7/18/2019 Existence of solution for a coupled system ofVolterra type integro - differential equations with nonlocal conditions

http://slidepdf.com/reader/full/existence-of-solution-for-a-coupled-system-ofvolterra-type-integro-differential 1/8

ISSN 2347-1921

5033 | P a g e S e p t e m b e r 0 1 , 2 0 1 5

Existence of solution for a coupled system ofVolterra type integro -differential equations with nonlocal conditions

 A. M. A. El-Sayed, A. A. Hilal.Faculty of Science,Alexandria University,Egypt

Faculty of Science,Zagzig University , Egypt

Abstract

In this paper we study the existence of a unique solution for a boundary value problemof a coupled system of Volterratype integro-differential equations under nonlocal conditions.

Keywords: Nonlocal boundary value problems, integro - differential equation;coupled system; Lipschitz condition;

Banach fixed point theorem.

Council for Innovative Research 

Peer Review Research Publishing System

Journal:JOURNAL OF ADVANCES IN MATHEMATICS 

Vol.11, No.3

www.cirjam.com , editorjam@gmail.com

7/18/2019 Existence of solution for a coupled system ofVolterra type integro - differential equations with nonlocal conditions

http://slidepdf.com/reader/full/existence-of-solution-for-a-coupled-system-ofvolterra-type-integro-differential 2/8

ISSN 2347-1921

5034 | P a g e S e p t e m b e r 0 1 , 2 0 1 5

1 INTRODUCTION

The study of value problem with nonlocal conditions is of significance, since they have application in problems in physics,engineering, economics and other areas of applied Mathematics. This feature allows the study of several types of integralequations such as:Fredholm, Volterra, Hammerstein, Urysohn, for different classes of functionals [see(6),(7)]. The mainobject of this paper to study the existence of solution, ∈ [0,1]  and, ∈ [0,1]  for the coupled system of Volterraintegro- differential equations

= ₁ , ,

, ∈ (0,1)

0

 

(1)

= ₂ , ,

, ∈ (0,1)

0

 

with the nonlocal boundary conditions

= , ∈ 0,1, ∈ 0,1 , ≠ 1 (2)

and

= , ∈ 0,1, ∈ 0,1, ≠ 1 (3)

Let

= , and

=  in (1), we obtain

= ₁, , (), ∈ (0,1)

0

 

(4)

= ₂, , (), ∈ (0,1)0

 

where

() = (0) + ()

0 

(5)

() = (0) + ()

0 

(6)

Using the nonlocal boundary condition (2), we obtain

( ) = (0) + ()

0

, 

and

() = (0) + ()

0,

then

(0) =

1 − ()

0

− 1

1 −

0

 

Substituting in (5), we obtain

() =

1− ()

0 − 1

1− 0

 + ()

0

. (7)

 And using the nonlocal boundary condition (3), we obtain

( ) = (0) + ()

0 , 

7/18/2019 Existence of solution for a coupled system ofVolterra type integro - differential equations with nonlocal conditions

http://slidepdf.com/reader/full/existence-of-solution-for-a-coupled-system-ofvolterra-type-integro-differential 3/8

ISSN 2347-1921

5035 | P a g e S e p t e m b e r 0 1 , 2 0 1 5

and

 () = (0) + ()

0

, 

then

(0) =

1 −

(

)

0 −

1

1 −

0 . 

Substituting in (6), we obtain

() =

1− ()

0 − 1

1− 0

 + ()

0

. (8)

3. Existence of a unique continuous solution

Here, we study the existence of a unique continuous solution of the coupled system of integral equations (4) , under thefollowing assumptions:

(1) ∶   [0,1] × [0,1] + →   are continuous, and satisfy the Lipschitz

condition

│ , ,   −   , , │  ≤   , │ −  │ ,   = 1 , 2. 

where

∶   [0,1] × [0,1] → +are integral in (, ).

(2)   (, ) 1

0

  ≤   ,   ∈   [0, 1],   = 1,2.

Let    = = , : ,   ∈   0, 1, and its norm defined as

║, ║  = ║║  + ║║  =  ││  +  ││,   ∈   [0, 1].

Now, for the existence of a unique continuous solution for the coupled system of the integral equations (4) , we have thefollowing theorem.

