119894)

ℎ (1199052119894 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591


(1199052119894) + 119892 (119905

2119894) ]

(30)

Let us set V2119894= 1199100(1199052119894) and 119906

2119894= 119910119898minus1

(1199052119894) and

119905V2119894 =

1199052119894 1199052119894minus2

le 1199100(1199052119894) lt 1199052119894

119905V2119894 1199050le 1199100(1199052119894) lt 1199052119894minus2

1199051199062119894

=

1199052119894 1199052119894minus2

le 119910119898minus1

(1199052119894) lt 1199052119894

1199051199062119894 1199050le 119910119898minus1

(1199052119894) lt 1199052119894minus2

(31)


Δ119910119898(1199052119894) =

1

119867 (1199052119894)

times [int

119905V2119894

1199100(1199052119894)

ℎ (1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

+

119894minus1

sum

119895=V2119894

int

1199052119895+2

1199052119895

ℎ (1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

+ int

1199051199062119894

119910119898minus1(1199052119894)

ℎ (1199052119894 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591

+

119894minus1

sum

119895=1199062119894

int

1199052119895+2

1199052119895

ℎ (1199052119894 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591


(1199052119894) + 119892 (119905

2119894) ]

(32)



Case 1When V2119894

= 2119894 and 1199062119894

= 2119894 we have

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (119905V2119894 minus 1199100(1199052119894))

times (ℎ (1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+

119894minus1

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 4ℎ (1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ ℎ (1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

)

+ 05 (1199051199062119894


(1199052119894))

times (ℎ (1199052119894 1199051199062119894) log 10038161003816100381610038161003816119909119898minus1 (1199051199062119894)

10038161003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816(119909119898minus1 (119910119898minus1 (1199052119894)))

1003816100381610038161003816 )

+

119894minus1

sum

119895=1199062119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895) log (119909

119898minus1(1199052119895))

+ 4ℎ (1199052119894 1199052119895+1

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+1)10038161003816100381610038161003816

+ ℎ (1199052119894 1199052119895+2

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+2)10038161003816100381610038161003816)


(1199052119894) + 119892 (119905

2119894) ]

]

(33)

Case 2 If V2119894= 2119894 and 119906

2119894= 2119894 then

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (1199052119894minus 1199100(1199052119894))

times (ℎ (1199052119894 1199052119894)

Δ119909119898(119905V2119894)

1199090(1199052119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+ 05 (1199051199062119894


(1199052119894))

times (ℎ (1199052119894 1199051199062119894) log 10038161003816100381610038161003816119909119898minus1 (1199051199062119894)

10038161003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )

+

119894minus1

sum

119895=1199062119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895) log 10038161003816100381610038161003816119909119898minus1 (1199052119895)

10038161003816100381610038161003816

+ 4ℎ (1199052119894 1199052119895+1

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+1)10038161003816100381610038161003816

+ ℎ (1199052119894 1199052119895+2

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+2)10038161003816100381610038161003816)


(1199052119894) + 119892 (119905

2119894) ]

]

(34)

Case 3When V2119894

= 2119894 and 1199062119894= 2119894 we get

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (119905V2119894 minus 1199100(1199052119894))

times (ℎ (1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)


+

119894minus1

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 4ℎ (1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ ℎ (1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

)

+ 05 (1199052119894minus 119910119898minus1

(1199052119894))

times (ℎ (1199052119894 1199052119894) log 1003816100381610038161003816119909119898minus1 (1199052119894)

1003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )


(1199052119894) + 119892 (119905

2119894) ]

]

(35)

Case 4 If V2119894= 2119894 and 119906

2119894= 2119894 then

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (1199052119894minus 1199100(1199052119894))

times (ℎ (1199052119894 1199052119894)Δ119909119898(1199052119894)

1199090(1199052119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+ 05 (1199052119894minus 119910119898minus1

(1199052119894))

times (ℎ (1199052119894 1199052119894) log 1003816100381610038161003816119909119898minus1 (1199052119894)

1003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )


(1199052119894) + 119892 (119905

2119894) ]

]

(36)





(i) 119862[1199050 119879]


0 119879]

(ii) 119862[1199050 119879]times[1199050 119879]


0 119879] times [119905

0 119879]

(iii) 119862 = 119883 119883 = (119909(119905) 119910(119905)) 119909(119905) 119910(119905) isin 119862[1199050 119879]

(iv) 119862lt[1199050 119879]

= 119910(119905) isin 1198621

[1199050 119879] 119910(119905) lt 119905


119909 = max119905isin[1199050 119879]

|119909 (119905)|

Δ119883119862= max Δ119909

119862[1199050119879]1003817100381710038171003817Δ119910

1003817100381710038171003817119862[1199050119879]

1198831198621 = max 119909

119862[1199050119879]10038171003817100381710038171003817119909101584010038171003817100381710038171003817119862[1199050 119879]

1003817100381710038171003817100381711988310038171003817100381710038171003817119862

= max 119909119862[1199050119879]

1003817100381710038171003817119910

1003817100381710038171003817119862[1199050 119879]

ℎ (119905 120591) = 1198671

10038171003817100381710038171003817ℎ1015840

120591(119905 120591)

10038171003817100381710038171003817= 1198671015840

1

119896 (119905 120591) = 1198672

100381710038171003817100381710038171198961015840

120591(119905 120591)

10038171003817100381710038171003817= 1198671015840

2

10038171003817100381710038171003817100381710038171003817

1

1199090

10038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

10038161003816100381610038161003816100381610038161003816

1

1199090(119905)

10038161003816100381610038161003816100381610038161003816

= 1198881

100381710038171003817100381710038171003817100381710038171003817

1

1199092

0

100381710038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

100381610038161003816100381610038161003816100381610038161003816

1

1199092

0(119905)

100381610038161003816100381610038161003816100381610038161003816

= 1198882

10038171003817100381710038171003817100381710038171003817

1

119866 (119905)

10038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

10038161003816100381610038161003816100381610038161003816

1

119866 (119905)

10038161003816100381610038161003816100381610038161003816

= 1198883

100381710038171003817100381711990901003817100381710038171003817 = max119905isin[1199050119879]

10038161003816100381610038161199090 (119905)1003816100381610038161003816 = 1198673

100381710038171003817100381710038171199091015840

0

10038171003817100381710038171003817= max119905isin[1199050119879]

100381610038161003816100381610038161199091015840

0(119905)

10038161003816100381610038161003816= 1198671015840

3

min119905isin[1199050119879]

10038161003816100381610038161199100 (119905)1003816100381610038161003816 = 1198674

1003817100381710038171003817log1003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816log (119909 (119905))1003816100381610038161003816 = 1198675

10038171003817100381710038171198921003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816119892 (119905)1003816100381610038161003816 = 1198676

10038171003817100381710038171198911003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816119891 (119905)1003816100381610038161003816 = 1198677

(37)

Let

1205781= max 119867

11198882(119879 minus 119867

4) 11986711198881 1198671015840

11198675+ 11986711198671015840

31198881

11986721198882(119879 minus 119867

4) 11986721198881 1198671015840

21198675+ 11986721198671015840

31198881

(38)



120595 (119905) = 119870 (119905 minus 1199050)2

minus (1 + 119870120578) (119905 minus 1199050) + 120578 (39)



0 119879]) 119883 minus 119883

0 le 119877 and has


0

119879]) 119883 minus 1198830 le 119903 where 119879 = 119905

0+ 119903 le 119905



(1) Γ0119875(1198830) le 120578(1 + 119870120578)

(2) Γ011987510158401015840(119883) le 2119870(1+119870120578) when 119883minus119883

0 le 119905minus119905

0le 119903



119905 = 120601 (119905) 120601 (119905) = 119905 + 1198880120595 (119905) (40)

119883 = 119878 (119883) 119878 (119883) = 119883 minus Γ0119875 (119883) (41)

where 1198880= minus1120595

1015840(1199050) = 1(1 + 119870120578) and Γ

0= [1198751015840(1198830)]minus1


1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817 =1003817100381710038171003817minusΓ0119875 (119883

0)1003817100381710038171003817 le

120578

1 + 119870120578= 120601 (119905

0) minus 1199050 (42)



1 page 504] we have

100381710038171003817100381710038171198781015840(119883)

10038171003817100381710038171003817=100381710038171003817100381710038171198781015840(119883) minus 119878

1015840(1198830)10038171003817100381710038171003817

le int

119883

1198830

1003817100381710038171003817100381711987810158401015840(119883)

10038171003817100381710038171003817119889119883 = int

119883

1198830

10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817119889119883

le int

119905

1199050

119888012059510158401015840(120591) 119889120591 = int

119905

1199050

2119870

1 + 119870120578119889120591

=2119870

1 + 119870120578(119905 minus 1199050) = 1206011015840(119905)

(43)



1199090(119905) isin 119862

1[1199050

119879] 1199090(1199100(119905)) = 0 1199092


1198621

[1199050 119879]times[1199050 119879]and (119909

0(119905) 1199100(119905)) isin Ω

0 then



0 le sum

infin

119895=1(11988811198671+

119888111988831198672)119895((119879 minus 119867

4)119895minus1

(119895 minus 1)) = 1205782

(2) Δ119883 le 120578(1 + 119870120578)

(3) 11987510158401015840(119883) le 1205781

(4) 120578 gt 1119870 and 119903 lt 120578 + 1199050



119898(119905) =


Δ119910119898(119905) =

1

119867 (119905)[int

119905

1199100(119905)

ℎ (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

+ int

119905

119910119898minus1(119905)

ℎ (119905 120591) log 1003816100381610038161003816119909119898minus1 (120591)1003816100381610038161003816 119889120591

minusΔ119909119898(119905) minus 119909

119898minus1(119905) + 119892 (119905) ]

Δ119909119898(119905) minus int

119905

1199100(119905)

1198961(119905 120591)

Δ119909119898(120591)

1199090(120591)

119889120591 = 119865119898minus1

(119905)

(44)

whereΔ119909119898(119905) = 119909

119898(119905)minus119909119898minus1

(119905) andΔ119910119898(119905) = 119910

119898(119905)minus119910119898minus1

(119905)119898 = 2 3 and 119883



1003817100381710038171003817119883lowastminus 119883119898

1003817100381710038171003817 le (2

1 + 119870120578)

119898

(1

119870) (45)


0(119905)) = 0 forall119905 isin [119905






119880 from 119862[1199050 119879] rarr 119862[119905


119885 = 119880 (Δ119909) 119885 (119905) = int

119905

1199100(119905)

1198962(119905 120591) Δ119909 (120591) 119889120591 (46)

where 1198962(119905 120591) = 119896

1(119905 120591)119909

0(120591) and 119896

1(119905 120591) is defined in (18)


Δ119909 minus 119880 (Δ119909) = 1198650 (47)


0by means

of the formula

Δ119909lowast= 1198650+ 119861 (119865

0) (48)


119861 (1198650) = 119880 (119865

0) + 1198802(1198650) + sdot sdot sdot + 119880

119899(1198650) + sdot sdot sdot (49)


119885119899= 119880119899 119885119899(119905) = int

119905

1199100(119905)

119896(119899)

2(119905 120591) Δ119909 (120591) 119889120591

(119899 = 1 2 )

(50)

where 119896(119899)2




Δ119909lowast= 1198650(119905) +

infin

sum

119895=1

int

119905

1199100(119905)

119896(119895)

2(119905 120591) 119865

0(120591) 119889120591 (51)


0 119879] Since

10038161003816100381610038161198962 (119905 120591)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

1198961(119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

ℎ (119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119896 (119905 120591)

1199090(120591) 119866 (119905)

10038161003816100381610038161003816100381610038161003816

le 11988811198671+ 119888111988831198672

(52)


get

10038161003816100381610038161003816119896(2)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

10038161003816100381610038161198962 (119905 119906) 1198962 (119906 120591)1003816100381610038161003816 119889119906 le

1198722(119905 minus 119867

4)

(1)

10038161003816100381610038161003816119896(3)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(2)

2(119906 120591)

10038161003816100381610038161003816119889119906 le

1198723(119905 minus 119867

4)2

(2)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(119899minus1)

2(119906 120591)

10038161003816100381610038161003816119889119906

le119872119899(119905 minus 119867

4)119899minus1

(119899 minus 1)

(119899 = 1 2 )

(53)

then

10038171003817100381710038171198801198991003817100381710038171003817 = max119905isin[1199050119879]

int

119905

1199100(119905)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816119889120591 le

119872119899(119879 minus 119867

4)(119899minus1)

(119899 minus 1)

(54)


119899radic119880119899 le

119872(119879 minus 1198674)1minus1119899

119899radic(119899 minus 1)

997888997888997888rarr119899rarrinfin

0 (55)




Δ119909lowast= 1198650+ int

119905

1199100(119905)

Γ0(119905 120591) 119865

0(120591) 119889120591 (56)

where

Γ0(119905 120591) =

infin

sum

119895=1

119896(119895)

2(119905 120591) (57)


1003817100381710038171003817Γ01003817100381710038171003817 =

1003817100381710038171003817119861 (1198650)1003817100381710038171003817 le

infin

sum

119895=1

1003817100381710038171003817100381711988011989510038171003817100381710038171003817

le

infin

sum

119895=1

119872119895 (119879 minus 119867)

119895minus1

(119895 minus 1)le 1205782 (58)


119875 (119883) = 0 (59)


119883119899+1

= 119878 (119883119899) (119899 = 0 1 2 ) (60)


119878 (1198830) = 119883

0minus Γ0119875 (1198830) (61)


1003817100381710038171003817Γ0119875 (1198830)1003817100381710038171003817 =

1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817

=10038171003817100381710038171198831 minus 119883

0

1003817100381710038171003817 = Δ119883 le120578

1 + 119870120578

(62)


119883 isin Ω0where 120578





11987510158401015840(1198830)(119883119883) = 119861(119883

0 119883119883) where

11987510158401015840(1198830) (119883119883)

= lim119904rarr0

1

119904[1198751015840(1199090+ 119904119883) minus 119875

1015840(1198830)]

= lim119904rarr0

1

119904[(

1205971198751

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119909(1199090 1199100)) 119909

+ (1205971198751

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119910(1199090 1199100)) 119910]

lim119904rarr0

1

119904[(

1205971198752

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119909(1199090 1199100)) 119909

+ (1205971198752

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119910(1199090 1199100)) 119910]


= lim119904rarr0

1

119904[(

12059721198751

1205971199092(1199090 1199100) 119904119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198751

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198751

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198751

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

lim119904rarr0

1

119904[(

12059721198752

1205971199092(1199090 1199100) 119904119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198752

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198752

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198752

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

= (12059721198751

1205971199092(1199090 1199100) 119909119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119910119909

+12059721198751

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198751

1205971199102(1199090 1199100) 119910119909

12059721198752

1205971199092(1199090 1199100) 119909119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119910119909

