Ulam - Hyers Stability of a 2- Variable AC - Mixed Type Functional Equation

Journal of Modern Mathematics Frontier Sept. 2012, Vol. 1 Iss. 3, PP. 10-26

10

Ulam - Hyers Stability of a 2- Variable AC - Mixed

Type Functional Equation: Direct and Fixed Point

Methods M. Arunkumar

1, Matina J. Rassias

2, Yanhui Zhang

3

1Department of Mathematics, Government Arts College, Tiruvannamalai - 606 603, TamilNadu, India 2Department of Statistical Science, University College London, 1-19 Torrington Place, #140, London, WC1E 7HB, UK

3Department of Mathematics, Beijing Technology and Business University, China [email protected]; [email protected]; [email protected]

Abstract- In this paper, we obtain the general solution and

generalized Ulam - Hyers stability of a 2 variable AC mixed

type functional equation

2 , 2 2 , 2

4 , , 6 ,

f x y z w f x y z w

f x y z w f x y z w f y w

using direct and fixed point methods.

Keywords- Additive Functional Equations; Cubic

Functional Equation; Mixed Type AC Functional Equation;

Ulam – Hyers Stability

I. INTRODUCTION

The study of stability problems for functional

equations is related to a question of Ulam [30] concerning

the stability of group homomorphisms and affirmatively

answered for Banach spaces by Hyers [9]. It was further

generalized and excellent results obtained by number of

authors [2], [6], [16], [20], [23].

Over the last six or seven deades, the above problem

was tackled by numerous authors and its solutions via

various forms of functional equations like additive,

quadratic, cubic, quartic, mixed type functional equations

which involve only these types of functional equations

were discussed. We refer the interested readers for more

information on such problems to the monographs [1], [5],

[8], [10], [12], [13], [15], [17], [19], [21], [24], [25], [26],

[27], [28], [29], [31], [32] and [33].

In 2006, K.W. Jun and H.M. Kim [11] introduced the

following generalized additive and quadratic type of

functional equation

1 1 1

2n n n

i i i j

i i i j n

f x n f x f x x

(1.1)

in the class of function between real vector spaces. The general solution and Ulam stability of mixed type additive

and cubic functional equation of the form

3 ( ) ( ) ( )

( ) 4[ ( ) ( ) ( )]

f x y z f x y z f x y z

f x y z f x f y f z

4[ ( ) ( ) ( )]f x y f x z f y z (1.2)

was introduced by J.M. Rassias [18]. The stability of

generalized mixed type functional equation of the form

( ) ( )f x ky f x ky

2 2[ ( ) ( )] 2(1 ) ( )k f x y f x y k f x (1.3)

for fixed integers k and 0, 1k in quasi-Banach spaces

was introduced by M. Eshaghi Gordji and H. Khodaie [7].

The mixed type (I.3) is having the property additive,

quadratic and cubic.

J.H. Bae and W.G. Park proved the general solution of

the 2- variable quadratic functional equation

( , ) ( , )f x y z w f x y z w

2 ( , ) 2 ( , )f x z f y w (1.4)

and investigated the generalized Hyers-Ulam-Rassias

stability of (I.4). The above functional equation has

solution

2 2( , )f x y ax bxy cy (1.5)

The stability of the functional equation (I.4) in fuzzy

normed space was investigated by M. Arunkumar et., al

[3]. Using the ideas in [3], the general solution and generalized Hyers-Ulam- Rassias stability of a 3- variable

quadratic functional equation

( , , ) ( , , )f x y z w u v f x y z w u v

2 ( , , ) 2 ( , , )f x z u f y w v (1.6)

was discussed by K. Ravi and M. Arunkumar [22]. The

solution of the (I.6) is of the form

2 2 2( , , )f x y z ax by cz dxy eyz fzx (1.7)

In this paper, we obtain the general solution and

generalized Ulam - Hyers stability of a 2 variable AC

mixed type functional equation

2 ,2 2 ,2f x y z w f x y z w

4 , , 6 ,f x y z w f x y z w f y w (1.8)

having solutions

( , )f x y ax by (1.9)

and

3 2 2 3( , )f x y ax bx y cxy dy (1.10)


11

In Section 2, we present the general solution of the

(I.8). The generalized Ulam-Hyers stability using direct

and fixed point method are discussed in Section 3 and

Section 4, respectively.

II. GENERAL SOLUTION

In this section, we present the solution of the (I.8).

Through out this section let U and V be real vector

spaces.

Lemma 2.1: If 2:f U V is a mapping satisfying (I.8)

and let 2:g U V be a mapping given by

( , ) (2 ,2 ) 8 ( , )g x x f x x f x x (II.1)

for all x U then

(2 ,2 ) 2 ( , )g x x g x x (II.2)

for all x U such that g is additive.

Proof: Letting ( , , , )x y z w by (0,0,0,0) in (I.8), we get

(0,0) 0.f (II.3)

Setting ( , , , )x y z w by (0, ,0, )y y in (I.8), we obtain

( , ) ( , )f y y f y y (II.4)

for all y U . Replacing ( , , , )x y z w by ( , , , )x x x x in (I.8),

we arrive

(3 ,3 ) 4 (2 ,2 ) 5 ( , )f x x f x x f x x (II.5)

for all x U . Again replacing ( , , , )x y z w by ( ,2 , ,2 )x x x x in

(I.8) and using (II.3), (II.4) and (II.5), we have

(4 ,4 ) 10 (3 ,3 ) 16 ( , )f x x f x x f x x (II.6)

for all x U . From (II.1), we establish

(2 ,2 ) 2 ( , ) (4 ,4 ) 10 (2 ,2 ) 16 ( , )g x x g x x f x x f x x f x x (II.7)

for all x U . Using (II.6) in (II.7), we desired our result.

Lemma 2.2: If 2:f U V be a mapping satisfying (I.8)

and let 2:h U V be a mapping given by

( , ) (2 ,2 ) 2 ( , )h x x f x x f x x (II.8)

for all x U then

(2 ,2 ) 8 ( , )h x x h x x (II.9)

for all x U such that h is cubic.

Proof: It follows from (II.8) that

(2 ,2 ) 8 ( , ) (4 ,4 ) 10 (2 ,2 ) 16 ( , )h x x g x x f x x f x x f x x

(II.10)

for all x U . Using (II.6) in (II.10), we desired our result.

Remark 2.3: If 2:f U V be a mapping satisfying (I.8)

and let 2, :g h U V be a mapping defined in (II.1) and

(II.8) then

1

( , ) ( , ) ( , )6

f x x h x x g x x (II.11)

for all x U .

Lemma 2.4: If 2:f U V is a mapping satisfying (I.8)

and let :t U V be a mapping given by

( ) ( , )t x f x x (II.12)

for all x U , then t satisfies

(2 ) (2 ) 4[ ( ) ( )] 6 ( )t x y t x y t x y t x y t y

(II.13)

for all ,x y U .

Proof: From (I.8) and (II.12), we get

(2 ) (2 )

(2 ,2 ) (2 ,2 )

4[ ( , ) ( , )] 6 ( , )

4[ ( ) ( )] 6 ( )

t x y t x y

f x y x y f x y x y

f x y x y f x y x y f y y

t x y t x y t y

for all ,x y U .

III. STABILITY RESULTS: DIRECT METHOD

In this section, we investigate the generalized Ulam-

Hyers stability of (I.8) using direct method.

Through out this section let U be a normed space and

V be a Banach space. Define a mapping 2:F U V by

( , , , ) 2 ,2 2 ,2F x y z w f x y z w f x y z w

4 , , 6 ,f x y z w f x y z w f y w

for all , , ,x y z w U .

Theorem 3.1: Let 1j . Let 2:f U V be a mapping for

which there exist a function 4: (0, ]U with the

condition

1lim (2 ,2 ,2 ,2 ) 0

2

nj nj nj nj

njnx y z w

(III.1)

such that the functional inequality

( , , , ) ( , , , )F x y z w x y z w (III.2)

for all , , ,x y z w U . Then there exists a unique 2-variable

additive mapping 2:A U V satisfying (I.8) and

1

2

1(2 ,2 ) 8 ( , ) ( , ) (2 )

2

kj

jk

f x x f x x A x x x

(III.3)

for all x U . The mapping (2 )kj x and ( , )A x x are

defined by

( 1) ( 1)

( 1) ( 1)

(2 ) 4 (2 ,2 ,2 ,2 ) (2 ,2 ,2 ,2 )

1( , ) lim (2 ,2 ) 8 (2 ,2 )

2

kj kj kj kj kj kj k j kj k j

n j n j nj nj

njn

x x x x x x x x x

A x x f x x f x x

(III.4)

for all x U .

