ulam - hyers stability of a 2- variable ac - mixed type functional equation
DESCRIPTION
http://www.sjmmf.org Abstract. In this paper, we obtain the general solution and generalized Ulam - Hyers stability of a 2 - variable AC - mixed type functional equation f(2x + y; 2z + w) - f(2x - y; 2z - w) = 4[f(x + y; z + w) - f(x - y; z - w)] - 6f(y;w) using direct and fixed point methods.TRANSCRIPT
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)
Journal of Modern Mathematics Frontier Sept. 2012, Vol. 1 Iss. 3, PP. 10-26
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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
Journal of Modern Mathematics Frontier Sept. 2012, Vol. 1 Iss. 3, PP. 10-26
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(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)
for all x U . From (III.12), we arrive
(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
Journal of Modern Mathematics Frontier Sept. 2012, Vol. 1 Iss. 3, PP. 10-26
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(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)
Journal of Modern Mathematics Frontier Sept. 2012, Vol. 1 Iss. 3, PP. 10-26
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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 x y z w f x y z w
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 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 ,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 x y z w f x y z w
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 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 ) 4 ( , )
4 ( , ) 6 ( , )
(2 ,2 ) ln | 2 ,2 |
(2 ,2 ) ln | (2 ,2 ) |
4( , ) ln | , |
4( , ) ln | ( , ) | 6( ,
f x y z w f x y z w f x y z w
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
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 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
Journal of Modern Mathematics Frontier Sept. 2012, Vol. 1 Iss. 3, PP. 10-26
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 x y z w f x y z w
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.
Example 3.5: Let : R R be a function defined by
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
Then F satisfies the functional inequality
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
Therefore for each 0,1,........ 1n k , we have
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
For 0,1,........ 1n k . From the definition of f and (III.27),
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)
But we can choose a positive integer m with | |m c .
If1
10,
2mx
, then 2 (0,1)n x for all
0, 1,..... 1.n m For this x, we get
Journal of Modern Mathematics Frontier Sept. 2012, Vol. 1 Iss. 3, PP. 10-26
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
which contradicts (III.28). Therefore (I.8) is not stable in
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)
for all , , ,x y z w U . Then there exists a unique 2-variable
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)
for all x U . From (III.35), we arrive
(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).
Corollary 3.7: 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
, 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)
for all , , ,x y z w U , then there exists a unique 2- variable
cubic function 2:C U V such that
(2 ,2 ) 2 ( , ) ( , )f x x f x x C x x
Journal of Modern Mathematics Frontier Sept. 2012, Vol. 1 Iss. 3, PP. 10-26
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.
Example 3.8: Let : R R be a function defined by
3, | | 1( )
,
x if xx
otherwise
where 0 is a constant, and define a function 2:f R R by
0
(2 )( , ) .
8
n
nn
xf x x for all x R
Then F satisfies the functional inequality
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
Therefore we see that f is bounded. We are going to prove
that f satisfies (III.42).
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 .
Then there exists a positive integer k such that
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
Therefore for each 0,1,........ 1n k , we have
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
For 0,1,........ 1n k . From the definition of f and (III.44),
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
Thus f satisfies (III.42) for all , , ,x y z w with
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
is bounded on any open interval containing the origin and
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)
But we can choose a positive integer m with | |m c .
If1
10,
2mx
, then 2 (0,1)n x for all
0, 1,..... 1.n m For this x, we get
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
which contradicts (III.45). Therefore (I.8) is not stable in
sense of Ulam, Hyers and Rassias if s =3, assumed in
the inequality (ii) of (III.41).
A counter example to illustrate the non stability in (iii)
of Corollary 3.7 is given in the following example.
Example 3.9: Let s be such that 3
04
s . Then there is a
function 2:F R R and a constant 0 satisfying
3 3
4 4 4 4( , , , ) | | | | | | | |s s s s
F x y z w x y z w
(III.46)
for all , , ,x y z w R and
Journal of Modern Mathematics Frontier Sept. 2012, Vol. 1 Iss. 3, PP. 10-26
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 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
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 x y z w f x y z w
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 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
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 x y z w f x y z w
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
2 , , 0,2 ,2 , , 0z w x y z w x y z w x 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 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
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 x y z w f x y z w
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
2 , , 0,2 ,2 , , 0z w x y z w x y z w x 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 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
Journal of Modern Mathematics Frontier Sept. 2012, Vol. 1 Iss. 3, PP. 10-26
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 x y z w f x y z w
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.