Theorem 1. Let the assumption (1)-(2) be satisfied. < 1,   = 1, 2  , then the coupled system of integral

equations (4) has a unique solution in  . 

Proof.  Define the operator  associated with the coupled system of integral equations (4) by

(, ) = (₁ , ₂ )

Where

₁  = ₁ , ,  

0

 

₂  = 2, , ,

0

 

Firstly prove that ∶     →   . 

Let

,   ∈  0, 1, ₁, ₂  ∈ [0,1], ₁  < ₂, and │₂_ ₁│ ≤  , now to prove

₁ :  0, 1   →   0, 1, then

│₁(₂) −  ₁(₁)│  = │ f ₁ (t ₂, s, v(s)) ds2

0- f ₁t ₁, s ,vsds │1

0 

=│ f ₁ (t ₂, s, v(s)) ds1

0+ f ₁ (t ₂, s, v(s)) ds

2

1 

-

f 

₁t 

₁, s ,v

s

ds │1

0 

≤│ f ₁ (t ₂, s, v(s)) ds1

0− f ₁t ₁, s ,vsds │

1

0 

7/18/2019 Existence of solution for a coupled system ofVolterra type integro - differential equations with nonlocal conditions

http://slidepdf.com/reader/full/existence-of-solution-for-a-coupled-system-ofvolterra-type-integro-differential 4/8

ISSN 2347-1921

5036 | P a g e S e p t e m b e r 0 1 , 2 0 1 5

+│ f ₁ (t ₂, s, v(s)) ds│2

1 

≤   │₁ ₂, , 1

0− ₁ ₁, , │ 

+   │2

1 f ₁ (t ₂, s,v(s))│. 

This prove that

₁(

)

∶ 

 [0,1]

→ 

 [0,1],

∀(

)

∈ 

 [0, 1].

 As done before, we obtain

₂() ∶   [0, 1] →   [0,1], ∀() ∈   [0,1].

Now since (, ) = (₁, ₂) 

(, )(₂) −  (, )(₁) = ((₂), (₂)) −  ((₁), (₁))

= (₁(₂) −  ₁(₁), ₂(₂) −  ₂(₁)).

Then

║(, )(₂) −  (, )(₁)║  = ║(₁(₂) −  ₁(₁)║  + ║₂(₂) ₂(₁))║. 

Hence

∶ 

  

→ 

 . 

Secondly to prove that  is a contraction, we have following.

Let

  = (, ) ∈   and  ₁  = (₁, ₁) ∈   , we have 

(, ) = (₁(), ₂())

and

(₁, ₁) = (₁₁(), ₂₁()),

then

│ ₁ () −  ₁ ₁() │  = │ ₁ (, , ())  

0

  −   ₁ (, , ₁())

0

 │ 

≤ │f ₁ t,s, s −  f ₁ t,s, ₁s│ ds

0

 

≤   k ₁(t,s) │ (s) − ₁(s) │ ds

0

 

≤ k ₁(t, s) sup  │(s) − ₁(s) │ds

0

 

≤ ║ () −  ₁() ║ k ₁(t,s) ds

0.

Then

║ ₁ () −  ₁ ₁() ║  ≤  ₁ ║  −  ₁║ .

Since ₁  < 1, then ₁ is a contraction.

 As done before, we obtain

│ ₂ () −  ₂₁ () │  ≤  ₂ ║  −  ₁║.

Since ₂  < 1, then₂ is a contraction.

Then

║(, ) −  (₁, ₁) ║  = ║ (₁, ₂) −   (₁₁, ₂₁) ║ 

7/18/2019 Existence of solution for a coupled system ofVolterra type integro - differential equations with nonlocal conditions

http://slidepdf.com/reader/full/existence-of-solution-for-a-coupled-system-ofvolterra-type-integro-differential 5/8

ISSN 2347-1921

5037 | P a g e S e p t e m b e r 0 1 , 2 0 1 5

= ║₁  −  ₁₁, ₂  −  ₂₁║ 

= ║₁  −  ₁₁║  + ║₂  −  ₂₁║ 

≤   ₁ , ₂ ║( , ) −   (₁ , ₁)║.

   =  ₁ , ₂ 

║(, ) −  (₁, ₁) ║  ≤   ║( , ) −   ( ₁, ₁)║.