Proof: Assume 1j . Letting ( , , , )x y z w by ( , , , )x x x x in

(III.2), we obtain

(3 ,3 ) 4 (2 ,2 ) 5 ( , ) ( , , , )f x x f x x f x x x x x x (III.5)

for all x U . Replacing ( , , , )x y z w by ( ,2 , ,2 )x x x x in

(III.2), we get


12

(4 ,4 ) 4 (3 ,3 ) 6 (2 ,2 ) 4 ( , )f x x f x x f x x f x x

( ,2 , ,2 )x x x x (III.6)

for all x U . Now, from (III.6) and (III.7), we have

(4 ,4 ) 10 (2 ,2 ) 16 ( , )

4 (3 ,3 ) 4 (2 ,2 ) 5 ( , )

(4 ,4 ) 4 (3 ,3 ) 6 (2 ,2 ) 4 ( , )

f x x f x x f x x

f x x f x x f x x

f x x f x x f x x f x x

4 ( , , , ) ( ,2 , ,2 )x x x x x x x x (III.7)

for all x U . From (III.8), we arrive

(4 ,4 ) 10 (2 ,2 ) 16 ( , ) ( )f x x f x x f x x x (III.8)

where

( ) 4 ( , , , ) ( ,2 , ,2 )x x x x x x x x x (III.9)

for all x U . It is easy to see from (III.9) that

(4 ,4 ) 8 (2 ,2 ) 2( (2 ,2 ) 8 ( ; )) ( )f x x f x x f x x f x x x (III.10)

for all x U . Using (II.1) in (III.11), we obtain

(2 ,2 ) 2 ( , ) ( )g x x g x x x (III.11)


(2 ,2 ) ( )( , )

2 2

g x x xg x x

(III.12)

for all x U . Now replacing x by 2x and dividing by 2 in (III.13), we get

2 2

2 2

(2 ,2 ) ( , ) (2 )

2 2 2

g x x g x x x (III.13)

for all x U . From (III.13) and (III.14), we obtain

2 2

2

2 2

2

(2 ,2 )( , )

2

(2 ,2 ) (2 ,2 ) (2 ,2 )( , )

2 2 2

g x xg x x

g x x g x x g x xg x x

1 (2 )

( )2 2

xx

(III.14)

for all x U . Proceeding further and using induction on a positive integer n , we get

1

0

(2 ,2 ) 1 (2 )( , )

2 2 2

n n kn

nk

g x x xg x x

(III.15)

0

1 (2 )

2 2

k

k

x

for all x U . In order to prove the convergence of the sequence

(2 ,2 )

2

n n

n

g x x

,

replacing x by 2m x and dividing by 2m in (III.16), for any m, n > 0 , we deduce

0

(2 ,2 ) (2 ,2 )

2 2

1 (2 2 ,2 2 )(2 ,2 )

2 2

1 (2 )0

2 2

n m n m m m

n m m

n m n mm m

m n

k

k mk

g x x g x x

g x xg x x

xas m

for all x U . This shows that the sequence (2 ,2 )

2

n n

n

g x x

is Cauchy sequence. Since V is complete, there exists a

mapping 2( , ) :A x x U V such that

(2 ,2 )( , ) lim

2

n n

nn

g x xA x x x U

.

Letting n in (III.16) and using (II.1), we see that

(III.3) holds for all x U . To show that A satisfies (I.8),

replacing ( , , , )x y z w by (2 ,2 ,2 ,2 )n n n nx y z w and dividing by

2n in (III.2), we obtain

1 1(2 ,2 ,2 ,2 ) (2 ,2 ,2 ,2 )

2 2

n n n n n n n n

n nF x y z w x y z w

for all , , ,x y z w U . Letting n in the above inequality

and using the definition of ( , )A x x , we see that

2 , 2 2 , 2

4 , , 6 ,

A x y z w A x y z w

A x y z w A x y z w A y w

Hence A satisfies (I.8) for all , , ,x y z w U . To prove A is

unique 2-variable additive function satisfying (I.8), we let

B(x,x) be another 2-variable additive mapping satisfying

(I.8) and (III.3), then

1 1

1 1

0

( , ) ( , )

1(2 ,2 ) (2 ,2 )

2

1(2 ,2 ) (2 ,2 ) 8 (2 ,2 )

2

(2 ,2 ) 8 (2 ,2 ) (2 ,2 )

(2 )

2

0

n n n n

n

n n n n n n

n

n n n n n n

k n

k nk

A x x B x x

A x x B x x

A x x f x x f x x

f x x f x x B x x

x

as n

for all n . Hence A is unique.

For 1j , we can prove a similar stability result. This

completes the proof of the theorem.

The following Corollary is an immediate consequence

of Theorem 3.1 concerning the stability of (I.8).

Corollary 3.2: Let 2:F U V be a mapping and there

exists real numbers and s such that

( , , , )F x y z w

4 4 4 4

, 1 1;

1 1, ;

4 4

,

1 1 ;

4 4

s s s s

s s s s

s s s s s s s s

x y z w s or s

x y z w s or s

x y z w x y z w

s or s

(III.17)

for all , , ,x y z w U , then there exists a unique 2- variable

additive function 2:A U V such that


13

(2 ,2 ) 8 ( , ) ( , )f x x f x x A x x

1

42

4

2 44

4 2

5

18 2

2 2

4 2

2 2

22 2 2 2

2 2 2 2

ss

s

ss

s

s ss

s s

x

x

x

(III.18)

for all x U .

Now we will provide an example to illustrate that (I.8) is

not stable for s = 1 in (ii) of Corollary 3.2.

Example 3.3: Let : R R be a function defined by

, | | 1( )

,

x if xx

otherwise

where 0 is a constant, and define a function 2:f R R by

0

(2 )( , ) .

2

n

nn

xf x x for all x R

Then F satisfies the functional inequality

( , , , ) 32 (| | | | | | | |)F x y z w x y z w (III.19)

for all , , ,x y z w R . Then there do not exist a additive

mapping 2:A R R and a constant 0 such that

(2 ,2 ) 8 ( , ) ( , ) | | .f x x f x x A x x x for all x R (III.20)

Proof: Now

0 0

(2 )( , ) 2

2 2

n

n nn n

xf x x

Therefore we see that f is bounded. We are going to prove

that f satisfies (III.19).

If x = y = z = w = 0 then (III.19) is trivial. If

1| | | | | | | |

2x y z w , then the left hand side of (III.19)

is less than 32 . Now suppose that

10 | | | | | | | |

2x y z w . Then there exists a positive

integer k such that

1

1 1| | | | | | | |

2 2k kx y z w

(III.21)

so that

1 1 1 11 1 1 12 ,2 ,2 ,2

2 2 2 2

k k k kx y z w

and consequently

1 1 1

1 1

2 ( , ),2 ( , ),2 ( , ),

2 (2 ,2 ),2 (2 ,2 ), ( 1,1).

k k k

k k

y w x y z w x y z w

x y z w x y z w

Therefore for each 0,1,........ 1n k , we have

2 ( , ),2 ( , ),2 ( , ),

2 (2 ,2 ),2 (2 ,2 ), ( 1,1).

n n n

n n

y w x y z w x y z w

x y z w x y z w

and

(2 (2 ,2 )) (2 (2 ,2 ))

4[ (2 ( , )) (2 ( , ))] 6 (2 ( , )) 0

n n

n n n

x y z w x y z w

x y z w x y z w y w

For 0,1,........ 1n k . From the definition of f and (III.21),

we obtain that

0

( , , , )

1(2 (2 ,2 )) (2 (2 ,2 ))

2

4[ (2 ( , )) (2 ( , ))] 6 (2 ( , ))

1(2 (2 ,2 )) (2 (2 ,2 ))

2

4[ (2 ( , )) (2 ( , ))] 6 (2 ( , ))

116 1

2

n n

nn

n n n

n n

nn k

n n n

nn k

F x y z w

x y z w x y z w

x y z w x y z w y w

x y z w x y z w

x y z w x y z w y w

2

6 32 (| | | | | | | |)2k

x y z w

Thus f satisfies (III.19) for all , , ,x y z w R with

10 | | | | | | | |

2x y z w

We claim that the additive (I.8) is not stable for s = 1 in (ii)

Corollary 3.2. Suppose on the contrary that there exist a

additive mapping 2:A R R and a constant 0

satisfying (III.20). Since f is bounded and continuous for

all x R , A is bounded on any open interval containing the

origin and continuous at the origin. In view of Theorem

3.1. A must have the form ( , ) A x x cx for any x in R .

Thus we obtain that

(2 ,2 ) 8 ( , ) ( | |) | |, f x x f x x c x (III.22)

But we can choose a positive integer m with | |m c .