+12059721198752

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198752

1205971199102(1199090 1199100) 119910119909)

(63)


11987510158401015840(1198830) (119883119883) = (119863

11198632) (

119909

119910)(

119909

119910) (64)

where

1198631= (

12059721198751

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

1198632= (

12059721198752

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

(65)


2have the

estimate100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

ℎ (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986711198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[ℎ1015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ ℎ (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

11198675+ 11986711198671015840

31198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

119896 (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986721198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881


100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[1198961015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ 119896 (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

21198675+ 11986721198671015840

31198881

(66)


10038171003817100381710038171003817le 1205781 (67)


10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817le

2119870

1 + 119870120578 (68)


119863 = 11987021205782minus 2119870120578 + 1 = (119896120578 minus 1)

2

(69)


0and 1199032= 120578+119905

0


2implies

120595 (119903) le 0 (70)



0 119879] and the


1003817100381710038171003817119883lowastminus 1198830

1003817100381710038171003817 le 119905lowastminus 1199050 (71)



119905119898+1

= 120601 (119905119898) 119898 = 0 1 2 (72)


imation 119905119898

119905lowastminus 119905119898= 120601 (119905

lowast) minus 120601 (119905


) (73)

where 119905119898isin (119905119898minus1

119905lowast) and

1206011015840(119905) = 1 + 119888

01205951015840(119905) =

2119870

1 + 119870120578(119905 minus 1199050) (74)

therefore

1206011015840(119905119898) =

2119870

1 + 119870120578(119905119898minus 1199050)

le2119870

1 + 119870120578(119905lowastminus 1199050) =

2

1 + 119870120578

(75)

then


2


)


le2


)


2

1 + 119870120578(119905lowastminus 1199050)

(76)

consequently

119905lowastminus 119905119898le (

2

1 + 119870120578)

1198981

119870 (77)

it implies

1003817100381710038171003817119883lowastminus 119883119898


2

1 + 119870120578)

1198981

119870 (78)

5 Numerical Example


119909 (119905) minus int

119905

119910(119905)

119905120591 log (|119909 (120591)|) 119889120591 = 119890119905minus1199052

3

int

119905

119910(119905)

120591 log (|119909 (120591)|) 119889120591 =119905

3 119905 isin [10 15]

(79)


119905

119910lowast(119905) =


(80)


1199090(119905) = 119890

10(119905 minus 9)

1199100(119905) = 06119905 + 4

(81)





119909= max

119905isin[1015]|119909119898(119905) minus 119909

lowast(119905)| and 120598

119910=

max119905isin[1015]

|119910119898(119905) minus 119910

lowast(119905)|

6 Conclusion




119873 = 20 ℎ = 025

119898 120598119909

120598119910

1 000 000292 000 43597119864 minus 006

3 000 31061119864 minus 008

4 000 10140119864 minus 009

5 000 12541119864 minus 010

6 000 39968119864 minus 011



Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Δ119910 (119905) =1

119867 (119905)[int

119905

1199100(119905)

ℎ (119905 120591) [Δ119909 (120591)

1199090(120591)

+ log 10038161003816100381610038161199090 (120591)1003816100381610038161003816] 119889120591

minus [Δ119909 (119905) + 1199090(119905) minus 119892 (119905)]]

119866 (119905) [int

119905

1199100(119905)

ℎ (119905 120591) [Δ119909 (120591)

1199090(120591)

+ log 10038161003816100381610038161199090 (120591)1003816100381610038161003816] 119889120591

minus [Δ119909 (119905) + 1199090(119905) minus 119892 (119905)]]

= int

119905

1199100(119905)

119896 (119905 120591)Δ119909 (120591)

1199090(120591)

119889120591

minus int

119905

1199100(119905)

119896 (119905 120591) log 10038161003816100381610038161199090 (120591)1003816100381610038161003816 119889120591 + 119891 (119905)

(16)

where 119866(119905) = 119896(119905 1199100(119905))ℎ(119905 119910

0(119905)) and 119867(119905) = 1[ℎ(119905

1199100(119905)) log |119909


Δ119909 (119905) minus int

119905

1199100(119905)

1198961(119905 120591)

Δ119909 (120591)

1199090(120591)

120591 = 1198650(119905) (17)

where

1198961(119905 120591) = ℎ (119905 120591) minus

119896 (119905 120591)

119866 (119905)

119866 (119905) =119896 (119905 119910

0(119905))

ℎ (119905 1199100(119905))

119896 (119905 1199100(119905)) = 0 forall119905 isin [119905

0 119879]

1198650(119905) = int

119905

1199100(119905)

1198961(119905 120591) log 10038161003816100381610038161199090 (120591)

1003816100381610038161003816 119889120591 minus 1199090(119905) + 119892 (119905) +

119891 (119905)

119866 (119905)

(18)



form

Δ119910119898(119905)

=1

119867 (119905)[int

119905

1199100(119905)

ℎ (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

+ int

119905

119910119898minus1(119905)

ℎ (119905 120591) log 1003816100381610038161003816119909119898minus1 (120591)1003816100381610038161003816 119889120591

minusΔ119909119898(119905) minus 119909

119898minus1(119905) + 119892 (119905)]

(19)

Δ119909119898(119905) minus int

119905

1199100(119905)

1198961(119905 120591)

Δ119909119898(120591)

1199090(120591)

119889120591 = 119865119898minus1

(119905) (20)

where

119865119898minus1

(119905) = int

119905

119910119898minus1(119905)

1198961(119905 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591 minus 119909119898minus1

(119905)

+ 119892 (119905) +119891 (119905)

119866 (119905)

(21)



0 119879] To do




Δ119909119898(1199050) = minus119909

119898minus1(1199050) + 119892 (119905

0) +

119891 (1199050)

119866 (1199050) (22)

Δ119909119898(1199052119894) minus int

1199052119894

1199100(1199052119894)

1198961(1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

= 119865119898minus1

(1199052119894) 119894 = 1 2 119873

(23)


119905V2119894 = 119905V2119894 119905

0le 1199100(1199052119894) lt 1199052119894minus2

1199052119894 1199052119894minus2

le 1199100(1199052119894) lt 1199052119894

(24)


Δ119909119898(1199052119894) minus int

119905V2119894

1199100(1199052119894)

1198961(1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

minus

119894minus1

sum

119895=V2119894

int

1199052119895+2

1199052119895

1198961(1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

= 119865119898minus1

(1199052119894) 119894 = 1 2 119873

(25)


Case 1When V2119894

= 2119894 119894 = 1 2 119873 then

Δ119909119898(1199052119894) =

119865119898minus1

(1199052119894) + 119860 (119894) + 119861 (119894) + 119862 (119894)


) 6 sdot 1199090(1199052119894)) 1198961(1199052119894 1199052119894)

(26)


where

119860 (119894) = 05 (119905V2119894 minus 1199100(1199052119894))

times [1198961(1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ 1198961(1199052119894 1199100(1199052119894))

times

Δ119909119898(119905V2119894) (119905V2119894 minus 119910

0(1199052119894))

(119905V2119894 minus 119905V2119894minus2) (1199090 (1199100 (1199052119894)))

+ 1198961(1199052119894 1199100(1199052119894))

times


(119905V2119894 minus 119905V2119894minus2) (1199090 (1199100 (1199052119894)))]

119861 (119894) =

119894minus2

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times [1198961(1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 41198961(1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ 1198961(1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

]

119862 (119894) =(1199052119894minus 1199052119894minus2

)

6[1198961(1199052119894 1199052119894minus2

)Δ119909119898(1199052119894minus2

)

1199090(1199052119894minus2

)

+ 41198961(1199052119894 1199052119894minus1

)Δ119909119898(1199052119894minus1

)

1199090(1199052119894minus1

)]

(27)

Case 2When V2119894= 2119894 119894 = 1 2 119873 then

Δ119909119898(1199052119894) =

1198631(119894)

1198632(119894)

(28)

where

1198631(119894) = 119865

119898minus1(1199052119894) + 05119896

1(1199052119894 1199100(1199052119894))

times [Δ119909119898(1199052119894minus2

)

1199090(1199100(1199052119894))


)

1199052119894minus 1199052119894minus2

]

1198632(119894) = [1 minus 05 (119905

2119894minus 1199100(1199052))

1198961(1199052119894 1199052119894)

1199090(1199052119894)

minus 051198961(1199052119894 1199100(1199052119894))

(1199052119894minus 1199100(1199052119894))2

1199090(1199100(1199052119894)) (1199052119894minus 1199052119894minus2

)]

(29)



Δ119910119898(1199052119894) =

1

119867 (1199052119894)

times [int

1199052119894

1199100(1199052119894)

ℎ (1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

+ int

1199052119894

119910119898minus1(1199052119894)

ℎ (1199052119894 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591


(1199052119894) + 119892 (119905

2119894) ]

(30)

Let us set V2119894= 1199100(1199052119894) and 119906

2119894= 119910119898minus1

(1199052119894) and

119905V2119894 =

1199052119894 1199052119894minus2

le 1199100(1199052119894) lt 1199052119894

119905V2119894 1199050le 1199100(1199052119894) lt 1199052119894minus2

1199051199062119894

=

1199052119894 1199052119894minus2

le 119910119898minus1

(1199052119894) lt 1199052119894

1199051199062119894 1199050le 119910119898minus1

(1199052119894) lt 1199052119894minus2

(31)


Δ119910119898(1199052119894) =

1

119867 (1199052119894)

times [int

119905V2119894

1199100(1199052119894)

ℎ (1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

+

119894minus1

sum

119895=V2119894

int

1199052119895+2

1199052119895

ℎ (1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

+ int

1199051199062119894

119910119898minus1(1199052119894)

ℎ (1199052119894 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591

+

119894minus1

sum

119895=1199062119894

int

1199052119895+2

1199052119895

ℎ (1199052119894 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591


(1199052119894) + 119892 (119905

2119894) ]

(32)



Case 1When V2119894

= 2119894 and 1199062119894

= 2119894 we have

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (119905V2119894 minus 1199100(1199052119894))

times (ℎ (1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+

119894minus1

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 4ℎ (1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ ℎ (1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

)

+ 05 (1199051199062119894


(1199052119894))

times (ℎ (1199052119894 1199051199062119894) log 10038161003816100381610038161003816119909119898minus1 (1199051199062119894)

10038161003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816(119909119898minus1 (119910119898minus1 (1199052119894)))

1003816100381610038161003816 )

+

119894minus1

sum

119895=1199062119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895) log (119909

119898minus1(1199052119895))

+ 4ℎ (1199052119894 1199052119895+1

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+1)10038161003816100381610038161003816

+ ℎ (1199052119894 1199052119895+2

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+2)10038161003816100381610038161003816)


(1199052119894) + 119892 (119905

2119894) ]

]

(33)

Case 2 If V2119894= 2119894 and 119906

2119894= 2119894 then

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (1199052119894minus 1199100(1199052119894))

times (ℎ (1199052119894 1199052119894)

Δ119909119898(119905V2119894)

1199090(1199052119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+ 05 (1199051199062119894


(1199052119894))

times (ℎ (1199052119894 1199051199062119894) log 10038161003816100381610038161003816119909119898minus1 (1199051199062119894)

10038161003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )

+

119894minus1

sum

119895=1199062119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895) log 10038161003816100381610038161003816119909119898minus1 (1199052119895)

10038161003816100381610038161003816

+ 4ℎ (1199052119894 1199052119895+1

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+1)10038161003816100381610038161003816

+ ℎ (1199052119894 1199052119895+2

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+2)10038161003816100381610038161003816)


(1199052119894) + 119892 (119905

2119894) ]

]

(34)

Case 3When V2119894

= 2119894 and 1199062119894= 2119894 we get

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (119905V2119894 minus 1199100(1199052119894))

times (ℎ (1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)


+

119894minus1

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 4ℎ (1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ ℎ (1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

)

+ 05 (1199052119894minus 119910119898minus1

(1199052119894))

times (ℎ (1199052119894 1199052119894) log 1003816100381610038161003816119909119898minus1 (1199052119894)

1003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )


(1199052119894) + 119892 (119905

2119894) ]

]

(35)

Case 4 If V2119894= 2119894 and 119906

2119894= 2119894 then

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (1199052119894minus 1199100(1199052119894))

times (ℎ (1199052119894 1199052119894)Δ119909119898(1199052119894)

1199090(1199052119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+ 05 (1199052119894minus 119910119898minus1

(1199052119894))

times (ℎ (1199052119894 1199052119894) log 1003816100381610038161003816119909119898minus1 (1199052119894)

1003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )


(1199052119894) + 119892 (119905

2119894) ]

]

(36)





(i) 119862[1199050 119879]


0 119879]

(ii) 119862[1199050 119879]times[1199050 119879]


0 119879] times [119905

0 119879]

(iii) 119862 = 119883 119883 = (119909(119905) 119910(119905)) 119909(119905) 119910(119905) isin 119862[1199050 119879]

(iv) 119862lt[1199050 119879]

= 119910(119905) isin 1198621

[1199050 119879] 119910(119905) lt 119905


119909 = max119905isin[1199050 119879]

|119909 (119905)|

Δ119883119862= max Δ119909

119862[1199050119879]1003817100381710038171003817Δ119910

1003817100381710038171003817119862[1199050119879]

1198831198621 = max 119909

119862[1199050119879]10038171003817100381710038171003817119909101584010038171003817100381710038171003817119862[1199050 119879]

1003817100381710038171003817100381711988310038171003817100381710038171003817119862

= max 119909119862[1199050119879]

1003817100381710038171003817119910

1003817100381710038171003817119862[1199050 119879]

ℎ (119905 120591) = 1198671

10038171003817100381710038171003817ℎ1015840

120591(119905 120591)

10038171003817100381710038171003817= 1198671015840

1

119896 (119905 120591) = 1198672

100381710038171003817100381710038171198961015840

120591(119905 120591)

10038171003817100381710038171003817= 1198671015840

2

10038171003817100381710038171003817100381710038171003817

1

1199090

10038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

10038161003816100381610038161003816100381610038161003816

1

1199090(119905)

10038161003816100381610038161003816100381610038161003816

= 1198881

100381710038171003817100381710038171003817100381710038171003817

1

1199092

0

100381710038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

100381610038161003816100381610038161003816100381610038161003816

1

1199092

0(119905)

100381610038161003816100381610038161003816100381610038161003816

= 1198882

10038171003817100381710038171003817100381710038171003817

1

119866 (119905)

10038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

10038161003816100381610038161003816100381610038161003816

1

119866 (119905)

10038161003816100381610038161003816100381610038161003816

= 1198883

100381710038171003817100381711990901003817100381710038171003817 = max119905isin[1199050119879]

10038161003816100381610038161199090 (119905)1003816100381610038161003816 = 1198673

100381710038171003817100381710038171199091015840

0

10038171003817100381710038171003817= max119905isin[1199050119879]

100381610038161003816100381610038161199091015840

0(119905)

