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An AQCQ-functional equation in paranormed spaces

Advances in Difference Equations20122012:63

https://doi.org/10.1186/1687-1847-2012-63

  • Received: 5 March 2012
  • Accepted: 17 May 2012
  • Published:

Abstract

In this article, we prove the Hyers-Ulam stability of an additive-quadratic-cubic-quartic functional equation in paranormed spaces.

Mathematics Subject Classification (2010): Primary 39B82; 39B52; 39B72; 46A99.

Keywords

  • Hyers-Ulam stability
  • paranormed space
  • additive-quadratic-cubic-quartic functional equation.

1. Introduction and preliminaries

The concept of statistical convergence for sequences of real numbers was introduced by Fast [1] and Steinhaus [2] independently and since then several generalizations and applications of this notion have been investigated by various authors (see [37]). This notion was defined in normed spaces by Kolk [8].

We recall some basic facts concerning Fré chet spaces.

Definition 1.1. [9] Let X be a vector space. A paranorm P : X → [0, ∞) is a function on X such that
  1. (1)

    P(0) = 0;

     
  2. (2)

    P(-x) = P(x);

     
  3. (3)

    P(x + y) ≤ P(x) + P(y) (triangle inequality)

     
  4. (4)

    If {t n } is a sequence of scalars with t n t and {x n } X with P(x n - x) → 0, then P(t n x n - tx) → 0 (continuity of multiplication).

     

The pair (X, P) is called a paranormed space if P is a paranorm on X.

The paranorm is called total if, in addition, we have
  1. (5)

    P(x) = 0 implies x = 0.

     

A Fréchet space is a total and complete paranormed space.

The stability problem of functional equations originated from a question of Ulam [10] concerning the stability of group homomorphisms. Hyers [11] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [12] for additive mappings and by Rassias [13] for linear mappings by considering an unbounded Cauchy difference. A generalization of Rassias' theorem was obtained by Găvruta [14] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias' approach.

In 1990, Rassias [15] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. In 1991, Gajda [16] following the same approach as in Rassias [13], gave an affirmative solution to this question for p > 1. It was shown by Gajda [16], as well as by Rassias and Šemrl [17] that one cannot prove a Rassias-type theorem when p = 1 (cf. the books of Czerwik [18], Hyers et al. [19]).

In 1982, Rassias [20] followed the innovative approach of the Rassias' theorem [13] in which he replaced the factor x p + y p by x p · y q for p, q with p + q ≠ 1.

The functional equation
f ( x + y ) + f ( x - y ) = 2 f ( x ) + 2 f ( y )

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [21] for mappings f : XY, where X is a normed space and Y is a Banach space. Cholewa [22] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [23] proved the Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have extensively been investigated by a number of authors and there are many interesting results concerning this problem (see [2430]).

Jun and Kim [31] considered the following cubic functional equation
f ( 2 x + y ) + f ( 2 x - y ) = 2 f ( x + y ) + 2 f ( x - y ) + 12 f ( x ) .
(1.1)

It is easy to show that the function f(x) = x3 satisfies the functional equation (1.1), which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic mapping.

Lee et al. [32] considered the following quartic functional equation
f ( 2 x + y ) + f ( 2 x - y ) = 4 f ( x + y ) + 4 f ( x - y ) + 24 f ( x ) - 6 f ( y ) .
(1.2)

It is easy to show that the function f(x) = x4 satisfies the functional equation (1.2), which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping.

Throughout this article, assume that (X, P) is a Fré chet space and that (Y, · ) is a Banach space.

In this article, we prove the Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation
f ( x + 2 y ) + f ( x - 2 y ) = 4 f ( x + y ) + 4 f ( x - y ) - 6 f ( x ) + f ( 2 y ) + f ( - 2 y ) - 4 f ( y ) - 4 f ( - y )
(1.3)

in paranormed spaces.

One can easily show that an odd mapping f : XY satisfies (1.3) if and only if the odd mapping f : XY is an additive-cubic mapping, i.e.,
f ( x + 2 y ) + f ( x - 2 y ) = 4 f ( x + y ) + 4 f ( x - y ) - 6 f ( x ) .