If 1

10,

2mx

, then 2 (0,1)n x for all 0, 1,..... 1.n m

For this x, we get

1

0 0

(2 ) (2 )(2 ,2 ) 8 ( , )

2 2

( | |)

n nm

n nn n

x xf x x f x x

m x c x

which contradicts (III.22). Therefore (I.8) is not stable in

sense of Ulam, Hyers and Rassias if s = 1, assumed in (ii)

of (III.17).

A counter example to illustrate the non stability in (iii)

of Corollary 3.2 is given in the following example.

Example 3.4: Let s be such that1

04

s . Then there is a

function 2:F R R and a constant 0 satisfying

1 3

4 4 4 4( , , , ) | | | | | | | |s s s s

F x y z w x y z w

(III.23)

for all , , ,x y z w R and

0

(2 ,2 ) 8 ( , ) ( , )supx

f x x f x x A x x

x

(III.24)


14

for every additive mapping 2( , ) :A x x R R .

Proof: If we take

( , )ln( , ), 0 ( , )

0, 0

x x x x if xf x x

if x

Then from (III.24), it follows that

0

0

0

0

(2 ,2 ) 8 ( , ) ( , )sup

(2 ,2 ) 8 ( , ) ( , )sup

(2,2)ln | 2 ,2 | 8 (1,1)ln | , | (1,1)sup

sup (2,2)ln | 2 ,2 | 8(1,1)ln | , | (1,1)

x

nn

nn

nn

f x x f x x A x x

x

f n n f n n A n n

n

n n n n n n nA

n

n n n n A

We have to prove (III.23) is true.

Case (i): If , , , 0x y z w in (III.23) then,

(2 ,2 ) (2 ,2 ) 4 ( , )

4 ( , ) 6 ( , )

(2 ,2 )ln | 2 ,2 |

(2 ,2 )ln | 2 ,2 |

4( , ) ln | , |

4( , ) ln | , | 6( , ) ln | , |

f x y z w f x y z w f x y z w

f x y z w f y w

x y z w x y z w

x y z w x y z w

x y z w x y z w

x y z w x y z w y w y w

Set 1 2 3 4, , ,x v y v z v w v it follows that

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4 2 4 2

(2 ,2 ) (2 ,2 ) 4 ( , )

4 ( , ) 6 ( , )

(2 ,2 )ln | 2 ,2 |

(2 ,2 )ln | 2 ,2 |

4( , ) ln | , |

4( , ) ln | , | 6( , ) ln | ,


f x y z w f y w

v v v v v v v v

v v v v v v v v

v v v v v v v v

v v v v v v v v v v v v

4

1 2 3 4 1 2 3 4

1 2 3 4

1 2 3 4 2 4

|

(2 ,2 ) (2 ,2 )

4 ( , )

4 ( , ) 6 ( , )

f v v v v f v v v v

f v v v v

f v v v v f v v

1 3

4 4 4 41 2 3 4

1 3

4 4 4 4

| | | | | | | |

| | | | | | | |

s s s s

s s s s

v v v v

x y z w

Case (ii): If , , , 0x y z w in (III.23) then,

(2 ,2 ) (2 ,2 ) 4 ( , )

4 ( , ) 6 ( , )

(2 ,2 )ln | 2 ,2 |

(2 ,2 )ln | 2 ,2 |

4( , ) ln | , |

4( , ) ln | , | 6( , ) ln | , |


f x y z w f y w

x y z w x y z w

x y z w x y z w

x y z w x y z w



1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4

(2 ,2 ) (2 ,2 ) 4 ( , )

4 ( , ) 6 ( , )

( 2 , 2 ) ln | 2 , 2 |

( 2 , 2 ) ln | 2 , 2 |

4( , ) ln | , |

4( , ) ln | , |


f x y z w f y w

v v v v v v v v

v v v v v v v v

v v v v v v v v

v v v v v v v v

2 4 2 4

1 2 3 4 1 2 3 4

1 2 3 4

1 2 3 4 2 4

1 3

4 4 4 41 2 3 4

1 3

4 4 4 4

6( , ) ln | , |

( 2 , 2 ) ( 2 , 2 )

4 ( , )

4 ( , ) 6 ( , )

| | | | | | | |

| | | | | | | |

s s s s

s s s s

v v v v

f v v v v f v v v v

f v v v v

f v v v v f v v

v v v v

x y z w

Case (iii): If , 0, , 0x z y w then 2 ,x y

2 , , 0,2 ,2 , , 0z w x y z w x y z w x y z w in (III.23)

then,

(2 ,2 ) (2 ,2 ) 4 ( , )

4 ( , ) 6 ( , )

(2 ,2 )ln | 2 ,2 |

(2 ,2 )ln | 2 ,2 |

4( , ) ln | , |

4( , ) ln | , | 6( , ) ln | , |


f x y z w f y w

x y z w x y z w

x y z w x y z w

x y z w x y z w



1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4 2 4

(2 ,2 ) (2 ,2 ) 4 ( , )

4 ( , ) 6 ( , )

(2 ,2 ) ln | 2 ,2 |

(2 ,2 ) ln | (2 ,2 ) |

4( , ) ln | , |

4( , ) ln | ( , ) | 6( ,


f x y z w f y w

v v v v v v v v

v v v v v v v v

v v v v v v v v

v v v v v v v v v v

2 4) ln | , |v v

1 2 3 4 1 2 3 4

1 2 3 4

1 2 3 4 2 4

1 3

4 4 4 41 2 3 4

1 3

4 4 4 4

(2 ,2 ) (2 ,2 )

4 ( , )

4 ( , ) 6 ( , )

| | | | | | | |

| | | | | | | |

s s s s

s s s s

f v v v v f v v v v

f v v v v

f v v v v f v v

v v v v

x y z w

Case (iv): If , 0, , 0x z y w then 2 ,x y


then,

(2 ,2 ) (2 ,2 ) 4 ( , )

4 ( , ) 6 ( , )

(2 ,2 )ln | 2 ,2 |

(2 ,2 )ln | 2 ,2 |

4( , ) ln | , |

4( , ) ln | , | 6( , ) ln | , |


f x y z w f y w

x y z w x y z w

x y z w x y z w

x y z w x y z w




15

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4 2 4

(2 ,2 ) (2 ,2 ) 4 ( , )

4 ( , ) 6 ( , )

(2 ,2 ) ln | (2 ,2 ) |

(2 ,2 ) ln | 2 ,2 |

4( , ) ln | ( , ) |

4( , ) ln | , | 6( ,


f x y z w f y w

v v v v v v v v

v v v v v v v v

v v v v v v v v

v v v v v v v v v v

2 4

1 2 3 4 1 2 3 4

1 2 3 4

1 2 3 4 2 4

1 3

4 4 4 41 2 3 4

1 3

4 4 4 4

) ln | , |

(2 ,2 ) (2 ,2 )

4 ( , )

4 ( , ) 6 ( , )

| | | | | | | |

| | | | | | | |

s s s s

s s s s

v v

f v v v v f v v v v

f v v v v

f v v v v f v v

v v v v

x y z w

Case (v): If 0x y z w in (III.23) then it is trivial.

Now we will provide an example to illustrate that the

(I.8) is not stable for 1

4s in (iv) of Corollary 3.2.