10038161003816100381610038161003816= 1198671015840

3

min119905isin[1199050119879]

10038161003816100381610038161199100 (119905)1003816100381610038161003816 = 1198674

1003817100381710038171003817log1003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816log (119909 (119905))1003816100381610038161003816 = 1198675

10038171003817100381710038171198921003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816119892 (119905)1003816100381610038161003816 = 1198676

10038171003817100381710038171198911003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816119891 (119905)1003816100381610038161003816 = 1198677

(37)

Let

1205781= max 119867

11198882(119879 minus 119867

4) 11986711198881 1198671015840

11198675+ 11986711198671015840

31198881

11986721198882(119879 minus 119867

4) 11986721198881 1198671015840

21198675+ 11986721198671015840

31198881

(38)



120595 (119905) = 119870 (119905 minus 1199050)2

minus (1 + 119870120578) (119905 minus 1199050) + 120578 (39)



0 119879]) 119883 minus 119883

0 le 119877 and has


0

119879]) 119883 minus 1198830 le 119903 where 119879 = 119905

0+ 119903 le 119905



(1) Γ0119875(1198830) le 120578(1 + 119870120578)

(2) Γ011987510158401015840(119883) le 2119870(1+119870120578) when 119883minus119883

0 le 119905minus119905

0le 119903



119905 = 120601 (119905) 120601 (119905) = 119905 + 1198880120595 (119905) (40)

119883 = 119878 (119883) 119878 (119883) = 119883 minus Γ0119875 (119883) (41)

where 1198880= minus1120595

1015840(1199050) = 1(1 + 119870120578) and Γ

0= [1198751015840(1198830)]minus1


1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817 =1003817100381710038171003817minusΓ0119875 (119883

0)1003817100381710038171003817 le

120578

1 + 119870120578= 120601 (119905

0) minus 1199050 (42)



1 page 504] we have

100381710038171003817100381710038171198781015840(119883)

10038171003817100381710038171003817=100381710038171003817100381710038171198781015840(119883) minus 119878

1015840(1198830)10038171003817100381710038171003817

le int

119883

1198830

1003817100381710038171003817100381711987810158401015840(119883)

10038171003817100381710038171003817119889119883 = int

119883

1198830

10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817119889119883

le int

119905

1199050

119888012059510158401015840(120591) 119889120591 = int

119905

1199050

2119870

1 + 119870120578119889120591

=2119870

1 + 119870120578(119905 minus 1199050) = 1206011015840(119905)

(43)



1199090(119905) isin 119862

1[1199050

119879] 1199090(1199100(119905)) = 0 1199092


1198621

[1199050 119879]times[1199050 119879]and (119909

0(119905) 1199100(119905)) isin Ω

0 then



0 le sum

infin

119895=1(11988811198671+

119888111988831198672)119895((119879 minus 119867

4)119895minus1

(119895 minus 1)) = 1205782

(2) Δ119883 le 120578(1 + 119870120578)

(3) 11987510158401015840(119883) le 1205781

(4) 120578 gt 1119870 and 119903 lt 120578 + 1199050



119898(119905) =


Δ119910119898(119905) =

1

119867 (119905)[int

119905

1199100(119905)

ℎ (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

+ int

119905

119910119898minus1(119905)

ℎ (119905 120591) log 1003816100381610038161003816119909119898minus1 (120591)1003816100381610038161003816 119889120591

minusΔ119909119898(119905) minus 119909

119898minus1(119905) + 119892 (119905) ]

Δ119909119898(119905) minus int

119905

1199100(119905)

1198961(119905 120591)

Δ119909119898(120591)

1199090(120591)

119889120591 = 119865119898minus1

(119905)

(44)

whereΔ119909119898(119905) = 119909

119898(119905)minus119909119898minus1

(119905) andΔ119910119898(119905) = 119910

119898(119905)minus119910119898minus1

(119905)119898 = 2 3 and 119883



1003817100381710038171003817119883lowastminus 119883119898

1003817100381710038171003817 le (2

1 + 119870120578)

119898

(1

119870) (45)


0(119905)) = 0 forall119905 isin [119905






119880 from 119862[1199050 119879] rarr 119862[119905


119885 = 119880 (Δ119909) 119885 (119905) = int

119905

1199100(119905)

1198962(119905 120591) Δ119909 (120591) 119889120591 (46)

where 1198962(119905 120591) = 119896

1(119905 120591)119909

0(120591) and 119896

1(119905 120591) is defined in (18)


Δ119909 minus 119880 (Δ119909) = 1198650 (47)


0by means

of the formula

Δ119909lowast= 1198650+ 119861 (119865

0) (48)


119861 (1198650) = 119880 (119865

0) + 1198802(1198650) + sdot sdot sdot + 119880

119899(1198650) + sdot sdot sdot (49)


119885119899= 119880119899 119885119899(119905) = int

119905

1199100(119905)

119896(119899)

2(119905 120591) Δ119909 (120591) 119889120591

(119899 = 1 2 )

(50)

where 119896(119899)2




Δ119909lowast= 1198650(119905) +

infin

sum

119895=1

int

119905

1199100(119905)

119896(119895)

2(119905 120591) 119865

0(120591) 119889120591 (51)


0 119879] Since

10038161003816100381610038161198962 (119905 120591)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

1198961(119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

ℎ (119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119896 (119905 120591)

1199090(120591) 119866 (119905)

10038161003816100381610038161003816100381610038161003816

le 11988811198671+ 119888111988831198672

(52)


get

10038161003816100381610038161003816119896(2)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

10038161003816100381610038161198962 (119905 119906) 1198962 (119906 120591)1003816100381610038161003816 119889119906 le

1198722(119905 minus 119867

4)

(1)

10038161003816100381610038161003816119896(3)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(2)

2(119906 120591)

10038161003816100381610038161003816119889119906 le

1198723(119905 minus 119867

4)2

(2)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(119899minus1)

2(119906 120591)

10038161003816100381610038161003816119889119906

le119872119899(119905 minus 119867

4)119899minus1

(119899 minus 1)

(119899 = 1 2 )

(53)

then

10038171003817100381710038171198801198991003817100381710038171003817 = max119905isin[1199050119879]

int

119905

1199100(119905)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816119889120591 le

119872119899(119879 minus 119867

4)(119899minus1)

(119899 minus 1)

(54)


119899radic119880119899 le

119872(119879 minus 1198674)1minus1119899

119899radic(119899 minus 1)

997888997888997888rarr119899rarrinfin

0 (55)




Δ119909lowast= 1198650+ int

119905

1199100(119905)

Γ0(119905 120591) 119865

0(120591) 119889120591 (56)

where

Γ0(119905 120591) =

infin

sum

119895=1

119896(119895)

2(119905 120591) (57)


1003817100381710038171003817Γ01003817100381710038171003817 =

1003817100381710038171003817119861 (1198650)1003817100381710038171003817 le

infin

sum

119895=1

1003817100381710038171003817100381711988011989510038171003817100381710038171003817

le

infin

sum

119895=1

119872119895 (119879 minus 119867)

119895minus1

(119895 minus 1)le 1205782 (58)


119875 (119883) = 0 (59)


119883119899+1

= 119878 (119883119899) (119899 = 0 1 2 ) (60)


119878 (1198830) = 119883

0minus Γ0119875 (1198830) (61)


1003817100381710038171003817Γ0119875 (1198830)1003817100381710038171003817 =

1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817

=10038171003817100381710038171198831 minus 119883

0

1003817100381710038171003817 = Δ119883 le120578

1 + 119870120578

(62)


119883 isin Ω0where 120578





11987510158401015840(1198830)(119883119883) = 119861(119883

0 119883119883) where

11987510158401015840(1198830) (119883119883)

= lim119904rarr0

1

119904[1198751015840(1199090+ 119904119883) minus 119875

1015840(1198830)]

= lim119904rarr0

1

119904[(

1205971198751

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119909(1199090 1199100)) 119909

+ (1205971198751

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119910(1199090 1199100)) 119910]

lim119904rarr0

1

119904[(

1205971198752

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119909(1199090 1199100)) 119909

+ (1205971198752

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119910(1199090 1199100)) 119910]


= lim119904rarr0

1

119904[(

12059721198751

1205971199092(1199090 1199100) 119904119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198751

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198751

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198751

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

lim119904rarr0

1

119904[(

12059721198752

1205971199092(1199090 1199100) 119904119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198752

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198752

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198752

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

= (12059721198751

1205971199092(1199090 1199100) 119909119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119910119909

+12059721198751

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198751

1205971199102(1199090 1199100) 119910119909

12059721198752

1205971199092(1199090 1199100) 119909119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119910119909

+12059721198752

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198752

1205971199102(1199090 1199100) 119910119909)

(63)


11987510158401015840(1198830) (119883119883) = (119863

11198632) (

119909

119910)(

119909

119910) (64)

where

1198631= (

12059721198751

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

1198632= (

12059721198752

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

(65)


2have the

estimate100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

ℎ (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986711198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[ℎ1015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ ℎ (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

11198675+ 11986711198671015840

31198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

119896 (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986721198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881


100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[1198961015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ 119896 (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

21198675+ 11986721198671015840

31198881

(66)


10038171003817100381710038171003817le 1205781 (67)


10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817le

2119870

1 + 119870120578 (68)


119863 = 11987021205782minus 2119870120578 + 1 = (119896120578 minus 1)

2

(69)


0and 1199032= 120578+119905

0


2implies

120595 (119903) le 0 (70)



0 119879] and the


1003817100381710038171003817119883lowastminus 1198830

1003817100381710038171003817 le 119905lowastminus 1199050 (71)



119905119898+1

= 120601 (119905119898) 119898 = 0 1 2 (72)


imation 119905119898

119905lowastminus 119905119898= 120601 (119905

lowast) minus 120601 (119905


) (73)

where 119905119898isin (119905119898minus1

119905lowast) and

1206011015840(119905) = 1 + 119888

01205951015840(119905) =

2119870

1 + 119870120578(119905 minus 1199050) (74)

therefore

1206011015840(119905119898) =

2119870

1 + 119870120578(119905119898minus 1199050)

le2119870

1 + 119870120578(119905lowastminus 1199050) =

2

1 + 119870120578

(75)

then


2


)


le2


)


2

1 + 119870120578(119905lowastminus 1199050)

(76)

consequently

119905lowastminus 119905119898le (

2

1 + 119870120578)

1198981

119870 (77)

it implies

1003817100381710038171003817119883lowastminus 119883119898


2

1 + 119870120578)

1198981

119870 (78)

5 Numerical Example


119909 (119905) minus int

119905

119910(119905)

119905120591 log (|119909 (120591)|) 119889120591 = 119890119905minus1199052

3

int

119905

119910(119905)

120591 log (|119909 (120591)|) 119889120591 =119905

3 119905 isin [10 15]

(79)


119905

119910lowast(119905) =


(80)


1199090(119905) = 119890

10(119905 minus 9)

1199100(119905) = 06119905 + 4

(81)





119909= max

119905isin[1015]|119909119898(119905) minus 119909

lowast(119905)| and 120598

119910=

max119905isin[1015]

|119910119898(119905) minus 119910

lowast(119905)|

6 Conclusion




119873 = 20 ℎ = 025

119898 120598119909

120598119910

1 000 000292 000 43597119864 minus 006

3 000 31061119864 minus 008

4 000 10140119864 minus 009

5 000 12541119864 minus 010

6 000 39968119864 minus 011



Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

119860 (119894) = 05 (119905V2119894 minus 1199100(1199052119894))

times [1198961(1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ 1198961(1199052119894 1199100(1199052119894))

times

Δ119909119898(119905V2119894) (119905V2119894 minus 119910

0(1199052119894))

(119905V2119894 minus 119905V2119894minus2) (1199090 (1199100 (1199052119894)))

+ 1198961(1199052119894 1199100(1199052119894))

times


(119905V2119894 minus 119905V2119894minus2) (1199090 (1199100 (1199052119894)))]

119861 (119894) =

119894minus2

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times [1198961(1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 41198961(1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ 1198961(1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

]

119862 (119894) =(1199052119894minus 1199052119894minus2

)

6[1198961(1199052119894 1199052119894minus2

)Δ119909119898(1199052119894minus2

)

1199090(1199052119894minus2

)

+ 41198961(1199052119894 1199052119894minus1

)Δ119909119898(1199052119894minus1

)

1199090(1199052119894minus1

)]

(27)

Case 2When V2119894= 2119894 119894 = 1 2 119873 then

Δ119909119898(1199052119894) =

1198631(119894)

1198632(119894)

(28)

where

1198631(119894) = 119865

119898minus1(1199052119894) + 05119896

1(1199052119894 1199100(1199052119894))

times [Δ119909119898(1199052119894minus2

)

1199090(1199100(1199052119894))


)

1199052119894minus 1199052119894minus2

]

1198632(119894) = [1 minus 05 (119905

2119894minus 1199100(1199052))

1198961(1199052119894 1199052119894)

1199090(1199052119894)

minus 051198961(1199052119894 1199100(1199052119894))

(1199052119894minus 1199100(1199052119894))2

1199090(1199100(1199052119894)) (1199052119894minus 1199052119894minus2

)]

(29)



Δ119910119898(1199052119894) =

1

119867 (1199052119894)

times [int

1199052119894

1199100(1199052119894)

ℎ (1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

+ int

1199052119894

119910119898minus1(1199052119894)

ℎ (1199052119894 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591


(1199052119894) + 119892 (119905

2119894) ]

(30)

Let us set V2119894= 1199100(1199052119894) and 119906

2119894= 119910119898minus1

(1199052119894) and

119905V2119894 =

1199052119894 1199052119894minus2

le 1199100(1199052119894) lt 1199052119894

119905V2119894 1199050le 1199100(1199052119894) lt 1199052119894minus2

1199051199062119894

=

1199052119894 1199052119894minus2

le 119910119898minus1

(1199052119894) lt 1199052119894

1199051199062119894 1199050le 119910119898minus1

(1199052119894) lt 1199052119894minus2

(31)


Δ119910119898(1199052119894) =

1

119867 (1199052119894)

times [int

119905V2119894

1199100(1199052119894)

ℎ (1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

+

119894minus1

sum

119895=V2119894

int

1199052119895+2

1199052119895

ℎ (1199052119894 120591)

Δ119909119898(120591)

1199090(120591)

119889120591

+ int

1199051199062119894

119910119898minus1(1199052119894)

ℎ (1199052119894 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591

+

119894minus1

sum

119895=1199062119894

int

1199052119895+2

1199052119895

ℎ (1199052119894 120591) log 1003816100381610038161003816119909119898minus1 (120591)

1003816100381610038161003816 119889120591


(1199052119894) + 119892 (119905

2119894) ]

(32)



Case 1When V2119894

= 2119894 and 1199062119894

= 2119894 we have

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (119905V2119894 minus 1199100(1199052119894))

times (ℎ (1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+

119894minus1

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 4ℎ (1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ ℎ (1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