Since  < 1, then  is a contraction, by using Banach fixed point Theorem[(5)], then there exists a unique solution in X forthe coupled system of the integral equations (4).

4 Solution of the problem (1)-(3)

Consider now the problem (1)-(3).

Theorem 2. Let the assumption of the theorem 1 be satisfied, then there exists a unique solution ,   ∈  [0, 1] of the

problem (1)-(3) . 

Proof.The solution of the problem (1) and (3) is given by

=

1 −

0

− 1

1 −

0

  +

0

  ∈ [0,1],

and

=

1− 0

− 1

1− 0

  + 0

  ∈[0,1].

Where

() = ₁, , , ∈ C0, 1.

0

 

= ₂, , , ∈ C0, 1.

0

 

Then from Theorem 1 we can deduce that there exists a unique continuous solution of the problem (1)-(3).

. Existence of a unique  −solution 

Here, we study the existence of integrable solution of the coupled system of integral equations (4) under the followingassumptions:

() ∶  [0, 1] × [0,1] × + →   are measurable in (, ), and satisfy the Lipschitz condition

│  , , −   , , │  ≤    │   −   │,   = 1,2.

(

)

(

,

 0)

∈ 

₁ [0, 1], and

│   0│  

1

0

  ≤  ,   ∈   0, 1,   = 1,2.

Let   = = (, ) ∶  ,   ∈ ₁ [0,1] , and its norm defined as

║(, )║  = ║║  + ║║  = │()│ 

1

0

  + ││ 

1

0

 

Now, for the existence of integrable solution for the coupled system of the integral equations(4), we have the followingtheorem.

Theorem 3.  Let the assumption (i)-(ii) be satisfied .   < 1,   = 1,2, then the coupledsystem of the integralequations (4) has a unique solution in  .

7/18/2019 Existence of solution for a coupled system ofVolterra type integro - differential equations with nonlocal conditions

http://slidepdf.com/reader/full/existence-of-solution-for-a-coupled-system-ofvolterra-type-integro-differential 6/8

ISSN 2347-1921

5038 | P a g e S e p t e m b e r 0 1 , 2 0 1 5

Proof.  Define the operator  associated with the coupled system of integralequations (4)by

(, ) = (₁ , ₂ ).

Where

₁ = ₁ , ,  

 

₂ = ₂, , .0  

Firstly to prove that ∶    →   , 

now to prove  ₁ ∶ ₁ [0, 1] →   ₁[0,1], then

│ ₁(, , ) │  −  │ ₁(, ,0) │  ≤ │ ₁(, , ) −  ₁(, , 0) │  ≤  ₁ │  │ 

and

│ ₁(, , ) │  ≤  ₁ │  │  + │ ₁(, , 0) │.

Hence

│₁(

)

│  =

│ 

₁(

,

,

(

))

│ ≤

0  

₁ 

│ 

 (

)

│ 

0

  +

│ 

₁(

,

,0)

│.

Integrating both sides with respect to t, we obtain

│  f ₁(t,s,  (s))ds│dt≤

0

1

0

 k ₁ │  (s)│ds dt 

0

1

0

+ │ f ₁(t, s,0) │1

0

 

≤ k ₁ │s│dt +1

0 │ f ₁(t,s,0) │1

0 

≤ k ₁ ║║1 + M₁.

Then║₁║1   ≤  k ₁ ║║1   + ₁.

This proves that  ₁ ∶ ₁ [0, 1] →   ₁[0,1].

 As done before, we obtain

║₂║1   ≤  k ₂ ║║1   + ₂.

This proves that ₂∶   ₁ [0, 1] →   ₁[0,1].

Hence

║ (, ) ║  = ║ ₁, ₂ ║ 

= ║ ₁ ║  + ║ ₂ ║ 

= k ₁ ║║1  + ₁  + k ₂ ║║1   + ₂.

This proves :   →  .

Secondly prove that  is a contraction.

Let   = , ∈    and ₁  = (₁, ₁) ∈   .

Then (, ) = (₁, ₂) and (₁, ₁) = (₁₁, ₂₁)

│ ₁  −  ₁₁│  = │₁, , −  ₁, , ₁│.