1, | |

4( )

,4

x if x

x

otherwise

where 0 is a constant, and define a function 2:f E R by

0

(2 )( , ) .

2

n

nn

xf x x for all x R


1 1 1 1

4 4 4 4

( , , , )

8 | | | | | | | | | | | | | | | |

F x y z w

x y z w x y z w

(III.25)

for all , , ,x y z w R . Then there do not exist a additive

mapping 2:A R R and a constant 0 such that

(2 ,2 ) 8 ( , ) ( , ) | | .f x x f x x A x x x for all x R (III.26)

Proof: Now

0 0

(2 ) 1( , )

4 22 2

n

n nn n

xf x x

.

Therefore we see that f is bounded. We are going to

prove that f satisfies (III.25).

If x = y = z = w = 0 then (III.25) is trivial.

If 1 1 1 1

4 4 4 41

| | | | | | | | | | | | | | | |2

x y z w x y z w , then the left

hand side of (III.25) is less than 8 . Now suppose that

1 1 1 1

4 4 4 41

0 | | | | | | | | | | | | | | | |2

x y z w x y z w .

Then there exists a positive integer k such that

1 1 1 1

4 4 4 41

1 1| | | | | | | | | | | | | | | |

2 2k kx y z w x y z w

(III.27)

so that

1 1 1 1

1 1 14 4 4 41 1 1

2 | | | | | | | | ,2 ,2 ,2 2 2

k k kx y z w x y

1 11 12 ,2

2 2

k kz w

and consequently

1 1 1

1 1

2 ( , ),2 ( , ),2 ( , ),

1 12 (2 ,2 ),2 (2 ,2 ), , .

4 4

k k k

k k

y w x y z w x y z w

x y z w x y z w


2 ( , ),2 ( , ),2 ( , ),

1 12 (2 ,2 ),2 (2 ,2 ) , .

4 4

n n n

n n

y w x y z w x y z w

x y z w x y z w

and

(2 (2 ,2 )) (2 (2 ,2 ))

4[ (2 ( , )) (2 ( , ))] 6 (2 ( , )) 0

n n

n n n

x y z w x y z w

x y z w x y z w y w


we obtain that

0

( , , , )

1(2 (2 ,2 )) (2 (2 ,2 ))

2

4[ (2 ( , )) (2 ( , ))] 6 (2 ( , ))

1(2 (2 ,2 )) (2 (2 ,2 ))

2

4[ (2 ( , )) (2 ( , ))] 6 (2 ( , ))

1 16

2 4

n n

nn

n n n

n n

nn k

n n n

nn k

F x y z w

x y z w x y z w

x y z w x y z w y w

x y z w x y z w

x y z w x y z w y w

1 1 1 1

4 4 4 4

24

2

8 | | | | | | | | | | | | | | | |

k

x y z w x y z w

Thus f satisfies (III.25) for all , , ,x y z w with

1 1 1 1

4 4 4 41

0 | | | | | | | | | | | | | | | |2

x y z w x y z w

We claim that (I.8) is not stable for 1

4s in (iii) Corollary

3.2. Suppose on the contrary that there exist a additive

mapping 2:A R R and a constant 0 satisfying

(III.26). Since f is bounded and continuous for all x R , A

is bounded on any open interval containing the origin and

continuous at the origin. In view of Theorem 3.1, A must

have the form ( , ) A x x cx for any x in R . Thus we obtain

that

(2 ,2 ) 8 ( , ) ( | |) | |, f x x f x x c x (III.28)


If1

10,

2mx

, then 2 (0,1)n x for all

0, 1,..... 1.n m For this x, we get


16

1

0 0

(2 ) (2 )(2 ,2 ) 8 ( , )

2 2

( | |)

n nm

n nn n

x xf x x f x x

m x c x


sense of Ulam, Hyers and Rassias if 1

4s , assumed in (iv)

of (III.17).

Theorem 3.6: Let 1j . Let 2:f U V be a mapping for

which there exist a function 4: (0, ]U with the

condition

1lim (2 ,2 ,2 ,2 ) 0

8

nj nj nj nj

njnx y z w

(III.29)

such that the functional inequality

( , , , ) ( , , , )F x y z w x y z w (III.30)


cubic mapping 2:C U V satisfying (I.8) and

1

2

1(2 ,2 ) 2 ( , ) ( , ) (2 )

8

kj

jk

f x x f x x C x x x

(III.31)

for all x U . The mapping (2 )kj x and ( , )C x x are

defined by

( 1) ( 1)

( 1) ( 1)

(2 ) 4 (2 ,2 ,2 ,2 ) (2 ,2 ,2 ,2 )

1( , ) lim (2 ,2 ) 2 (2 ,2 )

8

kj kj kj kj kj kj k j kj k j

n j n j nj nj

njn

x x x x x x x x x

C x x f x x f x x

(III.32)

for all x U .

Proof: It is easy to see from (III.9) that

(4 ,4 ) 2 (2 ,2 ) 8( (2 ,2 ) 2 ( ; )) ( )f x x f x x f x x f x x x

(III.34)

for all x U . Using (II.8) in (III.34), we obtain

(2 ,2 ) 8 ( , ) ( )h x x h x x x (III.35)


(2 ,2 ) ( )( , )

8 8

h x x xh x x

(III.36)

for all x U . Now replacing x by 2x and dividing by 8

in (III.36), we get

2 2

2 2

(2 ,2 ) ( , ) (2 )

8 8 8

h x x h x x x (III.37)

for all x U . From (III.36) and (III.37), we obtain

2 2

2

2 2

2

(2 ,2 )( , )

8

(2 ,2 ) (2 ,2 ) (2 ,2 )( , )

8 8 8

h x xh x x

h x x h x x h x xh x x

1 (2 )( )

8 8

xx

(III.38)

for all x U . Proceeding further and using induction on a

positive integer n , we get

1

0

(2 ,2 ) 1 (2 )( , )

8 8 8

n n kn

n kk

h x x xh x x

0

1 (2 )

8 8

k

kk

x

(III.39)

for all x U . In order to prove the convergence of the

sequence

(2 ,2 )

8

n n

n

h x x

,

replacing x by 2m x and dividing by 8m in (III.39), for any

m, n > 0 , we deduce

0

(2 ,2 ) (2 ,2 )

8 8

1 (2 2 ,2 2 )(2 ,2 )

8 8

1 (2 )

8 8

0

n m n m m m

n m m

n m n mm m

m n

k

k mk

h x x h x x

h x xh x x

x

as m

for all x U . This shows that the sequence (2 ,2 )

8

n n

n

h x x

is Cauchy sequence. Since V is complete, there exists a

mapping 2( , ) :C x x U V such that

(2 ,2 )( , ) lim

2

n n

nn

h x xC x x x U

.

Letting n in (III.39) and using (II.8), we see that

(III.31) holds for all x U . To show that C satisfies (I.8)

and it is unique the proof is similar to that of Theorem 3.1

The following Corollary is an immediate consequence

of Theorem 3.6 concerning the stability of (I.8).



( , , , )F x y z w

4 4 4 4

, 3 3;

3 3, ;

4 4

,

3 3 ;

4 4

s s s s

s s s s

s s s s

s s s s

x y z w s or s

x y z w s or s

x y z w

x y z w

s or s

(III.40)


cubic function 2:C U V such that

(2 ,2 ) 2 ( , ) ( , )f x x f x x C x x


17

1

42

4

2 44

4 2

5 / 7

18 2

7 8 2

4 2

7 8 2

22 2 2 2

7 8 2 7 8 2

ss

s

ss

s

s ss

s s

x

x

x

(III.41)

for all x U .

Now we will provide an example to illustrate that (I.8)

is not stable for s = 3 in (ii) of Corollary 3.6.


3, | | 1( )

,

x if xx

otherwise


0

(2 )( , ) .

8

n

nn

xf x x for all x R


33 3 3 316 8

( , , , ) (| | | | | | | | )7

F x y z w x y z w

(III.42)

for all , , ,x y z w R . Then there do not exist a cubic

mapping 2:C R R and a constant 0 such that

3(2 ,2 ) 2 ( , ) ( , ) | | .f x x f x x C x x x for all x R (III.43)

Proof: Now

0 0

(2 ) 8( , )

78 8

n

n nn n

xf x x



If x = y = z = w = 0, then (III.42) is trivial. If

3 3 3 3 1| | | | | | | |

8x y z w , then the left hand side of (III.42)

is less than 16 8

7

. Now suppose that

3 3 3 3 10 | | | | | | | |

8x y z w .