)

+ 05 (1199051199062119894


(1199052119894))

times (ℎ (1199052119894 1199051199062119894) log 10038161003816100381610038161003816119909119898minus1 (1199051199062119894)

10038161003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816(119909119898minus1 (119910119898minus1 (1199052119894)))

1003816100381610038161003816 )

+

119894minus1

sum

119895=1199062119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895) log (119909

119898minus1(1199052119895))

+ 4ℎ (1199052119894 1199052119895+1

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+1)10038161003816100381610038161003816

+ ℎ (1199052119894 1199052119895+2

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+2)10038161003816100381610038161003816)


(1199052119894) + 119892 (119905

2119894) ]

]

(33)

Case 2 If V2119894= 2119894 and 119906

2119894= 2119894 then

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (1199052119894minus 1199100(1199052119894))

times (ℎ (1199052119894 1199052119894)

Δ119909119898(119905V2119894)

1199090(1199052119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+ 05 (1199051199062119894


(1199052119894))

times (ℎ (1199052119894 1199051199062119894) log 10038161003816100381610038161003816119909119898minus1 (1199051199062119894)

10038161003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )

+

119894minus1

sum

119895=1199062119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895) log 10038161003816100381610038161003816119909119898minus1 (1199052119895)

10038161003816100381610038161003816

+ 4ℎ (1199052119894 1199052119895+1

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+1)10038161003816100381610038161003816

+ ℎ (1199052119894 1199052119895+2

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+2)10038161003816100381610038161003816)


(1199052119894) + 119892 (119905

2119894) ]

]

(34)

Case 3When V2119894

= 2119894 and 1199062119894= 2119894 we get

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (119905V2119894 minus 1199100(1199052119894))

times (ℎ (1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)


+

119894minus1

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 4ℎ (1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ ℎ (1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

)

+ 05 (1199052119894minus 119910119898minus1

(1199052119894))

times (ℎ (1199052119894 1199052119894) log 1003816100381610038161003816119909119898minus1 (1199052119894)

1003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )


(1199052119894) + 119892 (119905

2119894) ]

]

(35)

Case 4 If V2119894= 2119894 and 119906

2119894= 2119894 then

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (1199052119894minus 1199100(1199052119894))

times (ℎ (1199052119894 1199052119894)Δ119909119898(1199052119894)

1199090(1199052119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+ 05 (1199052119894minus 119910119898minus1

(1199052119894))

times (ℎ (1199052119894 1199052119894) log 1003816100381610038161003816119909119898minus1 (1199052119894)

1003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )


(1199052119894) + 119892 (119905

2119894) ]

]

(36)





(i) 119862[1199050 119879]


0 119879]

(ii) 119862[1199050 119879]times[1199050 119879]


0 119879] times [119905

0 119879]

(iii) 119862 = 119883 119883 = (119909(119905) 119910(119905)) 119909(119905) 119910(119905) isin 119862[1199050 119879]

(iv) 119862lt[1199050 119879]

= 119910(119905) isin 1198621

[1199050 119879] 119910(119905) lt 119905


119909 = max119905isin[1199050 119879]

|119909 (119905)|

Δ119883119862= max Δ119909

119862[1199050119879]1003817100381710038171003817Δ119910

1003817100381710038171003817119862[1199050119879]

1198831198621 = max 119909

119862[1199050119879]10038171003817100381710038171003817119909101584010038171003817100381710038171003817119862[1199050 119879]

1003817100381710038171003817100381711988310038171003817100381710038171003817119862

= max 119909119862[1199050119879]

1003817100381710038171003817119910

1003817100381710038171003817119862[1199050 119879]

ℎ (119905 120591) = 1198671

10038171003817100381710038171003817ℎ1015840

120591(119905 120591)

10038171003817100381710038171003817= 1198671015840

1

119896 (119905 120591) = 1198672

100381710038171003817100381710038171198961015840

120591(119905 120591)

10038171003817100381710038171003817= 1198671015840

2

10038171003817100381710038171003817100381710038171003817

1

1199090

10038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

10038161003816100381610038161003816100381610038161003816

1

1199090(119905)

10038161003816100381610038161003816100381610038161003816

= 1198881

100381710038171003817100381710038171003817100381710038171003817

1

1199092

0

100381710038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

100381610038161003816100381610038161003816100381610038161003816

1

1199092

0(119905)

100381610038161003816100381610038161003816100381610038161003816

= 1198882

10038171003817100381710038171003817100381710038171003817

1

119866 (119905)

10038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

10038161003816100381610038161003816100381610038161003816

1

119866 (119905)

10038161003816100381610038161003816100381610038161003816

= 1198883

100381710038171003817100381711990901003817100381710038171003817 = max119905isin[1199050119879]

10038161003816100381610038161199090 (119905)1003816100381610038161003816 = 1198673

100381710038171003817100381710038171199091015840

0

10038171003817100381710038171003817= max119905isin[1199050119879]

100381610038161003816100381610038161199091015840

0(119905)

10038161003816100381610038161003816= 1198671015840

3

min119905isin[1199050119879]

10038161003816100381610038161199100 (119905)1003816100381610038161003816 = 1198674

1003817100381710038171003817log1003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816log (119909 (119905))1003816100381610038161003816 = 1198675

10038171003817100381710038171198921003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816119892 (119905)1003816100381610038161003816 = 1198676

10038171003817100381710038171198911003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816119891 (119905)1003816100381610038161003816 = 1198677

(37)

Let

1205781= max 119867

11198882(119879 minus 119867

4) 11986711198881 1198671015840

11198675+ 11986711198671015840

31198881

11986721198882(119879 minus 119867

4) 11986721198881 1198671015840

21198675+ 11986721198671015840

31198881

(38)



120595 (119905) = 119870 (119905 minus 1199050)2

minus (1 + 119870120578) (119905 minus 1199050) + 120578 (39)



0 119879]) 119883 minus 119883

0 le 119877 and has


0

119879]) 119883 minus 1198830 le 119903 where 119879 = 119905

0+ 119903 le 119905



(1) Γ0119875(1198830) le 120578(1 + 119870120578)

(2) Γ011987510158401015840(119883) le 2119870(1+119870120578) when 119883minus119883

0 le 119905minus119905

0le 119903



119905 = 120601 (119905) 120601 (119905) = 119905 + 1198880120595 (119905) (40)

119883 = 119878 (119883) 119878 (119883) = 119883 minus Γ0119875 (119883) (41)

where 1198880= minus1120595

1015840(1199050) = 1(1 + 119870120578) and Γ

0= [1198751015840(1198830)]minus1


1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817 =1003817100381710038171003817minusΓ0119875 (119883

0)1003817100381710038171003817 le

120578

1 + 119870120578= 120601 (119905

0) minus 1199050 (42)



1 page 504] we have

100381710038171003817100381710038171198781015840(119883)

10038171003817100381710038171003817=100381710038171003817100381710038171198781015840(119883) minus 119878

1015840(1198830)10038171003817100381710038171003817

le int

119883

1198830

1003817100381710038171003817100381711987810158401015840(119883)

10038171003817100381710038171003817119889119883 = int

119883

1198830

10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817119889119883

le int

119905

1199050

119888012059510158401015840(120591) 119889120591 = int

119905

1199050

2119870

1 + 119870120578119889120591

=2119870

1 + 119870120578(119905 minus 1199050) = 1206011015840(119905)

(43)



1199090(119905) isin 119862

1[1199050

119879] 1199090(1199100(119905)) = 0 1199092


1198621

[1199050 119879]times[1199050 119879]and (119909

0(119905) 1199100(119905)) isin Ω

0 then



0 le sum

infin

119895=1(11988811198671+

119888111988831198672)119895((119879 minus 119867

4)119895minus1

(119895 minus 1)) = 1205782

(2) Δ119883 le 120578(1 + 119870120578)

(3) 11987510158401015840(119883) le 1205781

(4) 120578 gt 1119870 and 119903 lt 120578 + 1199050



119898(119905) =


Δ119910119898(119905) =

1

119867 (119905)[int

119905

1199100(119905)

ℎ (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

+ int

119905

119910119898minus1(119905)

ℎ (119905 120591) log 1003816100381610038161003816119909119898minus1 (120591)1003816100381610038161003816 119889120591

minusΔ119909119898(119905) minus 119909

119898minus1(119905) + 119892 (119905) ]

Δ119909119898(119905) minus int

119905

1199100(119905)

1198961(119905 120591)

Δ119909119898(120591)

1199090(120591)

119889120591 = 119865119898minus1

(119905)

(44)

whereΔ119909119898(119905) = 119909

119898(119905)minus119909119898minus1

(119905) andΔ119910119898(119905) = 119910

119898(119905)minus119910119898minus1

(119905)119898 = 2 3 and 119883



1003817100381710038171003817119883lowastminus 119883119898

1003817100381710038171003817 le (2

1 + 119870120578)

119898

(1

119870) (45)


0(119905)) = 0 forall119905 isin [119905






119880 from 119862[1199050 119879] rarr 119862[119905


119885 = 119880 (Δ119909) 119885 (119905) = int

119905

1199100(119905)

1198962(119905 120591) Δ119909 (120591) 119889120591 (46)

where 1198962(119905 120591) = 119896

1(119905 120591)119909

0(120591) and 119896

1(119905 120591) is defined in (18)


Δ119909 minus 119880 (Δ119909) = 1198650 (47)


0by means

of the formula

Δ119909lowast= 1198650+ 119861 (119865

0) (48)


119861 (1198650) = 119880 (119865

0) + 1198802(1198650) + sdot sdot sdot + 119880

119899(1198650) + sdot sdot sdot (49)


119885119899= 119880119899 119885119899(119905) = int

119905

1199100(119905)

119896(119899)

2(119905 120591) Δ119909 (120591) 119889120591

(119899 = 1 2 )

(50)

where 119896(119899)2




Δ119909lowast= 1198650(119905) +

infin

sum

119895=1

int

119905

1199100(119905)

119896(119895)

2(119905 120591) 119865

0(120591) 119889120591 (51)


0 119879] Since

10038161003816100381610038161198962 (119905 120591)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

1198961(119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

ℎ (119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119896 (119905 120591)

1199090(120591) 119866 (119905)

10038161003816100381610038161003816100381610038161003816

le 11988811198671+ 119888111988831198672

(52)


get

10038161003816100381610038161003816119896(2)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

10038161003816100381610038161198962 (119905 119906) 1198962 (119906 120591)1003816100381610038161003816 119889119906 le

1198722(119905 minus 119867

4)

(1)

10038161003816100381610038161003816119896(3)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(2)

2(119906 120591)

10038161003816100381610038161003816119889119906 le

1198723(119905 minus 119867

4)2

(2)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(119899minus1)

2(119906 120591)

10038161003816100381610038161003816119889119906

le119872119899(119905 minus 119867

4)119899minus1

(119899 minus 1)

(119899 = 1 2 )

(53)

then

10038171003817100381710038171198801198991003817100381710038171003817 = max119905isin[1199050119879]

int

119905

1199100(119905)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816119889120591 le

119872119899(119879 minus 119867

4)(119899minus1)

(119899 minus 1)

(54)


119899radic119880119899 le

119872(119879 minus 1198674)1minus1119899

119899radic(119899 minus 1)

997888997888997888rarr119899rarrinfin

0 (55)




Δ119909lowast= 1198650+ int

119905

1199100(119905)

Γ0(119905 120591) 119865

0(120591) 119889120591 (56)

where

Γ0(119905 120591) =

infin

sum

119895=1

119896(119895)

2(119905 120591) (57)


1003817100381710038171003817Γ01003817100381710038171003817 =

1003817100381710038171003817119861 (1198650)1003817100381710038171003817 le

infin

sum

119895=1

1003817100381710038171003817100381711988011989510038171003817100381710038171003817

le

infin

sum

119895=1

119872119895 (119879 minus 119867)

119895minus1

(119895 minus 1)le 1205782 (58)


119875 (119883) = 0 (59)


119883119899+1

= 119878 (119883119899) (119899 = 0 1 2 ) (60)


119878 (1198830) = 119883

0minus Γ0119875 (1198830) (61)


1003817100381710038171003817Γ0119875 (1198830)1003817100381710038171003817 =

1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817

=10038171003817100381710038171198831 minus 119883

0

1003817100381710038171003817 = Δ119883 le120578

1 + 119870120578

(62)


119883 isin Ω0where 120578





11987510158401015840(1198830)(119883119883) = 119861(119883

0 119883119883) where

11987510158401015840(1198830) (119883119883)

= lim119904rarr0

1

119904[1198751015840(1199090+ 119904119883) minus 119875

1015840(1198830)]

= lim119904rarr0

1

119904[(

1205971198751

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119909(1199090 1199100)) 119909

+ (1205971198751

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119910(1199090 1199100)) 119910]

lim119904rarr0

1

119904[(

1205971198752

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119909(1199090 1199100)) 119909

+ (1205971198752

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119910(1199090 1199100)) 119910]


= lim119904rarr0

1

119904[(

12059721198751

1205971199092(1199090 1199100) 119904119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198751

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198751

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198751

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

lim119904rarr0

1

119904[(

12059721198752

1205971199092(1199090 1199100) 119904119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198752

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198752

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198752

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

= (12059721198751

1205971199092(1199090 1199100) 119909119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119910119909

+12059721198751

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198751

1205971199102(1199090 1199100) 119910119909

12059721198752

1205971199092(1199090 1199100) 119909119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119910119909

+12059721198752

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198752

1205971199102(1199090 1199100) 119910119909)

(63)


11987510158401015840(1198830) (119883119883) = (119863

11198632) (

119909

119910)(

119909

119910) (64)

where

1198631= (

12059721198751

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

1198632= (

12059721198752

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

(65)


2have the

estimate100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

ℎ (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986711198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[ℎ1015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ ℎ (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

11198675+ 11986711198671015840

31198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

119896 (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986721198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881


100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[1198961015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ 119896 (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

21198675+ 11986721198671015840

31198881

(66)


10038171003817100381710038171003817le 1205781 (67)


10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817le

2119870

1 + 119870120578 (68)


119863 = 11987021205782minus 2119870120578 + 1 = (119896120578 minus 1)

2

(69)


0and 1199032= 120578+119905

0


2implies

120595 (119903) le 0 (70)



0 119879] and the


1003817100381710038171003817119883lowastminus 1198830

1003817100381710038171003817 le 119905lowastminus 1199050 (71)



119905119898+1

= 120601 (119905119898) 119898 = 0 1 2 (72)


imation 119905119898

119905lowastminus 119905119898= 120601 (119905

lowast) minus 120601 (119905


) (73)

where 119905119898isin (119905119898minus1

119905lowast) and

1206011015840(119905) = 1 + 119888

01205951015840(119905) =

2119870

1 + 119870120578(119905 minus 1199050) (74)

therefore

1206011015840(119905119898) =

2119870

1 + 119870120578(119905119898minus 1199050)

le2119870

1 + 119870120578(119905lowastminus 1199050) =

2

1 + 119870120578

(75)

then


2


)


le2


)