It was shown in [[33], Lemma 2.2] that g(x) := f(2x) - 2f(x) and h(x) := f(2x) - 8f(x) are cubic and additive, respectively, and that f ( x ) = 1 6 g ( x ) - 1 6 h ( x ) .

One can easily show that an even mapping f : XY satisfies (1.3) if and only if the even mapping f : XY is a quadratic-quartic mapping, i.e.,
f ( x + 2 y ) + f ( x - 2 y ) = 4 f ( x + y ) + 4 f ( x - y ) - 6 f ( x ) + 2 f ( 2 y ) - 8 f ( y ) .

It was shown in [[34], Lemma 2.1] that g(x) := f(2x) - 4f(x) and h(x) := f(2x) - 16f(x) are quartic and quadratic, respectively, and that f ( x ) = 1 12 g ( x ) - 1 12 h ( x ) .

2. Hyers-Ulam stability of the functional equation (1.3): an odd mapping case

For a given mapping f, we define
D f ( x , y ) : = f ( x + 2 y ) + f ( x - 2 y ) - 4 f ( x + y ) - 4 f ( x - y ) + 6 f ( x ) - f ( 2 y ) - f ( - 2 y ) + 4 f ( y ) + 4 f ( - y ) .

In this section, we prove the Hyers-Ulam stability of the functional equation Df(x, y) = 0 in paranormed spaces: an odd mapping case.

Note that P(2x) ≤ 2P(x) for all x Y.

Theorem 2.1. Let r, θ be positive real numbers with r > 1, and let f : YX be an odd mapping such that
P ( D f ( x , y ) ) θ ( x r + y r )
(2.1)
for all x, y Y. Then there exists a unique additive mapping A : YX such that
P ( f ( 2 x ) - 8 f ( x ) - A ( x ) ) 2 r + 9 2 r - 2 θ x r
(2.2)

for all x Y.

Proof. Letting x = y in (2.1), we get
P ( f ( 3 y ) - 4 f ( 2 y ) + 5 f ( y ) ) 2 θ y r
(2.3)

for all y Y.

Replacing x by 2y in (2.1), we get
P ( f ( 4 y ) - 4 f ( 3 y ) + 6 f ( 2 y ) - 4 f ( y ) ) ( 2 p + 1 ) θ y r
(2.4)

for all y Y.

By (2.3) and (2.4),
P ( f ( 4 y ) - 10 f ( 2 y ) + 16 f ( y ) ) P ( 4 ( f ( 3 y ) - 4 f ( 2 y ) + 5 f ( y ) ) ) + P ( f ( 4 y ) - 4 f ( 3 y ) + 6 f ( 2 y ) - 4 f ( y ) ) 4 P ( f ( 3 y ) - 4 f ( 2 y ) + 5 f ( y ) ) + P ( f ( 4 y ) - 4 f ( 3 y ) + 6 f ( 2 y ) - 4 f ( y ) ) 8 θ y r + ( 2 r + 1 ) θ y r = ( 2 r + 9 ) θ y r
(2.5)
for all y Y. Replacing y by x 2 and letting g(x) := f(2x) - 8f(x) in (2.5), we get
P g ( x ) - 2 g x 2 2 r + 9 2 r θ x r
for all x Y. Hence
P 2 l g x 2 l - 2 m g x 2 m j = l m - 1 ( 2 r + 9 ) 2 j 2 r j + r θ x r
(2.6)
for all nonnegative integers m and l with m > l and all x Y. It follows from (2.6) that the sequence { 2 k g ( x 2 k ) } is Cauchy for all x Y. Since X is complete, the sequence { 2 k g ( x 2 k ) } converges. So one can define the mapping A : YX by
A ( x ) : = lim k 2 k g x 2 k

for all x Y.

By (2.1),
P ( D A ( x , y ) ) = lim k P 2 k D g x 2 k , y 2 k 2 k θ 2 r k ( 2 r + 8 ) ( x r + y r ) = 0

for all x, y Y. So DA(x, y) = 0. Since g : YX is odd, A : YX is odd. So the mapping A : YX is additive. Moreover, letting l = 0 and passing the limit m → ∞ in (2.6), we get (2.2). So there exists an additive mapping A : YX satisfying (2.2).