3 3 3 3

2 1

1 1| | | | | | | |

8 8k kx y z w

(III.44)

so that

1 3 1 3 1 3 1 31 1 1 12 ,2 ,2 ,2

8 8 8 8

k k k kx y z w

and consequently

1 1 1

1 1

2 ( , ),2 ( , ),2 ( , ),

1 12 (2 ,2 ),2 (2 ,2 ) , .

2 2

k k k

k k

y w x y z w x y z w

x y z w x y z w


2 ( , ),2 ( , ),2 ( , ),

1 12 (2 ,2 ),2 (2 ,2 ) , .

2 2

n n n

n n

y w x y z w x y z w

x y z w x y z w

and

(2 (2 ,2 )) (2 (2 ,2 ))

4[ (2 ( , )) (2 ( , ))] 6 (2 ( , )) 0

n n

n n n

x y z w x y z w

x y z w x y z w y w


we obtain that

0

( , , , )

1(2 (2 ,2 )) (2 (2 ,2 ))

2

4[ (2 ( , )) (2 ( , ))] 6 (2 ( , ))

1(2 (2 ,2 )) (2 (2 ,2 ))

2

4[ (2 ( , )) (2 ( , ))] 6 (2 ( , ))

1 116

8

n n

nn

n n n

n n

nn k

n n n

nn k

F x y z w

x y z w x y z w

x y z w x y z w y w

x y z w x y z w

x y z w x y z w y w

3

3 3 3 3

6 8 1

7 8

16 8(| | | | | | | | )

7

k

x y z w


3 3 3 3 10 | | | | | | | |

8x y z w

We claim that (I.8) is not stable for s =3 in (ii)

Corollary 3.7. Suppose on the contrary that there exist a

cubic mapping 2:C R R and a constant 0 satisfying

(III.43). Since f is bounded and continuous for all x R , C


continuous at the origin. In view of Theorem 3.6, C must

have the form 3( , ) C x x cx for any x in R . Thus we

obtain that

3(2 ,2 ) 2 ( , ) ( | |) | | , f x x f x x c x (III.45)


If1

10,

2mx



31

0 0

3 3

(2 ) (2 )(2 ,2 ) 2 ( , )

8 8

( | |)

n nm

n nn n

x xf x x f x x

m x c x


sense of Ulam, Hyers and Rassias if s =3, assumed in

the inequality (ii) of (III.41).



Example 3.9: Let s be such that 3

04

s . Then there is a


3 3

4 4 4 4( , , , ) | | | | | | | |s s s s

F x y z w x y z w

(III.46)



18

30

(2 ,2 ) 2 ( , ) ( , )supx

f x x f x x C x x

x

(III.47)

for every cubic mapping 2( , ) :C x x R R .

Proof: If we take

3( , ) ln( , ), 0 ( , )

0, 0

x x x x if xf x x

if x

Then from (III.47), it follows that

30

3

0

3 3 3 3 3

3

0

3 3

0

(2 ,2 ) 2 ( , ) ( , )sup

(2 ,2 ) 2 ( , ) ( , )sup

(2,2) ln | 2 ,2 | 2 (1,1) ln | , | (1,1)sup

sup (2,2) ln | 2 ,2 | 8(1,1) ln | , | (1,1)

x

nn

nn

nn

f x x f x x C x x

x

f n n f n n C n n

n

n n n n n n n C

n

n n n n C

We have to prove (III.46) is true.

Case (i): If , , , 0x y z w in (III.46) then,

3

3

3

3 3

(2 ,2 ) (2 ,2 ) 4 ( , )

4 ( , ) 6 ( , )

(2 ,2 ) ln | 2 ,2 |

(2 ,2 ) ln | 2 ,2 |

4( , ) ln | , |

4( , ) ln | , | 6( , ) ln | , |


f x y z w f y w

x y z w x y z w

x y z w x y z w

x y z w x y z w



3

1 2 3 4 1 2 3 4

3

1 2 3 4 1 2 3 4

3

1 2 3 4 1 2 3 4

3 3

1 2 3 4 1 2 3 4 2 4

(2 ,2 ) (2 ,2 ) 4 ( , )

4 ( , ) 6 ( , )

(2 ,2 ) ln | 2 ,2 |

(2 ,2 ) ln | 2 ,2 |

4( , ) ln | , |

4( , ) ln | , | 6( , ) ln


f x y z w f y w

v v v v v v v v

v v v v v v v v

v v v v v v v v

v v v v v v v v v v

2 4

1 2 3 4 1 2 3 4

1 2 3 4

1 2 3 4 2 4

1 3

4 4 4 41 2 3 4

1 3

4 4 4 4

| , |

(2 ,2 ) (2 ,2 )

4 ( , )

4 ( , ) 6 ( , )

| | | | | | | |

| | | | | | | |

s s s s

s s s s

v v

f v v v v f v v v v

f v v v v

f v v v v f v v

v v v v

x y z w

Case (ii): If , , , 0x y z w in (III.46) then,

3

3

3

3 3

(2 ,2 ) (2 ,2 ) 4 ( , )

4 ( , ) 6 ( , )

(2 ,2 ) ln | 2 ,2 |

(2 ,2 ) ln | 2 ,2 |

4( , ) ln | , |

4( , ) ln | , | 6( , ) ln | , |


f x y z w f y w

x y z w x y z w

x y z w x y z w

x y z w x y z w



3

1 2 3 4 1 2 3 4

3

1 2 3 4 1 2 3 4

3

1 2 3 4 1 2 3 4

3

1 2 3 4 1 2 3

(2 ,2 ) (2 ,2 ) 4 ( , )

4 ( , ) 6 ( , )

( 2 , 2 ) ln | 2 , 2 |

( 2 , 2 ) ln | 2 , 2 |

4( , ) ln | , |

4( , ) ln | ,


f x y z w f y w

v v v v v v v v

v v v v v v v v

v v v v v v v v

v v v v v v v

3

4 2 4 2 4

1 2 3 4 1 2 3 4

1 2 3 4

1 2 3 4 2 4

1 3

4 4 4 41 2 3 4

1 3

4 4 4 4

| 6( , ) ln | , |

( 2 , 2 ) ( 2 , 2 )

4 ( , )

4 ( , ) 6 ( , )

| | | | | | | |

| | | | | | | |

s s s s

s s s s

v v v v v

f v v v v f v v v v

f v v v v

f v v v v f v v

v v v v

x y z w

Case (iii): If , 0, , 0x z y w then 2 ,x y


then,

3

3

3

3 3

(2 ,2 ) (2 ,2 ) 4 ( , )

4 ( , ) 6 ( , )

(2 ,2 ) ln | 2 ,2 |

(2 ,2 ) ln | 2 ,2 |

4( , ) ln | , |

4( , ) ln | , | 6( , ) ln | , |


f x y z w f y w

x y z w x y z w

x y z w x y z w

x y z w x y z w



3

1 2 3 4 1 2 3 4

3

1 2 3 4 1 2 3 4

3

1 2 3 4 1 2 3 4

3

1 2 3 4 1 2 3 4 2

(2 ,2 ) (2 ,2 ) 4 ( , )

4 ( , ) 6 ( , )

(2 ,2 ) ln | 2 ,2 |

(2 ,2 ) ln | (2 ,2 ) |

4( , ) ln | , |

4( , ) ln | ( , ) | 6(


f x y z w f y w

v v v v v v v v

v v v v v v v v

v v v v v v v v

v v v v v v v v v

3

4 2 4

1 2 3 4 1 2 3 4

1 2 3 4

1 2 3 4 2 4

1 3

4 4 4 41 2 3 4

1 3

4 4 4 4

, ) ln | , |

(2 ,2 ) (2 ,2 )

4 ( , )