2

1 + 119870120578(119905lowastminus 1199050)

(76)

consequently

119905lowastminus 119905119898le (

2

1 + 119870120578)

1198981

119870 (77)

it implies

1003817100381710038171003817119883lowastminus 119883119898


2

1 + 119870120578)

1198981

119870 (78)

5 Numerical Example


119909 (119905) minus int

119905

119910(119905)

119905120591 log (|119909 (120591)|) 119889120591 = 119890119905minus1199052

3

int

119905

119910(119905)

120591 log (|119909 (120591)|) 119889120591 =119905

3 119905 isin [10 15]

(79)


119905

119910lowast(119905) =


(80)


1199090(119905) = 119890

10(119905 minus 9)

1199100(119905) = 06119905 + 4

(81)





119909= max

119905isin[1015]|119909119898(119905) minus 119909

lowast(119905)| and 120598

119910=

max119905isin[1015]

|119910119898(119905) minus 119910

lowast(119905)|

6 Conclusion




119873 = 20 ℎ = 025

119898 120598119909

120598119910

1 000 000292 000 43597119864 minus 006

3 000 31061119864 minus 008

4 000 10140119864 minus 009

5 000 12541119864 minus 010

6 000 39968119864 minus 011



Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Case 1When V2119894

= 2119894 and 1199062119894

= 2119894 we have

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (119905V2119894 minus 1199100(1199052119894))

times (ℎ (1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+

119894minus1

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 4ℎ (1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ ℎ (1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

)

+ 05 (1199051199062119894


(1199052119894))

times (ℎ (1199052119894 1199051199062119894) log 10038161003816100381610038161003816119909119898minus1 (1199051199062119894)

10038161003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816(119909119898minus1 (119910119898minus1 (1199052119894)))

1003816100381610038161003816 )

+

119894minus1

sum

119895=1199062119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895) log (119909

119898minus1(1199052119895))

+ 4ℎ (1199052119894 1199052119895+1

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+1)10038161003816100381610038161003816

+ ℎ (1199052119894 1199052119895+2

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+2)10038161003816100381610038161003816)


(1199052119894) + 119892 (119905

2119894) ]

]

(33)

Case 2 If V2119894= 2119894 and 119906

2119894= 2119894 then

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (1199052119894minus 1199100(1199052119894))

times (ℎ (1199052119894 1199052119894)

Δ119909119898(119905V2119894)

1199090(1199052119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+ 05 (1199051199062119894


(1199052119894))

times (ℎ (1199052119894 1199051199062119894) log 10038161003816100381610038161003816119909119898minus1 (1199051199062119894)

10038161003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )

+

119894minus1

sum

119895=1199062119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895) log 10038161003816100381610038161003816119909119898minus1 (1199052119895)

10038161003816100381610038161003816

+ 4ℎ (1199052119894 1199052119895+1

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+1)10038161003816100381610038161003816

+ ℎ (1199052119894 1199052119895+2

) log 10038161003816100381610038161003816119909119898minus1 (1199052119895+2)10038161003816100381610038161003816)


(1199052119894) + 119892 (119905

2119894) ]

]

(34)

Case 3When V2119894

= 2119894 and 1199062119894= 2119894 we get

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (119905V2119894 minus 1199100(1199052119894))

times (ℎ (1199052119894 119905V2119894)

Δ119909119898(119905V2119894)

1199090(119905V2119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)


+

119894minus1

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 4ℎ (1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ ℎ (1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

)

+ 05 (1199052119894minus 119910119898minus1

(1199052119894))

times (ℎ (1199052119894 1199052119894) log 1003816100381610038161003816119909119898minus1 (1199052119894)

1003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )


(1199052119894) + 119892 (119905

2119894) ]

]

(35)

Case 4 If V2119894= 2119894 and 119906

2119894= 2119894 then

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (1199052119894minus 1199100(1199052119894))

times (ℎ (1199052119894 1199052119894)Δ119909119898(1199052119894)

1199090(1199052119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+ 05 (1199052119894minus 119910119898minus1

(1199052119894))

times (ℎ (1199052119894 1199052119894) log 1003816100381610038161003816119909119898minus1 (1199052119894)

1003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )


(1199052119894) + 119892 (119905

2119894) ]

]

(36)





(i) 119862[1199050 119879]


0 119879]

(ii) 119862[1199050 119879]times[1199050 119879]


0 119879] times [119905

0 119879]

(iii) 119862 = 119883 119883 = (119909(119905) 119910(119905)) 119909(119905) 119910(119905) isin 119862[1199050 119879]

(iv) 119862lt[1199050 119879]

= 119910(119905) isin 1198621

[1199050 119879] 119910(119905) lt 119905


119909 = max119905isin[1199050 119879]

|119909 (119905)|

Δ119883119862= max Δ119909

119862[1199050119879]1003817100381710038171003817Δ119910

1003817100381710038171003817119862[1199050119879]

1198831198621 = max 119909

119862[1199050119879]10038171003817100381710038171003817119909101584010038171003817100381710038171003817119862[1199050 119879]

1003817100381710038171003817100381711988310038171003817100381710038171003817119862

= max 119909119862[1199050119879]

1003817100381710038171003817119910

1003817100381710038171003817119862[1199050 119879]

ℎ (119905 120591) = 1198671

10038171003817100381710038171003817ℎ1015840

120591(119905 120591)

10038171003817100381710038171003817= 1198671015840

1

119896 (119905 120591) = 1198672

100381710038171003817100381710038171198961015840

120591(119905 120591)

10038171003817100381710038171003817= 1198671015840

2

10038171003817100381710038171003817100381710038171003817

1

1199090

10038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

10038161003816100381610038161003816100381610038161003816

1

1199090(119905)

10038161003816100381610038161003816100381610038161003816

= 1198881

100381710038171003817100381710038171003817100381710038171003817

1

1199092

0

100381710038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

100381610038161003816100381610038161003816100381610038161003816

1

1199092

0(119905)

100381610038161003816100381610038161003816100381610038161003816

= 1198882

10038171003817100381710038171003817100381710038171003817

1

119866 (119905)

10038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

10038161003816100381610038161003816100381610038161003816

1

119866 (119905)

10038161003816100381610038161003816100381610038161003816

= 1198883

100381710038171003817100381711990901003817100381710038171003817 = max119905isin[1199050119879]

10038161003816100381610038161199090 (119905)1003816100381610038161003816 = 1198673

100381710038171003817100381710038171199091015840

0

10038171003817100381710038171003817= max119905isin[1199050119879]

100381610038161003816100381610038161199091015840

0(119905)

10038161003816100381610038161003816= 1198671015840

3

min119905isin[1199050119879]

10038161003816100381610038161199100 (119905)1003816100381610038161003816 = 1198674

1003817100381710038171003817log1003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816log (119909 (119905))1003816100381610038161003816 = 1198675

10038171003817100381710038171198921003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816119892 (119905)1003816100381610038161003816 = 1198676

10038171003817100381710038171198911003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816119891 (119905)1003816100381610038161003816 = 1198677

(37)

Let

1205781= max 119867

11198882(119879 minus 119867

4) 11986711198881 1198671015840

11198675+ 11986711198671015840

31198881

11986721198882(119879 minus 119867

4) 11986721198881 1198671015840

21198675+ 11986721198671015840

31198881

(38)



120595 (119905) = 119870 (119905 minus 1199050)2

minus (1 + 119870120578) (119905 minus 1199050) + 120578 (39)



0 119879]) 119883 minus 119883

0 le 119877 and has


0

119879]) 119883 minus 1198830 le 119903 where 119879 = 119905

0+ 119903 le 119905



(1) Γ0119875(1198830) le 120578(1 + 119870120578)

(2) Γ011987510158401015840(119883) le 2119870(1+119870120578) when 119883minus119883

0 le 119905minus119905

0le 119903



119905 = 120601 (119905) 120601 (119905) = 119905 + 1198880120595 (119905) (40)

119883 = 119878 (119883) 119878 (119883) = 119883 minus Γ0119875 (119883) (41)

where 1198880= minus1120595

1015840(1199050) = 1(1 + 119870120578) and Γ

0= [1198751015840(1198830)]minus1


1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817 =1003817100381710038171003817minusΓ0119875 (119883

0)1003817100381710038171003817 le

120578

1 + 119870120578= 120601 (119905

0) minus 1199050 (42)



1 page 504] we have

100381710038171003817100381710038171198781015840(119883)

10038171003817100381710038171003817=100381710038171003817100381710038171198781015840(119883) minus 119878

1015840(1198830)10038171003817100381710038171003817

le int

119883

1198830

1003817100381710038171003817100381711987810158401015840(119883)

10038171003817100381710038171003817119889119883 = int

119883

1198830

10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817119889119883

le int

119905

1199050

119888012059510158401015840(120591) 119889120591 = int

119905

1199050

2119870

1 + 119870120578119889120591

=2119870

1 + 119870120578(119905 minus 1199050) = 1206011015840(119905)

(43)



1199090(119905) isin 119862

1[1199050

119879] 1199090(1199100(119905)) = 0 1199092


1198621

[1199050 119879]times[1199050 119879]and (119909

0(119905) 1199100(119905)) isin Ω

0 then



0 le sum

infin

119895=1(11988811198671+

119888111988831198672)119895((119879 minus 119867

4)119895minus1

(119895 minus 1)) = 1205782

(2) Δ119883 le 120578(1 + 119870120578)

(3) 11987510158401015840(119883) le 1205781

(4) 120578 gt 1119870 and 119903 lt 120578 + 1199050



119898(119905) =


Δ119910119898(119905) =

1

119867 (119905)[int

119905

1199100(119905)

ℎ (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

+ int

119905

119910119898minus1(119905)

ℎ (119905 120591) log 1003816100381610038161003816119909119898minus1 (120591)1003816100381610038161003816 119889120591

minusΔ119909119898(119905) minus 119909

119898minus1(119905) + 119892 (119905) ]

Δ119909119898(119905) minus int

119905

1199100(119905)

1198961(119905 120591)

Δ119909119898(120591)

1199090(120591)

119889120591 = 119865119898minus1

(119905)

(44)

whereΔ119909119898(119905) = 119909

119898(119905)minus119909119898minus1

(119905) andΔ119910119898(119905) = 119910

119898(119905)minus119910119898minus1

(119905)119898 = 2 3 and 119883



1003817100381710038171003817119883lowastminus 119883119898

1003817100381710038171003817 le (2

1 + 119870120578)

119898

(1

119870) (45)


0(119905)) = 0 forall119905 isin [119905






119880 from 119862[1199050 119879] rarr 119862[119905


119885 = 119880 (Δ119909) 119885 (119905) = int

119905

1199100(119905)

1198962(119905 120591) Δ119909 (120591) 119889120591 (46)

where 1198962(119905 120591) = 119896

1(119905 120591)119909

0(120591) and 119896

1(119905 120591) is defined in (18)


Δ119909 minus 119880 (Δ119909) = 1198650 (47)


0by means

of the formula

Δ119909lowast= 1198650+ 119861 (119865

0) (48)


119861 (1198650) = 119880 (119865

0) + 1198802(1198650) + sdot sdot sdot + 119880

119899(1198650) + sdot sdot sdot (49)


119885119899= 119880119899 119885119899(119905) = int

119905

1199100(119905)

119896(119899)

2(119905 120591) Δ119909 (120591) 119889120591

(119899 = 1 2 )

(50)

where 119896(119899)2




Δ119909lowast= 1198650(119905) +

infin

sum

119895=1

int

119905

1199100(119905)

119896(119895)

2(119905 120591) 119865

0(120591) 119889120591 (51)


0 119879] Since

10038161003816100381610038161198962 (119905 120591)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

1198961(119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

ℎ (119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119896 (119905 120591)

1199090(120591) 119866 (119905)

10038161003816100381610038161003816100381610038161003816

le 11988811198671+ 119888111988831198672

(52)


get

10038161003816100381610038161003816119896(2)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

10038161003816100381610038161198962 (119905 119906) 1198962 (119906 120591)1003816100381610038161003816 119889119906 le

1198722(119905 minus 119867

4)

(1)

10038161003816100381610038161003816119896(3)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(2)

2(119906 120591)

10038161003816100381610038161003816119889119906 le

1198723(119905 minus 119867

4)2

(2)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(119899minus1)

2(119906 120591)

10038161003816100381610038161003816119889119906

le119872119899(119905 minus 119867

4)119899minus1

(119899 minus 1)

(119899 = 1 2 )

(53)

then

10038171003817100381710038171198801198991003817100381710038171003817 = max119905isin[1199050119879]

int

119905

1199100(119905)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816119889120591 le

119872119899(119879 minus 119867

4)(119899minus1)

(119899 minus 1)

(54)


119899radic119880119899 le

119872(119879 minus 1198674)1minus1119899

119899radic(119899 minus 1)

997888997888997888rarr119899rarrinfin

0 (55)




Δ119909lowast= 1198650+ int

119905

1199100(119905)

Γ0(119905 120591) 119865

0(120591) 119889120591 (56)

where

Γ0(119905 120591) =

infin

sum

119895=1

119896(119895)

2(119905 120591) (57)


1003817100381710038171003817Γ01003817100381710038171003817 =

1003817100381710038171003817119861 (1198650)1003817100381710038171003817 le

infin

sum

119895=1

1003817100381710038171003817100381711988011989510038171003817100381710038171003817

le

infin

sum

119895=1

119872119895 (119879 minus 119867)

119895minus1

(119895 minus 1)le 1205782 (58)


119875 (119883) = 0 (59)


119883119899+1

= 119878 (119883119899) (119899 = 0 1 2 ) (60)


119878 (1198830) = 119883

0minus Γ0119875 (1198830) (61)


1003817100381710038171003817Γ0119875 (1198830)1003817100381710038171003817 =

1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817

=10038171003817100381710038171198831 minus 119883

0

1003817100381710038171003817 = Δ119883 le120578

1 + 119870120578

(62)


119883 isin Ω0where 120578





11987510158401015840(1198830)(119883119883) = 119861(119883

0 119883119883) where

11987510158401015840(1198830) (119883119883)

= lim119904rarr0

1

119904[1198751015840(1199090+ 119904119883) minus 119875

1015840(1198830)]

= lim119904rarr0

1

119904[(

1205971198751

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119909(1199090 1199100)) 119909

+ (1205971198751

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119910(1199090 1199100)) 119910]

lim119904rarr0

1

119904[(

1205971198752

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119909(1199090 1199100)) 119909

+ (1205971198752

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119910(1199090 1199100)) 119910]


= lim119904rarr0

1

119904[(

12059721198751

1205971199092(1199090 1199100) 119904119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198751