Now we will provide an example to illustrate that (I.8)
is not stable for 3
4s in (iv) of Corollary 3.7.
Example 3.10: Let : R R be a function defined by
3 3, | |
4( )
3,
4
x if x
x
otherwise
where 0 is a constant, and define a function 2:f R R by
0
(2 )( , ) .
8
n
nn
xf x x for all x R
Then F satisfies the functional inequality
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
3(2 ,2 ) 2 ( , ) ( , ) | | .f x x f x x C x x x for all x R (III.49)
Proof: Now
0 0
(2 ) 1 3 6( , )
4 78 8
n
n nn n
xf x x
.
Therefore we see that f is bounded. We are going to prove
that f satisfies (III.48).
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 .
Then there exists a positive integer k such that
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
Therefore for each 0,1,........ 1n k , we have
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
For 0,1,........ 1n k . From the definition of f and (III.50),
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
Thus f satisfies (III.48) for all , , ,x y z w with
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
is bounded on any open interval containing the origin and
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)
But we can choose a positive integer m with | |m c .
Journal of Modern Mathematics Frontier Sept. 2012, Vol. 1 Iss. 3, PP. 10-26
20
If1
10,
2mx
, then 2 (0,1)n x for all
0, 1,..... 1.n m For this x, we get
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
which contradicts (III.51). Therefore (I.8) is not stable in
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)
for all , , ,x y z w U . 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
( , ) ( , ) ( , )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).
Corollary 3.12: 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
, 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
additive mapping 2:A U V unique 2-variable cubic
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 .
Now we will provide an example to illustrate that (I.8)
is not stable for 1s in (ii) of Corollary 3.12.
Example 3.13: Let : R R be a function defined by
3( ), | | 1( )
,
x 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
Journal of Modern Mathematics Frontier Sept. 2012, Vol. 1 Iss. 3, PP. 10-26
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)
A counter example to illustrate the non stability in (iii)
of Corollary 3.12 is given in the following example.
Example 3.14: 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.60)
for all , , ,x y z w R and
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 .
Now we will provide an example to illustrate that (I.8)
is not stable for 1
4s in (iv) of Corollary 3.12.
Example 3.15: Let : R R be a function defined by
3 1( ), | |
4( )
,4
x x if x
x
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
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
3(2 ,2 ) 2 ( , ) ( , ) | | .f x x f x x C x x x for all x R (III.63)
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.
Journal of Modern Mathematics Frontier Sept. 2012, Vol. 1 Iss. 3, PP. 10-26
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)
for all x U . Using (IV.3) for the case 1i it reduces to
( , ) 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
iLf x x f x x A x x x
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).
Corollary 4.3: 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
(IV.8)
for all , , ,x y z w U , then there exists a unique 2- variable
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
Journal of Modern Mathematics Frontier Sept. 2012, Vol. 1 Iss. 3, PP. 10-26
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.
Theorem 4.4: Let 2:f U V be a mapping for which
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)
for all , , ,x y z w U . If there exists 1L L i such that
the function
1
2x x x ,
has the property
3
i ix L x . (IV.12)
Journal of Modern Mathematics Frontier Sept. 2012, Vol. 1 Iss. 3, PP. 10-26
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)
for all x U . Using (IV.14) for the case 0i it reduces to
(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)
for all x U . Using (IV.14) for the case 1i it reduces to
( , ) 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
iLf x x f x x C x x x
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).
Corollary 4.5: 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;
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)
for all , , ,x y z w U , then there exists a unique 2- variable
cubic function 2:C U V such that
(2 ,2 ) 8 ( , ) ( , )f x x f x x A x x
Journal of Modern Mathematics Frontier Sept. 2012, Vol. 1 Iss. 3, PP. 10-26
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.
Theorem 4.6: Let 2:f U V be a mapping for which
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)
for all , , ,x y z w U . If there exists 1L L i such that
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
iLf x x f x x A x x x
L
(IV.21)
and
1
1(2 ,2 ) 2 ( , ) ( , )1
iLf x x f x x C x x x
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).
Corollary 4.7: 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
, 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
additive mapping 2:A U V unique 2-variable cubic
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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