4 ( , ) 6 ( , )

| | | | | | | |

| | | | | | | |

s s s s

s s s s

v v v

f v v v v f v v v v

f v v v v

f v v v v f v v

v v v v

x y z w

Case (iv): If , 0, , 0x z y w then 2 ,x y


then,

3

3

3

3 3

(2 ,2 ) (2 ,2 ) 4 ( , )

4 ( , ) 6 ( , )

(2 ,2 ) ln | 2 ,2 |

(2 ,2 ) ln | 2 ,2 |

4( , ) ln | , |

4( , ) ln | , | 6( , ) ln | , |


f x y z w f y w

x y z w x y z w

x y z w x y z w

x y z w x y z w




19

3

1 2 3 4 1 2 3 4

3

1 2 3 4 1 2 3 4

3

1 2 3 4 1 2 3 4

3

1 2 3 4 1 2 3 4 2

(2 ,2 ) (2 ,2 ) 4 ( , )

4 ( , ) 6 ( , )

(2 ,2 ) ln | (2 ,2 ) |

(2 ,2 ) ln | 2 ,2 |

4( , ) ln | ( , ) |

4( , ) ln | , | 6(


f x y z w f y w

v v v v v v v v

v v v v v v v v

v v v v v v v v

v v v v v v v v v

3

4 2 4

1 2 3 4 1 2 3 4

1 2 3 4

1 2 3 4 2 4

1 3

4 4 4 41 2 3 4

1 3

4 4 4 4

, ) ln | , |

(2 ,2 ) (2 ,2 )

4 ( , )

4 ( , ) 6 ( , )

| | | | | | | |

| | | | | | | |

s s s s

s s s s

v v v

f v v v v f v v v v

f v v v v

f v v v v f v v

v v v v

x y z w

Case (v): If 0x y z w in (III.46) then it is trivial.


is not stable for 3



3 3, | |

4( )

3,

4

x if x

x

otherwise


0

(2 )( , ) .

8

n

nn

xf x x for all x R


3 3 3 32

3 3 3 34 4 4 4

( , , , )

96 8 | | | | | | | | | | | | | | | |

7

F x y z w

x y z w x y z w

(III.48)

for all , , ,x y z w R . Then there do not exist a cubic

mapping 2:C R R and a constant 0 such that


Proof: Now

0 0

(2 ) 1 3 6( , )

4 78 8

n

n nn n

xf x x

.



If x = y = z = w = 0, then (III.48) is trivial.

If 3 3 3 3

3 3 3 34 4 4 41

| | | | | | | | | | | | | | | |8

x y z w x y z w , then the

left hand side of (III.48) is less than 96

7

. Now suppose

that

3 3 3 3

3 3 3 34 4 4 41

0 | | | | | | | | | | | | | | | |8

x y z w x y z w .


3 3 3 3

3 3 3 34 4 4 42 1

1 1| | | | | | | | | | | | | | | |

8 8k kx y z w x y z w

(III.50)

so that

1 1 1 1

1 1 14 4 4 41 1 1

2 | | | | | | | | ,2 ,2 ,2 2 2

k k kx y z w x y

1 11 12 ,2

2 2

k kz w

and consequently

1 1 1

1 1

2 ( , ),2 ( , ),2 ( , ),

1 12 (2 ,2 ),2 (2 ,2 ), , .

2 2

k k k

k k

y w x y z w x y z w

x y z w x y z w


2 ( , ),2 ( , ),2 ( , ),

1 12 (2 ,2 ),2 (2 ,2 ) , .

2 2

n n n

n n

y w x y z w x y z w

x y z w x y z w

and

(2 (2 ,2 )) (2 (2 ,2 ))

4[ (2 ( , )) (2 ( , ))] 6 (2 ( , )) 0

n n

n n n

x y z w x y z w

x y z w x y z w y w


we obtain that

0

( , , , )

1(2 (2 ,2 )) (2 (2 ,2 ))

8

4[ (2 ( , )) (2 ( , ))] 6 (2 ( , ))

1(2 (2 ,2 )) (2 (2 ,2 ))

8

4[ (2 ( , )) (2 ( , ))] 6 (2 ( , ))

1 16 3

8

n n

nn

n n n

n n

nn k

n n n

nn k

F x y z w

x y z w x y z w

x y z w x y z w y w

x y z w x y z w

x y z w x y z w y w

3 3 3 3

3 3 3 34 4 4 4

12 8 1

4 7 8

8 | | | | | | | | | | | | | | | |

k

x y z w x y z w


3 3 3 3

3 3 3 34 4 4 41

0 | | | | | | | | | | | | | | | |8

x y z w x y z w

We claim that (I.8) is not stable for 3

4s in (iii) Corollary

3.7. Suppose on the contrary that there exist a cubic

mapping 2:C R R and a constant 0 satisfying

(III.49). Since f is bounded and continuous for all x R , C


continuous at the origin. In view of Theorem 3.6, C must

have the form 3( , ) C x x cx for any x in R . Thus we

obtain that

3(2 ,2 ) 2 ( , ) ( | |) | | , f x x f x x c x (III.51)



20

If1

10,

2mx



31

0 0

3 3

(2 ) (2 )(2 ,2 ) 2 ( , )

8 8

( | |)

n nm

n nn n

x xf x x f x x

m x c x


sense of Ulam, Hyers and Rassias if 3

4s , assumed in (iv)

of (III.41).

Now, we are ready to prove our main stability results.

Theorem 3.11: Let 1j . Let 2:f U V be a mapping

for which there exist a function 4: (0, ]U with the

condition given in (III.1) and (III.29) respectively, such that the functional inequality

( , , , ) ( , , , )F x y z w x y z w (III.52)


additive mapping 2:A U V unique 2-variable cubic

mapping 2:C U V satisfying (I.8) and

( , ) ( , ) ( , )f x x A x x C x x

1 1

2 2

1 1 1(2 ) (2 )

6 2 8

kj kj

j jk k

x x

(III.53)

for all x U . The mapping (2 )kj x , ( , )A x x and ( , )C x x are

defined in (III.4), (III.5) and (III.33)

for all x U .

Proof: By Theorems 3.1 and 3.6, there exists a unique 2-

variable additive function 2

1 :A U V and a unique 2-

variable cubic function 2

1 :C U V such that

11

2

1(2 ,2 ) 8 ( , ) ( , ) (2 )

2

kj

jk

f x x f x x A x x x

(III.54)

and

11

2

1(2 ,2 ) 2 ( , ) ( , ) (2 )

8

kj

jk

f x x f x x C x x x

(III.55)

for all x U . Now from (III.54) and (III.55), one can see

that

1 1

1

1

1

1

1 1

2 2

1 1( , ) ( , ) ( , )

6 6

(2 ,2 ) 8 ( , ) ( , )

6 6 6

(2 ,2 ) 2 ( , ) ( , )

6 6 6

1(2 ,2 ) 8 ( , ) ( , )

6

(2 ,2 ) 8 ( , ) ( , )

1 1 1(2 ) (2 )

6 2 8

kj kj

j jk k

f x x A x x C x x

f x x f x x A x x

f x x f x x C x x

f x x f x x A x x

f x x f x x A x x

x x

for all x U . Thus we obtain (III.55) by defining

1

1( , ) ( , )

6A x x A x x

and

1

1( , ) ( , )

6C x x C x x , (2 )kj x , ( , )A x x

and ( , )C x x are defined in (III.4), (III.5) and (III.33)

for all x U .

The following corollary is the immediate consequence

of Theorem 3.11, using Corollaries 3.2 and 3.7 concerning

the stability of (I.8).



( , , , )F x y z w

4 4 4 4

, 3 3;

3 3, ;

4 4

,

3 3 ;

4 4

s s s s

s s s s

s s s s s s s s

x y z w s or s

x y z w s or s

x y z w x y z w

s or s

(III.56)

for all , , ,x y z w U , then there exists a unique 2-variable


mapping 2:C U V such that

( , ) ( , ) ( , )f x x A x x C x x

1

2

4

4 4

2

4

4 2

44

4 2

5 11

6 7

18 2 1 1

6 2 2 7 8 2

4 2 1 1

6 2 2 7 8 2

22 2 1 1

6 2 2 7 8 2

2 2 1 1

6 2 2 7 8 2

s

s

s s

s

s

s s

s

s

s s

ss

s s

x

x

x

x

(III.57)

for all x U .


is not stable for 1s in (ii) of Corollary 3.12.