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198751

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198751

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

lim119904rarr0

1

119904[(

12059721198752

1205971199092(1199090 1199100) 119904119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198752

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198752

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198752

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

= (12059721198751

1205971199092(1199090 1199100) 119909119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119910119909

+12059721198751

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198751

1205971199102(1199090 1199100) 119910119909

12059721198752

1205971199092(1199090 1199100) 119909119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119910119909

+12059721198752

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198752

1205971199102(1199090 1199100) 119910119909)

(63)


11987510158401015840(1198830) (119883119883) = (119863

11198632) (

119909

119910)(

119909

119910) (64)

where

1198631= (

12059721198751

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

1198632= (

12059721198752

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

(65)


2have the

estimate100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

ℎ (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986711198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[ℎ1015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ ℎ (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

11198675+ 11986711198671015840

31198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

119896 (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986721198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881


100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[1198961015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ 119896 (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

21198675+ 11986721198671015840

31198881

(66)


10038171003817100381710038171003817le 1205781 (67)


10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817le

2119870

1 + 119870120578 (68)


119863 = 11987021205782minus 2119870120578 + 1 = (119896120578 minus 1)

2

(69)


0and 1199032= 120578+119905

0


2implies

120595 (119903) le 0 (70)



0 119879] and the


1003817100381710038171003817119883lowastminus 1198830

1003817100381710038171003817 le 119905lowastminus 1199050 (71)



119905119898+1

= 120601 (119905119898) 119898 = 0 1 2 (72)


imation 119905119898

119905lowastminus 119905119898= 120601 (119905

lowast) minus 120601 (119905


) (73)

where 119905119898isin (119905119898minus1

119905lowast) and

1206011015840(119905) = 1 + 119888

01205951015840(119905) =

2119870

1 + 119870120578(119905 minus 1199050) (74)

therefore

1206011015840(119905119898) =

2119870

1 + 119870120578(119905119898minus 1199050)

le2119870

1 + 119870120578(119905lowastminus 1199050) =

2

1 + 119870120578

(75)

then


2


)


le2


)


2

1 + 119870120578(119905lowastminus 1199050)

(76)

consequently

119905lowastminus 119905119898le (

2

1 + 119870120578)

1198981

119870 (77)

it implies

1003817100381710038171003817119883lowastminus 119883119898


2

1 + 119870120578)

1198981

119870 (78)

5 Numerical Example


119909 (119905) minus int

119905

119910(119905)

119905120591 log (|119909 (120591)|) 119889120591 = 119890119905minus1199052

3

int

119905

119910(119905)

120591 log (|119909 (120591)|) 119889120591 =119905

3 119905 isin [10 15]

(79)


119905

119910lowast(119905) =


(80)


1199090(119905) = 119890

10(119905 minus 9)

1199100(119905) = 06119905 + 4

(81)





119909= max

119905isin[1015]|119909119898(119905) minus 119909

lowast(119905)| and 120598

119910=

max119905isin[1015]

|119910119898(119905) minus 119910

lowast(119905)|

6 Conclusion




119873 = 20 ℎ = 025

119898 120598119909

120598119910

1 000 000292 000 43597119864 minus 006

3 000 31061119864 minus 008

4 000 10140119864 minus 009

5 000 12541119864 minus 010

6 000 39968119864 minus 011



Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+

119894minus1

sum

119895=V2119894

(1199052119895+2

minus 1199052119895)

6

times (ℎ (1199052119894 1199052119895)

Δ119909119898(1199052119895)

1199090(1199052119895)

+ 4ℎ (1199052119894 1199052119895+1

)

Δ119909119898(1199052119895+1

)

1199090(1199052119895+1

)

+ ℎ (1199052119894 1199052119895+2

)

Δ119909119898(1199052119895+2

)

1199090(1199052119895+2

)

)

+ 05 (1199052119894minus 119910119898minus1

(1199052119894))

times (ℎ (1199052119894 1199052119894) log 1003816100381610038161003816119909119898minus1 (1199052119894)

1003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )


(1199052119894) + 119892 (119905

2119894) ]

]

(35)

Case 4 If V2119894= 2119894 and 119906

2119894= 2119894 then

Δ119910119898(1199052119894)

=1

119867 (1199052119894)

times [

[

05 (1199052119894minus 1199100(1199052119894))

times (ℎ (1199052119894 1199052119894)Δ119909119898(1199052119894)

1199090(1199052119894)

+ ℎ (1199052119894 1199100(1199052119894))

Δ119909119898(1199100(1199052119894))

1199090(1199100(1199052119894))

)

+ 05 (1199052119894minus 119910119898minus1

(1199052119894))

times (ℎ (1199052119894 1199052119894) log 1003816100381610038161003816119909119898minus1 (1199052119894)

1003816100381610038161003816

+ ℎ (1199052119894 119910119898minus1

(1199052119894)) log 1003816100381610038161003816119909119898minus1 (119910119898minus1 (1199052119894))

1003816100381610038161003816 )


(1199052119894) + 119892 (119905

2119894) ]

]

(36)





(i) 119862[1199050 119879]


0 119879]

(ii) 119862[1199050 119879]times[1199050 119879]


0 119879] times [119905

0 119879]

(iii) 119862 = 119883 119883 = (119909(119905) 119910(119905)) 119909(119905) 119910(119905) isin 119862[1199050 119879]

(iv) 119862lt[1199050 119879]

= 119910(119905) isin 1198621

[1199050 119879] 119910(119905) lt 119905


119909 = max119905isin[1199050 119879]

|119909 (119905)|

Δ119883119862= max Δ119909

119862[1199050119879]1003817100381710038171003817Δ119910

1003817100381710038171003817119862[1199050119879]

1198831198621 = max 119909

119862[1199050119879]10038171003817100381710038171003817119909101584010038171003817100381710038171003817119862[1199050 119879]

1003817100381710038171003817100381711988310038171003817100381710038171003817119862

= max 119909119862[1199050119879]

1003817100381710038171003817119910

1003817100381710038171003817119862[1199050 119879]

ℎ (119905 120591) = 1198671

10038171003817100381710038171003817ℎ1015840

120591(119905 120591)

10038171003817100381710038171003817= 1198671015840

1

119896 (119905 120591) = 1198672

100381710038171003817100381710038171198961015840

120591(119905 120591)

10038171003817100381710038171003817= 1198671015840

2

10038171003817100381710038171003817100381710038171003817

1

1199090

10038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

10038161003816100381610038161003816100381610038161003816

1

1199090(119905)

10038161003816100381610038161003816100381610038161003816

= 1198881

100381710038171003817100381710038171003817100381710038171003817

1

1199092

0

100381710038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

100381610038161003816100381610038161003816100381610038161003816

1

1199092

0(119905)

100381610038161003816100381610038161003816100381610038161003816

= 1198882

10038171003817100381710038171003817100381710038171003817

1

119866 (119905)

10038171003817100381710038171003817100381710038171003817

= max119905isin[1199050 119879]

10038161003816100381610038161003816100381610038161003816

1

119866 (119905)

10038161003816100381610038161003816100381610038161003816

= 1198883

100381710038171003817100381711990901003817100381710038171003817 = max119905isin[1199050119879]

10038161003816100381610038161199090 (119905)1003816100381610038161003816 = 1198673

100381710038171003817100381710038171199091015840

0

10038171003817100381710038171003817= max119905isin[1199050119879]

100381610038161003816100381610038161199091015840

0(119905)

10038161003816100381610038161003816= 1198671015840

3

min119905isin[1199050119879]

10038161003816100381610038161199100 (119905)1003816100381610038161003816 = 1198674

1003817100381710038171003817log1003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816log (119909 (119905))1003816100381610038161003816 = 1198675

10038171003817100381710038171198921003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816119892 (119905)1003816100381610038161003816 = 1198676

10038171003817100381710038171198911003817100381710038171003817 = max119905isin[1199050119879]

1003816100381610038161003816119891 (119905)1003816100381610038161003816 = 1198677

(37)

Let

1205781= max 119867

11198882(119879 minus 119867

4) 11986711198881 1198671015840

11198675+ 11986711198671015840

31198881

11986721198882(119879 minus 119867

4) 11986721198881 1198671015840

21198675+ 11986721198671015840

31198881

(38)



120595 (119905) = 119870 (119905 minus 1199050)2

minus (1 + 119870120578) (119905 minus 1199050) + 120578 (39)



0 119879]) 119883 minus 119883

0 le 119877 and has


0

119879]) 119883 minus 1198830 le 119903 where 119879 = 119905

0+ 119903 le 119905



(1) Γ0119875(1198830) le 120578(1 + 119870120578)

(2) Γ011987510158401015840(119883) le 2119870(1+119870120578) when 119883minus119883

0 le 119905minus119905

0le 119903



119905 = 120601 (119905) 120601 (119905) = 119905 + 1198880120595 (119905) (40)

119883 = 119878 (119883) 119878 (119883) = 119883 minus Γ0119875 (119883) (41)

where 1198880= minus1120595

1015840(1199050) = 1(1 + 119870120578) and Γ

0= [1198751015840(1198830)]minus1


1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817 =1003817100381710038171003817minusΓ0119875 (119883

0)1003817100381710038171003817 le

120578

1 + 119870120578= 120601 (119905

0) minus 1199050 (42)



1 page 504] we have

100381710038171003817100381710038171198781015840(119883)

10038171003817100381710038171003817=100381710038171003817100381710038171198781015840(119883) minus 119878

1015840(1198830)10038171003817100381710038171003817

le int

119883

1198830

1003817100381710038171003817100381711987810158401015840(119883)

10038171003817100381710038171003817119889119883 = int

119883

1198830

10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817119889119883

le int

119905

1199050

119888012059510158401015840(120591) 119889120591 = int

119905

1199050

2119870

1 + 119870120578119889120591

=2119870

1 + 119870120578(119905 minus 1199050) = 1206011015840(119905)

(43)



1199090(119905) isin 119862

1[1199050

119879] 1199090(1199100(119905)) = 0 1199092


1198621

[1199050 119879]times[1199050 119879]and (119909

0(119905) 1199100(119905)) isin Ω

0 then



0 le sum

infin

119895=1(11988811198671+

119888111988831198672)119895((119879 minus 119867

4)119895minus1

(119895 minus 1)) = 1205782

(2) Δ119883 le 120578(1 + 119870120578)

(3) 11987510158401015840(119883) le 1205781

(4) 120578 gt 1119870 and 119903 lt 120578 + 1199050



119898(119905) =


Δ119910119898(119905) =

1

119867 (119905)[int

119905

1199100(119905)

ℎ (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

+ int

119905

119910119898minus1(119905)

ℎ (119905 120591) log 1003816100381610038161003816119909119898minus1 (120591)1003816100381610038161003816 119889120591

minusΔ119909119898(119905) minus 119909

119898minus1(119905) + 119892 (119905) ]

Δ119909119898(119905) minus int

119905

1199100(119905)

1198961(119905 120591)

Δ119909119898(120591)

1199090(120591)

119889120591 = 119865119898minus1

(119905)

(44)

whereΔ119909119898(119905) = 119909

119898(119905)minus119909119898minus1

(119905) andΔ119910119898(119905) = 119910

119898(119905)minus119910119898minus1

(119905)119898 = 2 3 and 119883



1003817100381710038171003817119883lowastminus 119883119898

1003817100381710038171003817 le (2

1 + 119870120578)

119898

(1

119870) (45)


0(119905)) = 0 forall119905 isin [119905






119880 from 119862[1199050 119879] rarr 119862[119905


119885 = 119880 (Δ119909) 119885 (119905) = int

119905

1199100(119905)

1198962(119905 120591) Δ119909 (120591) 119889120591 (46)

where 1198962(119905 120591) = 119896

1(119905 120591)119909

0(120591) and 119896

1(119905 120591) is defined in (18)


Δ119909 minus 119880 (Δ119909) = 1198650 (47)


0by means

of the formula

Δ119909lowast= 1198650+ 119861 (119865

0) (48)


119861 (1198650) = 119880 (119865

0) + 1198802(1198650) + sdot sdot sdot + 119880

119899(1198650) + sdot sdot sdot (49)


119885119899= 119880119899 119885119899(119905) = int

119905

1199100(119905)

119896(119899)

2(119905 120591) Δ119909 (120591) 119889120591

(119899 = 1 2 )

(50)

where 119896(119899)2




Δ119909lowast= 1198650(119905) +

infin

sum

119895=1

int

119905

1199100(119905)

119896(119895)

2(119905 120591) 119865

0(120591) 119889120591 (51)


0 119879] Since

10038161003816100381610038161198962 (119905 120591)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

1198961(119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

ℎ (119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119896 (119905 120591)

1199090(120591) 119866 (119905)

10038161003816100381610038161003816100381610038161003816

le 11988811198671+ 119888111988831198672

(52)


get

10038161003816100381610038161003816119896(2)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

10038161003816100381610038161198962 (119905 119906) 1198962 (119906 120591)1003816100381610038161003816 119889119906 le

1198722(119905 minus 119867

4)

(1)

10038161003816100381610038161003816119896(3)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(2)

2(119906 120591)

10038161003816100381610038161003816119889119906 le

1198723(119905 minus 119867

4)2

(2)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(119899minus1)

2(119906 120591)

10038161003816100381610038161003816119889119906

le119872119899(119905 minus 119867

4)119899minus1

(119899 minus 1)

(119899 = 1 2 )

(53)

then

10038171003817100381710038171198801198991003817100381710038171003817 = max119905isin[1199050119879]

int

119905

1199100(119905)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816119889120591 le

119872119899(119879 minus 119867

4)(119899minus1)

(119899 minus 1)

(54)


119899radic119880119899 le

119872(119879 minus 1198674)1minus1119899

119899radic(119899 minus 1)

997888997888997888rarr119899rarrinfin

0 (55)




Δ119909lowast= 1198650+ int

119905

1199100(119905)

Γ0(119905 120591) 119865

0(120591) 119889120591 (56)

where

Γ0(119905 120591) =

infin

sum

119895=1

119896(119895)

2(119905 120591) (57)


1003817100381710038171003817Γ01003817100381710038171003817 =

1003817100381710038171003817119861 (1198650)1003817100381710038171003817 le

infin

sum

119895=1

1003817100381710038171003817100381711988011989510038171003817100381710038171003817

le

infin

sum

119895=1

119872119895 (119879 minus 119867)

119895minus1

(119895 minus 1)le 1205782 (58)


119875 (119883) = 0 (59)


119883119899+1

= 119878 (119883119899) (119899 = 0 1 2 ) (60)


119878 (1198830) = 119883

0minus Γ0119875 (1198830) (61)


1003817100381710038171003817Γ0119875 (1198830)1003817100381710038171003817 =

1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817

=10038171003817100381710038171198831 minus 119883

0

1003817100381710038171003817 = Δ119883 le120578

1 + 119870120578

(62)