0

0

 

Integrating both sides with respect to t, we

│ G₁  −  G₁₁│dt1

0

≤ │ f ₁t,s, s −  f ₁t,s, ₁s

0

│1

0

 

≤ k 

₁│ 

s

− ₁s

│ ds

0

1

0

dt

7/18/2019 Existence of solution for a coupled system ofVolterra type integro - differential equations with nonlocal conditions

http://slidepdf.com/reader/full/existence-of-solution-for-a-coupled-system-ofvolterra-type-integro-differential 7/8

ISSN 2347-1921

5039 | P a g e S e p t e m b e r 0 1 , 2 0 1 5

≤ k ₁ │ s − ₁s│dt1

0

 

≤ ₁ ║   −  ₁ ║₁ .

Then

║G₁  −  G₁₁║1 ≤  ₁║   −  ₁ ║₁ .

Since₁  < 1, then ₁ is a contraction.

 As done before, we obtain

║G₂  −  G₂₁║1 ≤  ₂║   −  ₁ ║₁ .

Since ₂  < 1, then ₂ is a contraction.

Hence

║(, ) −  (₁, ₁)║  = ║(₁, ₂) −   (₁₁, ₂₁)║ 

= ║

₁ 

− 

₁₁,

₂ 

− 

₂₁║ 

≤  ₁ , ₂║, − (₁, ₁)║1.  

Let   =  ₁ , ₂.

Then

║, −  (₁, ₁)║1 ≤ ║, − (₁, ₁)║1.  

Since < 1, then  s a contraction, by using Banach fixed point Theorem [(5)], then there exists of solution in   for thecoupled system of the integral equations (4).

6. Solution of the problem (1)-(3)

Consider now the problem (1)-(3).

Theorem 4. Let the assumption of the theorem 3 be satisfied, then there exists a uniquesolution ,   ∈   [0, 1] of the

boundary value problem (1)-(3).

Proof. The solution of the problem (1) - (3) is given by

=

1− 0

− 1

1− 0

  + 0

∈ [0,1] ,

and

=

1 −

0

− 1

1 −

0

  +

0

  ∈ 0,1.

Where

() =  ₁, ,   ∈ ₁ [0, 1]1

0

 

(t) = ₂, ,   ∈ ₁ [0, 1] .1

0

 

Then from Theorem 3 we can deduce that there exists a unique solution of the problem (1) - (3).

References

[1] D. ORegan, M.Meehan, Existence theory for nonlinear integral and integro-differential equations, Kluwer Acad. Pulbel.Dordrecht, 1998.

[2] A. M. A. El-Sayed and E. O. Bin-Taher, An arbitraty fractional order differential equation with internal nonlocal andintegral conditions, Vol.1, No.3, pp. 59-62, (2011).

7/18/2019 Existence of solution for a coupled system ofVolterra type integro - differential equations with nonlocal conditions

http://slidepdf.com/reader/full/existence-of-solution-for-a-coupled-system-ofvolterra-type-integro-differential 8/8

ISSN 2347-1921

5040 | P a g e S e p t e m b e r 0 1 , 2 0 1 5

[3] A. M. A. El-Sayed and E. O. Bin-Taher, A nonlocal problem for a multi-term frac- tional order differential equation,Journal of Math Analysis, Vol. 5, No.29, PP.1445-1451, (2011).

[4] A. M. A. El-Sayed and E. O. Bin-Taher, a multi-term fractional- order differential equation with nonlocal condition,Egy.Chin.J Comp.App.Math. , Vol. 1, No.1, PP.54-60, (2012).

[5] Goebel, K. and Kirk W. A., Topics in Metric Fixed point theory, Cambridge University Press, Cambridge (1990)

[6] Ibrahim Abouelfarag Ibrahim, on the existence of solution of functional integral equation of Urysohn type, Computersand Mathematics with Applications, 1609-1614, 57, (2009).

[7] J. Banas, Integrable solutions of Hammerstein and Urysohn integral equation, J. Austral. Math. Soc (Series A), pp.61-68, 46, (1989).

[8]R.F.Apolaya, H. R. Clark and A.J. Feitosa, on a nonlinear coupled system with internal damping, nonlinear, taxas stateuniversity, Vol., No. 64, pp.1-17,(2000).