3( ), | | 1( )

,

x x if xx

otherwise


0

(2 )( , ) .

2

n

nn

xf x x for all x R



21

( , , , ) 32 (| | | | | | | |)F x y z w x y z w (III.58)

for all , , ,x y z w R . Then there do not exist a 2-variable

additive mapping 2:A U V and 2-variable cubic

mapping 2:C U V and a constant 0 such that

( , ) ( , ) ( , ) | | .f x x A x x C x x x for all x R (III.59)



Example 3.14: Let s be such that1

04

s . Then there is a


1 3

4 4 4 4( , , , ) | | | | | | | |s s s s

F x y z w x y z w

(III.60)


0

( , ) ( , ) ( , )supx

f x x A x x C x x

x

(III.61)

for every additive mapping 2( , ) :A x x R R , and for every

cubic mapping 2( , ) :C x x R R .


is not stable for 1



3 1( ), | |

4( )

,4

x x if x

x

otherwise


0

(2 )( , ) .

2

n

nn

xf x x for all x R


1 1 1 1

4 4 4 4

( , , , )

8 | | | | | | | | | | | | | | | |

F x y z w

x y z w x y z w

(III.62)

for all , , ,x y z w R . Then there do not exist a 2-variable

additive mapping 2:A U V and a cubic mapping 2:C R R and a constant 0 such that


IV. STABILITY RESULTS:FIXED POINT METHOD

In this section, we apply a fixed point method for

achieving stability of the 2-variable AC (I.8).

Now, we present the following theorem due to B.

Margolis and J.B. Diaz [14] for fixed point Theory.

Theorem 4.1: Suppose that for a complete generalized

metric space ,d and a strictly contractive mapping

:T with Lipschitz constant L . Then, for each

given x , either 1,n nd T x T x for all 0n ,or there

exists a natural number 0n such that

i) 1,n nd T x T x for all 0n n ;

ii) The sequence nT x is convergent to a fixed to a fixed

point *y of T

iii) *y is the unique fixed point of T in the set

0: , ;n

y d T x y

iv) * 1, ,

1d y y d y Ty

L

for all y .

Using the above theorem, we now obtain the generalized

Hyers-Ulam-Rassias stability of (I.8).

Thorough out this section, let U be a vector space and V

Banach space. Define a mapping 2:F U V by

( , , , ) 2 ,2 2 ,2F x y z w f x y z w f x y z w

4 , , 6 ,f x y z w f x y z w f y w

for all , , ,x y z w U .

Theorem 4.2: Let 2:f U V be a mapping for which

there exists a function 4: (0, ]U with the condition

1lim ( , , , ) 0ni

n n n n

i i i in

x y z w

(IV.1)

with 2i if 0i and 1

2i if 1i such that the

functional inequality

( , , , ) ( , , , )F x y z w x y z w (IV.2)

for all , , ,x y z w U . If there exists 1L L i such that

the function

1

2x x x ,

has the property

i ix L x . (IV.3)

Then there exists a unique 2-variable additive mapping 2:A U V satisfying (I.8) and

1

(2 ,2 ) 8 ( , ) ( , )1

iLf x x f x x A x x x

L

(IV.4)

for all x U . The mapping (2 )kj x and ( , )A x x are

defined in (III.10) and (III.5) respectively for all x U .

Proof: Consider the set

2: , 0 0p p U V p

and introduce the generalized metric on ,

, ,

inf 0, ,

d g h d g h

K p x q x K x x U

It is easy to see that ,d is complete.


22

Define 2:T by

1

, ,i i

i

Tp x x p x x

for all x U . Now for all ,p q ,

,

, , , ,

1 1 1, , , ,

1 1, , , ,

, , , ,

, .

i i i i i

i i

i i i i

i i

d g h K

p x x q x x K x x U

p x x q x x K x x U

p x x q x x LK x x U

T p x x T q x x LK x x U

d Tp Tq LK

This gives , ,d Tp Tq Ld p q , for all ,p q , i.e., T is a

strictly contractive mapping of , with Lipschitz constant

L . From (III.12), we arrive

(2 ,2 ) ( )( , )

2 2

g x x xg x x

(IV.5)

for all x U . Using (IV.3) for the case 0i it reduces to

(2 ,2 )( , ) ( )

2

g x xg x x L x for all x U

i.e., 1, ,d g Tg L d g Tg L L

Again replacing2

xx , in (IV.5), we get,

( , ) 2 ,2 2 2

x x xg x x g

(IV.6)


( , ) 2 , ( )2 2

x xg x x g x

for all x U ,

i.e., 0, 1 , 1d g Tg d g Tg L

Now from the fixed point alternative in both cases, it

follows that there exists a fixed point A of T in such

that

1 11( , ) lim ( , ) 8 ( , )n n n n

i i i inni

A x x f x x f x x

(IV.7)

for all x U , since lim , 0n

nd T g A

.

To prove 2:A U V is additive. Putting , , ,x y z w by

, , ,n n n n

i i i ix y z w in (IV.2) and divide by n

i . It follows

from (IV.1) that

, , ,, , , lim

, , ,lim 0

n n n n

i i i i

nni

n n n n

i i i i

nni

F x y z wA x y z w

x y z w

for all , , ,x y z w U , i.e., A satisfies (I.8).

According to the alternative fixed point, since A is the

unique fixed point of T in the set

: , ,p d f p A is the unique function such that

(2 ,2 ) 8 ( , ) ( , )f x x f x x A x x K x

for all x U and 0K . Again using the fixed point

alternative, we obtain

1

, ,1

d f A d f TfL

this implies

1

, ,1

iLd f A d f Tf

L

which yields

1

(2 ,2 ) 8 ( , ) ( , )1


L

this completes the proof of the theorem.

The following Corollary is an immediate consequence of

Theorem 4.2 concerning the stability of (I.8).



( , , , )F x y z w

4 4 4 4

, 1 1;

1 1, ;

4 4

,

1 1 ;

4 4

s s s s

s s s s

s s s s s s s s

x y z w s or s

x y z w s or s

x y z w x y z w

s or s

(IV.8)


additive function 2:A U V such that

(2 ,2 ) 8 ( , ) ( , )f x x f x x A x x

1 1

44 1 2

4

44 1 2 4

4

2 18 2

2 2

2 4 2

2 2

2 22 2 2 2

2 2

ss s

s

ss s

s

ss s s

s

x

x

x

(IV.9)

for all x U .

Proof: Setting

4 4 4 4

, ,

,

,

,

s s s s

s s s s

s s s s s s s s

x y z

x y z w

x y z w

x y z w x y z w


23

for all , , ,x y z w U . Then, for 2s if 0i and for 1s

if 1i we get

4 4 4 4

, , ,

,

,

,

0 as ,

0 as ,

0 as ,

n n n n

i i i i

n

i

s s s sn n n n

i i i in

i

s s s sn n n n

i i i in

i

s s s sn n n n

i i i in

i

s s s sn n n n

i i i i

x y z w

x y z w

x y z w

x y z w

x y z w

n

n

n

Thus (IV.1) is holds. But we have 1

2x x x , has

the property i ix L x , for all x U . Hence

2 4

2 4 4

1

2

1(4 ( , , , ) ( ,2 , ,2 ))

2

(18 || || 2 || 2 || )2

(4 2 ) || ||2

(22 2 2 2 ) || ||2

s s

s s

s s s

x x

x x x x x x x x

x x

x

x

Now

2 4

2 4 4

(18 || || 2 || 2 || )2

1(4 2 ) || ||

2

(22 2 2 2 ) || ||2

s s

i i

i

s s

i i

i i

s s s

i

i

x x

x x

x

4 2 4

4 2 4 4

(18 || || 2 || 2 || )2

(4 2 ) || ||2

(22 2 2 2 ) || ||2

s s s

i

i

s s s

i

i

s s s s

i

i

x x

x

x

1

4 1 2 4

4 1 2 4 4

1

4 1

4 1

(18 || || 2 || 2 || )2

(4 2 ) || ||2

(22 2 2 2 ) || ||2

( )

( )

( )

s s s

i

s s s

i

s s s s

i

s

i

s

i

s

i

x x

x

x

x

x

x

for all x U .