119883 isin Ω0where 120578





11987510158401015840(1198830)(119883119883) = 119861(119883

0 119883119883) where

11987510158401015840(1198830) (119883119883)

= lim119904rarr0

1

119904[1198751015840(1199090+ 119904119883) minus 119875

1015840(1198830)]

= lim119904rarr0

1

119904[(

1205971198751

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119909(1199090 1199100)) 119909

+ (1205971198751

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119910(1199090 1199100)) 119910]

lim119904rarr0

1

119904[(

1205971198752

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119909(1199090 1199100)) 119909

+ (1205971198752

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119910(1199090 1199100)) 119910]


= lim119904rarr0

1

119904[(

12059721198751

1205971199092(1199090 1199100) 119904119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198751

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198751

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198751

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

lim119904rarr0

1

119904[(

12059721198752

1205971199092(1199090 1199100) 119904119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198752

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198752

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198752

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

= (12059721198751

1205971199092(1199090 1199100) 119909119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119910119909

+12059721198751

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198751

1205971199102(1199090 1199100) 119910119909

12059721198752

1205971199092(1199090 1199100) 119909119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119910119909

+12059721198752

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198752

1205971199102(1199090 1199100) 119910119909)

(63)


11987510158401015840(1198830) (119883119883) = (119863

11198632) (

119909

119910)(

119909

119910) (64)

where

1198631= (

12059721198751

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

1198632= (

12059721198752

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

(65)


2have the

estimate100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

ℎ (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986711198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[ℎ1015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ ℎ (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

11198675+ 11986711198671015840

31198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

119896 (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986721198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881


100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[1198961015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ 119896 (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

21198675+ 11986721198671015840

31198881

(66)


10038171003817100381710038171003817le 1205781 (67)


10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817le

2119870

1 + 119870120578 (68)


119863 = 11987021205782minus 2119870120578 + 1 = (119896120578 minus 1)

2

(69)


0and 1199032= 120578+119905

0


2implies

120595 (119903) le 0 (70)



0 119879] and the


1003817100381710038171003817119883lowastminus 1198830

1003817100381710038171003817 le 119905lowastminus 1199050 (71)



119905119898+1

= 120601 (119905119898) 119898 = 0 1 2 (72)


imation 119905119898

119905lowastminus 119905119898= 120601 (119905

lowast) minus 120601 (119905


) (73)

where 119905119898isin (119905119898minus1

119905lowast) and

1206011015840(119905) = 1 + 119888

01205951015840(119905) =

2119870

1 + 119870120578(119905 minus 1199050) (74)

therefore

1206011015840(119905119898) =

2119870

1 + 119870120578(119905119898minus 1199050)

le2119870

1 + 119870120578(119905lowastminus 1199050) =

2

1 + 119870120578

(75)

then


2


)


le2


)


2

1 + 119870120578(119905lowastminus 1199050)

(76)

consequently

119905lowastminus 119905119898le (

2

1 + 119870120578)

1198981

119870 (77)

it implies

1003817100381710038171003817119883lowastminus 119883119898


2

1 + 119870120578)

1198981

119870 (78)

5 Numerical Example


119909 (119905) minus int

119905

119910(119905)

119905120591 log (|119909 (120591)|) 119889120591 = 119890119905minus1199052

3

int

119905

119910(119905)

120591 log (|119909 (120591)|) 119889120591 =119905

3 119905 isin [10 15]

(79)


119905

119910lowast(119905) =


(80)


1199090(119905) = 119890

10(119905 minus 9)

1199100(119905) = 06119905 + 4

(81)





119909= max

119905isin[1015]|119909119898(119905) minus 119909

lowast(119905)| and 120598

119910=

max119905isin[1015]

|119910119898(119905) minus 119910

lowast(119905)|

6 Conclusion




119873 = 20 ℎ = 025

119898 120598119909

120598119910

1 000 000292 000 43597119864 minus 006

3 000 31061119864 minus 008

4 000 10140119864 minus 009

5 000 12541119864 minus 010

6 000 39968119864 minus 011



Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120595 (119905) = 119870 (119905 minus 1199050)2

minus (1 + 119870120578) (119905 minus 1199050) + 120578 (39)



0 119879]) 119883 minus 119883

0 le 119877 and has


0

119879]) 119883 minus 1198830 le 119903 where 119879 = 119905

0+ 119903 le 119905



(1) Γ0119875(1198830) le 120578(1 + 119870120578)

(2) Γ011987510158401015840(119883) le 2119870(1+119870120578) when 119883minus119883

0 le 119905minus119905

0le 119903



119905 = 120601 (119905) 120601 (119905) = 119905 + 1198880120595 (119905) (40)

119883 = 119878 (119883) 119878 (119883) = 119883 minus Γ0119875 (119883) (41)

where 1198880= minus1120595

1015840(1199050) = 1(1 + 119870120578) and Γ

0= [1198751015840(1198830)]minus1


1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817 =1003817100381710038171003817minusΓ0119875 (119883

0)1003817100381710038171003817 le

120578

1 + 119870120578= 120601 (119905

0) minus 1199050 (42)



1 page 504] we have

100381710038171003817100381710038171198781015840(119883)

10038171003817100381710038171003817=100381710038171003817100381710038171198781015840(119883) minus 119878

1015840(1198830)10038171003817100381710038171003817

le int

119883

1198830

1003817100381710038171003817100381711987810158401015840(119883)

10038171003817100381710038171003817119889119883 = int

119883

1198830

10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817119889119883

le int

119905

1199050

119888012059510158401015840(120591) 119889120591 = int

119905

1199050

2119870

1 + 119870120578119889120591

=2119870

1 + 119870120578(119905 minus 1199050) = 1206011015840(119905)

(43)



1199090(119905) isin 119862

1[1199050

119879] 1199090(1199100(119905)) = 0 1199092


1198621

[1199050 119879]times[1199050 119879]and (119909

0(119905) 1199100(119905)) isin Ω

0 then



0 le sum

infin

119895=1(11988811198671+

119888111988831198672)119895((119879 minus 119867

4)119895minus1

(119895 minus 1)) = 1205782

(2) Δ119883 le 120578(1 + 119870120578)

(3) 11987510158401015840(119883) le 1205781

(4) 120578 gt 1119870 and 119903 lt 120578 + 1199050



119898(119905) =


Δ119910119898(119905) =

1

119867 (119905)[int

119905

1199100(119905)

ℎ (119905 120591)Δ119909119898(120591)

1199090(120591)

119889120591

+ int

119905

119910119898minus1(119905)

ℎ (119905 120591) log 1003816100381610038161003816119909119898minus1 (120591)1003816100381610038161003816 119889120591

minusΔ119909119898(119905) minus 119909

119898minus1(119905) + 119892 (119905) ]

Δ119909119898(119905) minus int

119905

1199100(119905)

1198961(119905 120591)

Δ119909119898(120591)

1199090(120591)

119889120591 = 119865119898minus1

(119905)

(44)

whereΔ119909119898(119905) = 119909

119898(119905)minus119909119898minus1

(119905) andΔ119910119898(119905) = 119910

119898(119905)minus119910119898minus1

(119905)119898 = 2 3 and 119883



1003817100381710038171003817119883lowastminus 119883119898

1003817100381710038171003817 le (2

1 + 119870120578)

119898

(1

119870) (45)


0(119905)) = 0 forall119905 isin [119905






119880 from 119862[1199050 119879] rarr 119862[119905


119885 = 119880 (Δ119909) 119885 (119905) = int

119905

1199100(119905)

1198962(119905 120591) Δ119909 (120591) 119889120591 (46)

where 1198962(119905 120591) = 119896

1(119905 120591)119909

0(120591) and 119896

1(119905 120591) is defined in (18)


Δ119909 minus 119880 (Δ119909) = 1198650 (47)


0by means

of the formula

Δ119909lowast= 1198650+ 119861 (119865

0) (48)


119861 (1198650) = 119880 (119865

0) + 1198802(1198650) + sdot sdot sdot + 119880

119899(1198650) + sdot sdot sdot (49)


119885119899= 119880119899 119885119899(119905) = int

119905

1199100(119905)

119896(119899)

2(119905 120591) Δ119909 (120591) 119889120591

(119899 = 1 2 )

(50)

where 119896(119899)2




Δ119909lowast= 1198650(119905) +

infin

sum

119895=1

int

119905

1199100(119905)

119896(119895)

2(119905 120591) 119865

0(120591) 119889120591 (51)


0 119879] Since

10038161003816100381610038161198962 (119905 120591)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

1198961(119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

ℎ (119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119896 (119905 120591)

1199090(120591) 119866 (119905)

10038161003816100381610038161003816100381610038161003816

le 11988811198671+ 119888111988831198672

(52)


get

10038161003816100381610038161003816119896(2)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

10038161003816100381610038161198962 (119905 119906) 1198962 (119906 120591)1003816100381610038161003816 119889119906 le

1198722(119905 minus 119867

4)

(1)

10038161003816100381610038161003816119896(3)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(2)

2(119906 120591)

10038161003816100381610038161003816119889119906 le

1198723(119905 minus 119867

4)2

(2)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(119899minus1)

2(119906 120591)

10038161003816100381610038161003816119889119906

le119872119899(119905 minus 119867

4)119899minus1

(119899 minus 1)

(119899 = 1 2 )

(53)

then

10038171003817100381710038171198801198991003817100381710038171003817 = max119905isin[1199050119879]

int

119905

1199100(119905)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816119889120591 le

119872119899(119879 minus 119867

4)(119899minus1)

(119899 minus 1)

(54)


119899radic119880119899 le

119872(119879 minus 1198674)1minus1119899

119899radic(119899 minus 1)

997888997888997888rarr119899rarrinfin

0 (55)




Δ119909lowast= 1198650+ int

119905

1199100(119905)

Γ0(119905 120591) 119865

0(120591) 119889120591 (56)

where

Γ0(119905 120591) =

infin

sum

119895=1

119896(119895)

2(119905 120591) (57)


1003817100381710038171003817Γ01003817100381710038171003817 =

1003817100381710038171003817119861 (1198650)1003817100381710038171003817 le

infin

sum

119895=1

1003817100381710038171003817100381711988011989510038171003817100381710038171003817

le

infin

sum

119895=1

119872119895 (119879 minus 119867)

119895minus1

(119895 minus 1)le 1205782 (58)


119875 (119883) = 0 (59)


119883119899+1

= 119878 (119883119899) (119899 = 0 1 2 ) (60)


119878 (1198830) = 119883

0minus Γ0119875 (1198830) (61)


1003817100381710038171003817Γ0119875 (1198830)1003817100381710038171003817 =

1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817

=10038171003817100381710038171198831 minus 119883

0

1003817100381710038171003817 = Δ119883 le120578

1 + 119870120578

(62)


119883 isin Ω0where 120578





11987510158401015840(1198830)(119883119883) = 119861(119883

0 119883119883) where

11987510158401015840(1198830) (119883119883)

= lim119904rarr0

1

119904[1198751015840(1199090+ 119904119883) minus 119875

1015840(1198830)]

= lim119904rarr0

1

119904[(

1205971198751

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119909(1199090 1199100)) 119909

+ (1205971198751

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119910(1199090 1199100)) 119910]

lim119904rarr0

1

119904[(

1205971198752

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119909(1199090 1199100)) 119909

+ (1205971198752

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119910(1199090 1199100)) 119910]


= lim119904rarr0

1

119904[(

12059721198751

1205971199092(1199090 1199100) 119904119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198751

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198751

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198751

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

lim119904rarr0

1

119904[(

12059721198752

1205971199092(1199090 1199100) 119904119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198752

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198752

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198752

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

= (12059721198751

1205971199092(1199090 1199100) 119909119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119910119909

+12059721198751

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198751

1205971199102(1199090 1199100) 119910119909

12059721198752

1205971199092(1199090 1199100) 119909119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119910119909

+12059721198752

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198752

1205971199102(1199090 1199100) 119910119909)

(63)


11987510158401015840(1198830) (119883119883) = (119863

11198632) (

119909

119910)(

119909

119910) (64)

where

1198631= (

12059721198751

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

1198632= (

12059721198752

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

(65)


2have the

estimate100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

ℎ (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986711198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[ℎ1015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ ℎ (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

11198675+ 11986711198671015840

31198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

119896 (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986721198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881


100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[1198961015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ 119896 (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

21198675+ 11986721198671015840

31198881

(66)


10038171003817100381710038171003817le 1205781 (67)


10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817le

2119870

1 + 119870120578 (68)


119863 = 11987021205782minus 2119870120578 + 1 = (119896120578 minus 1)

2

(69)


0and 1199032= 120578+119905

0


2implies

120595 (119903) le 0 (70)



0 119879] and the


1003817100381710038171003817119883lowastminus 1198830

1003817100381710038171003817 le 119905lowastminus 1199050 (71)



119905119898+1

= 120601 (119905119898) 119898 = 0 1 2 (72)


imation 119905119898

119905lowastminus 119905119898= 120601 (119905

lowast) minus 120601 (119905


) (73)

where 119905119898isin (119905119898minus1

119905lowast) and

1206011015840(119905) = 1 + 119888

01205951015840(119905) =

2119870

1 + 119870120578(119905 minus 1199050) (74)

therefore

1206011015840(119905119898) =

2119870

1 + 119870120578(119905119898minus 1199050)

le2119870

1 + 119870120578(119905lowastminus 1199050) =

2

1 + 119870120578

(75)

then


2


)


le2


)


2

1 + 119870120578(119905lowastminus 1199050)

(76)

consequently

119905lowastminus 119905119898le (

2

1 + 119870120578)

1198981

119870 (77)

it implies

1003817100381710038171003817119883lowastminus 119883119898


2

1 + 119870120578)

1198981

119870 (78)

5 Numerical Example


119909 (119905) minus int

119905

119910(119905)

119905120591 log (|119909 (120591)|) 119889120591 = 119890119905minus1199052

3

int

119905

119910(119905)

120591 log (|119909 (120591)|) 119889120591 =119905

3 119905 isin [10 15]

(79)


119905

119910lowast(119905) =


(80)


1199090(119905) = 119890

10(119905 minus 9)

1199100(119905) = 06119905 + 4

(81)





119909= max

119905isin[1015]|119909119898(119905) minus 119909

lowast(119905)| and 120598

119910=

max119905isin[1015]

|119910119898(119905) minus 119910

lowast(119905)|

6 Conclusion




119873 = 20 ℎ = 025

119898 120598119909

120598119910

1 000 000292 000 43597119864 minus 006

3 000 31061119864 minus 008

4 000 10140119864 minus 009

5 000 12541119864 minus 010

6 000 39968119864 minus 011



Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Δ119909lowast= 1198650(119905) +

infin

sum

119895=1

int

119905

1199100(119905)

119896(119895)

2(119905 120591) 119865

0(120591) 119889120591 (51)


0 119879] Since

10038161003816100381610038161198962 (119905 120591)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

1198961(119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

ℎ (119905 120591)

1199090(120591)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119896 (119905 120591)

1199090(120591) 119866 (119905)

10038161003816100381610038161003816100381610038161003816

le 11988811198671+ 119888111988831198672

(52)


get

10038161003816100381610038161003816119896(2)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

10038161003816100381610038161198962 (119905 119906) 1198962 (119906 120591)1003816100381610038161003816 119889119906 le

1198722(119905 minus 119867

4)

(1)

10038161003816100381610038161003816119896(3)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(2)

2(119906 120591)

10038161003816100381610038161003816119889119906 le

1198723(119905 minus 119867

4)2

(2)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816le int

119905

1199100(119905)

100381610038161003816100381610038161198962(119905 119906) 119896

(119899minus1)

2(119906 120591)

10038161003816100381610038161003816119889119906

le119872119899(119905 minus 119867

4)119899minus1

(119899 minus 1)

(119899 = 1 2 )

(53)

then

10038171003817100381710038171198801198991003817100381710038171003817 = max119905isin[1199050119879]

int

119905

1199100(119905)

10038161003816100381610038161003816119896(119899)

2(119905 120591)

10038161003816100381610038161003816119889120591 le

119872119899(119879 minus 119867

4)(119899minus1)

(119899 minus 1)

(54)


119899radic119880119899 le

119872(119879 minus 1198674)1minus1119899

119899radic(119899 minus 1)

997888997888997888rarr119899rarrinfin

0 (55)




Δ119909lowast= 1198650+ int

119905

1199100(119905)

Γ0(119905 120591) 119865

0(120591) 119889120591 (56)

where

Γ0(119905 120591) =

infin

sum

119895=1

119896(119895)

2(119905 120591) (57)


1003817100381710038171003817Γ01003817100381710038171003817 =

1003817100381710038171003817119861 (1198650)1003817100381710038171003817 le

infin

sum

119895=1

1003817100381710038171003817100381711988011989510038171003817100381710038171003817

le

infin

sum

119895=1

119872119895 (119879 minus 119867)

119895minus1

(119895 minus 1)le 1205782 (58)


119875 (119883) = 0 (59)


119883119899+1

= 119878 (119883119899) (119899 = 0 1 2 ) (60)


119878 (1198830) = 119883

0minus Γ0119875 (1198830) (61)


1003817100381710038171003817Γ0119875 (1198830)1003817100381710038171003817 =

1003817100381710038171003817119878 (1198830) minus 1198830

1003817100381710038171003817

=10038171003817100381710038171198831 minus 119883

0

1003817100381710038171003817 = Δ119883 le120578

1 + 119870120578

(62)


119883 isin Ω0where 120578





11987510158401015840(1198830)(119883119883) = 119861(119883

0 119883119883) where

11987510158401015840(1198830) (119883119883)

= lim119904rarr0

1

119904[1198751015840(1199090+ 119904119883) minus 119875

1015840(1198830)]

= lim119904rarr0

1

119904[(

1205971198751

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119909(1199090 1199100)) 119909

+ (1205971198751

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198751

120597119910(1199090 1199100)) 119910]

lim119904rarr0

1

119904[(

1205971198752

120597119909(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119909(1199090 1199100)) 119909

+ (1205971198752

120597119910(1199090+ 119904119909 119910

0+ 119904119910) minus

1205971198752

120597119910(1199090 1199100)) 119910]


= lim119904rarr0

1

119904[(

12059721198751

1205971199092(1199090 1199100) 119904119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198751

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198751

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198751

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

lim119904rarr0

1

119904[(

12059721198752

1205971199092(1199090 1199100) 119904119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198752

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198752

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198752

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

= (12059721198751

1205971199092(1199090 1199100) 119909119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119910119909

+12059721198751

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198751

1205971199102(1199090 1199100) 119910119909

12059721198752

1205971199092(1199090 1199100) 119909119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119910119909

+12059721198752

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198752

1205971199102(1199090 1199100) 119910119909)

(63)


11987510158401015840(1198830) (119883119883) = (119863

11198632) (

119909

119910)(

119909

119910) (64)

where

1198631= (

12059721198751

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

1198632= (

12059721198752

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

(65)


2have the

estimate100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

ℎ (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986711198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[ℎ1015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ ℎ (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

11198675+ 11986711198671015840

31198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

119896 (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986721198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881


100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[1198961015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ 119896 (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

21198675+ 11986721198671015840

31198881

(66)


10038171003817100381710038171003817le 1205781 (67)


10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817le

2119870

1 + 119870120578 (68)


119863 = 11987021205782minus 2119870120578 + 1 = (119896120578 minus 1)

2

(69)


0and 1199032= 120578+119905

0


2implies

120595 (119903) le 0 (70)



0 119879] and the


1003817100381710038171003817119883lowastminus 1198830

1003817100381710038171003817 le 119905lowastminus 1199050 (71)



119905119898+1

= 120601 (119905119898) 119898 = 0 1 2 (72)


imation 119905119898

119905lowastminus 119905119898= 120601 (119905

lowast) minus 120601 (119905


) (73)

where 119905119898isin (119905119898minus1

119905lowast) and

1206011015840(119905) = 1 + 119888

01205951015840(119905) =

2119870

1 + 119870120578(119905 minus 1199050) (74)

therefore

1206011015840(119905119898) =

2119870

1 + 119870120578(119905119898minus 1199050)

le2119870

1 + 119870120578(119905lowastminus 1199050) =

2

1 + 119870120578

(75)

then


2


)


le2


)


2

1 + 119870120578(119905lowastminus 1199050)

(76)

consequently

119905lowastminus 119905119898le (

2

1 + 119870120578)

1198981

119870 (77)

it implies

1003817100381710038171003817119883lowastminus 119883119898


2

1 + 119870120578)

1198981

119870 (78)

5 Numerical Example


119909 (119905) minus int

119905

119910(119905)

119905120591 log (|119909 (120591)|) 119889120591 = 119890119905minus1199052

3

int

119905

119910(119905)

120591 log (|119909 (120591)|) 119889120591 =119905

3 119905 isin [10 15]

(79)


119905

119910lowast(119905) =


(80)


1199090(119905) = 119890

10(119905 minus 9)

1199100(119905) = 06119905 + 4

(81)





119909= max

119905isin[1015]|119909119898(119905) minus 119909

lowast(119905)| and 120598

119910=

max119905isin[1015]

|119910119898(119905) minus 119910

lowast(119905)|

6 Conclusion




119873 = 20 ℎ = 025

119898 120598119909

120598119910

1 000 000292 000 43597119864 minus 006

3 000 31061119864 minus 008

4 000 10140119864 minus 009

5 000 12541119864 minus 010

6 000 39968119864 minus 011



Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= lim119904rarr0

1

119904[(

12059721198751

1205971199092(1199090 1199100) 119904119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198751

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198751

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198751

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198751

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198751

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198751

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

lim119904rarr0

1

119904[(

12059721198752

1205971199092(1199090 1199100) 119904119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119904119910

+1

2(12059731198752

1205971199093(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199102120597119909

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119910))119909

+ (12059721198752

120597119909120597119910(1199090 1199100) 119904119909 +

12059721198752

1205971199102(1199090 1199100) 119904119910

+1

2(

12059731198752

1205971199092120597119910

(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

21199092

+ 212059731198752

1205971199091205971199102(1199090+ 120579119904119909 119910

0+ 120575119904119910) 119904

2119909119910

+12059731198752

1205971199103(1199090+ 120579119904119909 120575119904119910) 119904

21199102))119910]

= (12059721198751

1205971199092(1199090 1199100) 119909119909 +

12059721198751

120597119910120597119909(1199090 1199100) 119910119909

+12059721198751

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198751

1205971199102(1199090 1199100) 119910119909

12059721198752

1205971199092(1199090 1199100) 119909119909 +

12059721198752

120597119910120597119909(1199090 1199100) 119910119909

+12059721198752

120597119909120597119910(1199090 1199100) 119909119910 +

12059721198752

1205971199102(1199090 1199100) 119910119909)

(63)


11987510158401015840(1198830) (119883119883) = (119863

11198632) (

119909

119910)(

119909

119910) (64)

where

1198631= (

12059721198751

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198751

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

1198632= (

12059721198752

1205971199092

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119910120597119909

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

120597119909120597119910

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

12059721198752

1205971199102

100381610038161003816100381610038161003816100381610038161003816(1199090 1199100)

)

(65)


2have the

estimate100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

ℎ (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986711198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

ℎ (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986711198881

100381710038171003817100381710038171003817100381710038171003817

12059721198751

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[ℎ1015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ ℎ (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

11198675+ 11986711198671015840

31198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199092

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

int

119905

1199100(119905)

119896 (119905 120591)119909 (120591)

1199092

0(120591)

119909 (120591) 119889120591

100381610038161003816100381610038161003816100381610038161003816

le 11986721198882(119879 minus 119867

4)

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119909120597119910

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881

100381710038171003817100381710038171003817100381710038171003817

12059721198752

120597119910120597119909

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

minus119896 (119905 1199100(119905))

119909 (1199100(119905))

1199090(1199100(119905))

119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 11986721198881


100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[1198961015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ 119896 (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

21198675+ 11986721198671015840

31198881

(66)


10038171003817100381710038171003817le 1205781 (67)


10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817le

2119870

1 + 119870120578 (68)


119863 = 11987021205782minus 2119870120578 + 1 = (119896120578 minus 1)

2

(69)


0and 1199032= 120578+119905

0


2implies

120595 (119903) le 0 (70)



0 119879] and the


1003817100381710038171003817119883lowastminus 1198830

1003817100381710038171003817 le 119905lowastminus 1199050 (71)



119905119898+1

= 120601 (119905119898) 119898 = 0 1 2 (72)


imation 119905119898

119905lowastminus 119905119898= 120601 (119905

lowast) minus 120601 (119905


) (73)

where 119905119898isin (119905119898minus1

119905lowast) and

1206011015840(119905) = 1 + 119888

01205951015840(119905) =

2119870

1 + 119870120578(119905 minus 1199050) (74)

therefore

1206011015840(119905119898) =

2119870

1 + 119870120578(119905119898minus 1199050)

le2119870

1 + 119870120578(119905lowastminus 1199050) =

2

1 + 119870120578

(75)

then


2


)


le2


)


2

1 + 119870120578(119905lowastminus 1199050)

(76)

consequently

119905lowastminus 119905119898le (

2

1 + 119870120578)

1198981

119870 (77)

it implies

1003817100381710038171003817119883lowastminus 119883119898


2

1 + 119870120578)

1198981

119870 (78)

5 Numerical Example


119909 (119905) minus int

119905

119910(119905)

119905120591 log (|119909 (120591)|) 119889120591 = 119890119905minus1199052

3

int

119905

119910(119905)

120591 log (|119909 (120591)|) 119889120591 =119905

3 119905 isin [10 15]

(79)


119905

119910lowast(119905) =


(80)


1199090(119905) = 119890

10(119905 minus 9)

1199100(119905) = 06119905 + 4

(81)





119909= max

119905isin[1015]|119909119898(119905) minus 119909

lowast(119905)| and 120598

119910=

max119905isin[1015]

|119910119898(119905) minus 119910

lowast(119905)|

6 Conclusion




119873 = 20 ℎ = 025

119898 120598119909

120598119910

1 000 000292 000 43597119864 minus 006

3 000 31061119864 minus 008

4 000 10140119864 minus 009

5 000 12541119864 minus 010

6 000 39968119864 minus 011



Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










100381710038171003817100381710038171003817100381710038171003817

12059721198752

1205971199102

100381710038171003817100381710038171003817100381710038171003817

= max119883le1119883le1

100381610038161003816100381610038161003816100381610038161003816

[1198961015840

120591(119905 1199100(119905)) log 10038161003816100381610038161199090 (1199100 (119905))

1003816100381610038161003816

+ 119896 (119905 1199100(119905))

1199091015840

0(1199100(119905))

1199090(1199100(119905))

]

times 119910 (119905) 119910 (119905)

100381610038161003816100381610038161003816100381610038161003816

le 1198671015840

21198675+ 11986721198671015840

31198881

(66)


10038171003817100381710038171003817le 1205781 (67)


10038171003817100381710038171003817Γ011987510158401015840(119883)

10038171003817100381710038171003817le

2119870

1 + 119870120578 (68)


119863 = 11987021205782minus 2119870120578 + 1 = (119896120578 minus 1)

2

(69)


0and 1199032= 120578+119905

0


2implies

120595 (119903) le 0 (70)



0 119879] and the


1003817100381710038171003817119883lowastminus 1198830

1003817100381710038171003817 le 119905lowastminus 1199050 (71)



119905119898+1

= 120601 (119905119898) 119898 = 0 1 2 (72)


imation 119905119898

119905lowastminus 119905119898= 120601 (119905

lowast) minus 120601 (119905


) (73)

where 119905119898isin (119905119898minus1

119905lowast) and

1206011015840(119905) = 1 + 119888

01205951015840(119905) =

2119870

1 + 119870120578(119905 minus 1199050) (74)

therefore

1206011015840(119905119898) =

2119870

1 + 119870120578(119905119898minus 1199050)

le2119870

1 + 119870120578(119905lowastminus 1199050) =

2

1 + 119870120578

(75)

then


2


)


le2


)


2

1 + 119870120578(119905lowastminus 1199050)

(76)

consequently

119905lowastminus 119905119898le (

2

1 + 119870120578)

1198981

119870 (77)

it implies

1003817100381710038171003817119883lowastminus 119883119898


2

1 + 119870120578)

1198981

119870 (78)

5 Numerical Example


119909 (119905) minus int

119905

119910(119905)

119905120591 log (|119909 (120591)|) 119889120591 = 119890119905minus1199052

3

int

119905

119910(119905)

120591 log (|119909 (120591)|) 119889120591 =119905

3 119905 isin [10 15]

(79)


119905

119910lowast(119905) =


(80)


1199090(119905) = 119890

10(119905 minus 9)

1199100(119905) = 06119905 + 4

(81)





119909= max

119905isin[1015]|119909119898(119905) minus 119909

lowast(119905)| and 120598

119910=

max119905isin[1015]

|119910119898(119905) minus 119910

lowast(119905)|

6 Conclusion




119873 = 20 ℎ = 025

119898 120598119909

120598119910

1 000 000292 000 43597119864 minus 006

3 000 31061119864 minus 008

4 000 10140119864 minus 009

5 000 12541119864 minus 010

6 000 39968119864 minus 011



Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119873 = 20 ℎ = 025

119898 120598119909

120598119910

1 000 000292 000 43597119864 minus 006

3 000 31061119864 minus 008

4 000 10140119864 minus 009

5 000 12541119864 minus 010

6 000 39968119864 minus 011



Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article On Newton-Kantorovich Method for Solving the...

Documents

Transcript of Research Article On Newton-Kantorovich Method for Solving the...