Hence the (IV.3) holds either, 12sL for 1s if 0i and

1

1

2sL

for 1s if 1i .

Now from (IV.4), we prove the following cases for (i).

Case (i): 12sL for 1s if 0i .

1 01 1

1

1 1

1

1 1

11 1

(2 ,2 ) 8 ( , ) ( , )

2 18 2|| ||

1 2 2

2 18 2|| ||

1 2 2

2 2 18 2|| ||

2 2 2

2 18 2|| ||

2 2

s ss

s

s ss

s

s ss

s

s s

s

s

f x x f x x A x x

x

x

x

x

Case (ii): 1

1

2sL

for 1s if 1i .

1 1

11

1

1 1

1

1 1

1 1

(2 ,2 ) 8 ( , ) ( , )

1

18 22|| ||

1 21

2

2 18 2|| ||

2 1 2

2 2 18 2|| ||

2 2 2

2 18 2|| ||

2 2

sss

s

s ss

s

s ss

s

s s

s

s

f x x f x x A x x

x

x

x

x

In similar manner we can prove the following cases

4 12 sL for 1s if 0i and 4 1

1

2 sL

for 1s if 1i ,

and 4 12 sL for 1s if 0i and 4 1

1

2 sL

for 1s if

1i for (ii) and (iii) respectively. Hence the proof is

complete.


there exist a function 4: (0, ]U with the condition

3

1lim ( , , , ) 0n n n n

i i i inni

x y z w

(IV.10)

with 2i if 0i and 1

2i if 1i such that the

functional inequality

( , , , ) ( , , , )F x y z w x y z w (IV.11)


the function

1

2x x x ,

has the property

3

i ix L x . (IV.12)


24

Then there exists a unique 2-variable cubic mapping 2:C U V satisfying (I.8) and

1

(2 ,2 ) 2 ( , ) ( , )1

iLf x x f x x C x x x

L

(IV.4)

for all x U . The mapping (2 )kj x and ( , )C x x are

defined in (III.10) and (III.33) respectively for all x U .

Proof: Consider the set

2: , 0 0p p U V p

and introduce the generalized metric on ,

, ,

inf 0, ,

d g h d g h

K p x q x K x x U

It is easy to see that ,d is complete.

Define 2:T by

3

1, ,i i

i

Tp x x p x x

for all x U . Now for all ,p q ,

3 3 3

3 3

,

, , , ,

1 1 1, , , ,

1 1, , , ,

, , , ,

, .

i i i i i

i i i

i i i i

i i

d g h K

p x x q x x K x x U

p x x q x x K x x U

p x x q x x LK x x U

T p x x T q x x LK x x U

d Tp Tq LK

This gives , ,d Tp Tq Ld p q , for all ,p q , i.e., T is a

strictly contractive mapping of , with Lipschitz constant L .

From (III.36), we arrive

(2 ,2 ) ( )( , )

8 8

h x x xh x x

(IV.14)


(2 ,2 )( , ) ( )

8

h x xh x x L x

for all x U ,

i.e., 1, ,d h Th L d h Th L L

Again replacing2

xx , in (IV.14), we get,

( , ) 8 ,2 2 2

x x xh x x h

(IV.15)


( , ) 8 , ( )2 2

x xh x x h x

for all x U ,

i.e., 0, 1 , 1d h Th d h Th L

Now from the fixed point alternative in both cases, it

follows that there exists a fixed point C of T in such

that

1 1

3

1( , ) lim ( , ) 2 ( , )n n n n

i i i inni

C x x f x x f x x

(IV.16)

for all x U , since lim , 0n

nd T h C

.

To prove 2:C U V is cubic. Putting , , ,x y z w by

, , ,n n n n

i i i ix y z w in (IV.11) and divide by 3n

i . It follows

from (IV.1) that

3

3

, , ,, , , lim

, , ,lim 0

n n n n

i i i i

nni

n n n n

i i i i

nni

F x y z wC x y z w

x y z w

for all , , ,x y z w U , i.e., C satisfies (I.8).

According to the alternative fixed point, since C is the

unique fixed point of T in the set

: , ,p d f p C is the unique function such that

(2 ,2 ) 8 ( , ) ( , )f x x f x x A x x K x

for all x U and 0K . Again using the fixed point

alternative, we obtain

1

, ,1

d f C d f TfL

this implies

1

, ,1

iLd f C d f Tf

L

which yields

1

(2 ,2 ) 2 ( , ) ( , )1


L

this completes the proof of the theorem.

The following Corollary is an immediate consequence of

Theorem 4.4 concerning the stability of (I.8).



( , , , )F x y z w

4 4 4 4

, 1 1;

3 3, ;

4 4

,

3 3 ;

4 4

s s s s

s s s s

s s s s s s s s

x y z w s or s

x y z w s or s

x y z w x y z w

s or s

(IV.17)


cubic function 2:C U V such that

(2 ,2 ) 8 ( , ) ( , )f x x f x x A x x


25

1 1

44 1 2

4

44 1 2 4

4

2 18 2

8 2

2 4 2

8 2

2 22 2 2 2

8 2

ss s

s

ss s

s

ss s s

s

x

x

x

(IV.18)

for all x U .

Proof: The proof of the corollary is similar tracing as that

of corollary 4.3.


there exist a function 4: (0, ]U with (IV.1) and

(IV.10) where 2i if 0i and 1

2i if 1i such that

the functional inequality

( , , , ) ( , , , )F x y z w x y z w (IV.19)


the function

1

2x x x ,

has the (IV.3) and (IV.12), then there exists a unique 2-

variable additive mapping 2:A U V unique 2-variable

cubic mapping 2:C U V satisfying (I.8) and

11

( , ) ( , ) ( , )31

iLf x x A x x C x x x

L

(IV.20)

for all x U . The mapping (2 )kj x , ( , )A x x and ( , )C x x are

defined in (III.5), (III.10) and (III.33) respectively for

all x U .

Proof: By Theorems 4.1 and 4.4, there exists a unique 2-

variable additive function 2

1 :A U V and a unique 2-

variable cubic function 2

1 :C U V such that

1

1(2 ,2 ) 8 ( , ) ( , )1


L

(IV.21)

and

1

1(2 ,2 ) 2 ( , ) ( , )1


L

(IV.22)

for all x U . Now from (III.21) and (III.22), one can see

that

1 1

1

1

1 1( , ) ( , ) ( , )

6 6

(2 ,2 ) 8 ( , ) ( , )

6 6 6

(2 ,2 ) 2 ( , ) ( , )

6 6 6

f x x A x x C x x

f x x f x x A x x

f x x f x x C x x

1

1

1 1

1(2 ,2 ) 8 ( , ) ( , )

6

(2 ,2 ) 8 ( , ) ( , )

1

6 1 1

i i

f x x f x x A x x

f x x f x x A x x

L Lx x

L L

for all x U . Thus we obtain (IV.21) by defining

1

1( , ) ( , )

6A x x A x x

and

1

1( , ) ( , )

6C x x C x x , (2 )kj x , ( , )A x x

and ( , )C x x are defined in (III.10), (III.5) and (III.33) for

all x U .

The following corollary is the immediate consequence

of Theorem 4.6, using Corollaries 4.3 and 4.5 concerning

the stability of (I.8).



( , , , )F x y z w

4 4 4 4

, 3 3;

3 3, ;

4 4

,

3 3 ;

4 4

s s s s

s s s s

s s s s s s s s

x y z w s or s

x y z w s or s

x y z w x y z w

s or s

(IV.23)

for all , , ,x y z w U , then there exists a unique 2-variable


mapping 2:C U V such that

( , ) ( , ) ( , )f x x A x x C x x

1

2

4

4 4

2 4

4

4 2

18 2 1 1

3 2 2 7 8 2

4 2 1 1

3 2 2 7 8 2

22 2 2 2 1 1

3 2 2 7 8 2

s

s

s s

s

s

s s

s s

s

s s

x

x

x

(III.24)

for all x U .

V. CONCLUSION

The (I.8) is stable in Banach spaces via direct and fixed

point method in sprit of Hyers, Ulam, Rasssias.

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Ulam - Hyers Stability of a 2- Variable AC - Mixed Type Functional Equation

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