Now, let T : YX be another additive mapping satisfying (2.2). Then we have
P ( A ( x ) - T ( x ) ) = P 2 p A x 2 q - 2 q T x 2 q P 2 q A x 2 q - g x 2 q + P 2 q T x 2 q - g x 2 q 2 ( 2 r + 9 ) 2 q ( 2 r - 2 ) 2 r q θ x r ,

which tends to zero as q → ∞ for all x Y. So we can conclude that A(x) = T(x) for all x Y. This proves the uniqueness of A. Thus the mapping A : YX is a unique additive mapping satisfying (2.2).

Theorem 2.2. Let r be a positive real number with r < 1, and let f : XY be an odd mapping such that
D f ( x , y ) P ( x ) r + P ( y ) r
(2.7)
for all x, y X. Then there exists a unique additive mapping A : XY such that
f ( 2 x ) - 8 f ( x ) - A ( x ) 9 + 2 r 2 - 2 r P ( x ) r
(2.8)

for all x X.

Proof. Letting x = y in (2.7), we get
f ( 3 y ) - 4 f ( 2 y ) + 5 f ( y ) 2 P ( y ) r
(2.9)

for all y X.

Replacing x by 2y in (2.7), we get
f ( 4 y ) - 4 f ( 3 y ) + 6 f ( 2 y ) - 4 f ( y ) ( 2 p + 1 ) P ( y ) r
(2.10)

for all y X.

By (2.9) and (2.10),
f ( 4 y ) - 10 f ( 2 y ) + 16 f ( y ) 4 ( f ( 3 y ) - 4 f ( 2 y ) + 5 f ( y ) ) + f ( 4 y ) - 4 f ( 3 y ) + 6 f ( 2 y ) - 4 f ( y ) 4 f ( 3 y ) - 4 f ( 2 y ) + 5 f ( y ) + f ( 4 y ) - 4 f ( 3 y ) + 6 f ( 2 y ) - 4 f ( y ) 8 P ( y ) r + ( 2 r + 1 ) P ( y ) r = ( 2 r + 9 ) P ( y ) r
(2.11)
for all y X. Replacing y by x and letting g(x) := f(2x) - 8f(x) in (2.11), we get
g ( x ) - 1 2 g ( 2 x ) 2 r + 9 2 P ( x ) r
for all x X. Hence
1 2 l g ( 2 l x ) - 1 2 m g ( 2 m x ) j = l m - 1 ( 2 r + 9 ) 2 r j 2 j + 1 P ( x ) r
(2.12)
for all nonnegative integers m and l with m > l and all x X. It follows from (2.12) that the sequence { 1 2 k g ( 2 k x ) } is Cauchy for all x X. Since Y is complete, the sequence { 1 2 k g ( 2 k x ) } converges. So one can define the mapping A : XY by
A ( x ) : = lim k 1 2 k g ( 2 k x )

for all x X.

By (2.7),
D A ( x , y ) = lim k 1 2 k D g ( 2 k x , 2 k y ) 2 r k 2 k ( 2 r + 8 ) ( P ( x ) r + P ( y ) r ) = 0

for all x, y X. So DA(x, y) = 0. Since g : XY is odd, A : XY is odd. So the mapping A : XY is additive. Moreover, letting l = 0 and passing the limit m → ∞ in (2.12), we get (2.8). So there exists an additive mapping A : XY satisfying (2.8).

Now, let T : XY be another additive mapping satisfying (2.8). Then we have
A ( x ) - T ( x ) = 1 2 q A ( 2 q x ) - 1 2 q T ( 2 q x ) 1 2 q ( A ( 2 q x ) - g ( 2 q x ) ) + 1 2 q ( T ( 2 q x ) - g ( 2 q x ) ) 2 ( 9 + 2 r ) 2 r q ( 2 - 2 r ) 2 q P ( x ) r ,

which tends to zero as q → ∞ for all x X. So we can conclude that A(x) = T(x) for all x X. This proves the uniqueness of A. Thus the mapping A : XY is a unique additive mapping satisfying (2.8).

Theorem 2.3. Let r, θ be positive real numbers with r > 3, and let f : YX be an odd mapping satisfying (2.1). Then there exists a unique cubic mapping C : YX such that
P ( f ( 2 x ) - 2 f ( x ) - C ( x ) ) 2 r + 9 2 r - 8 θ x r

for all x Y.

Proof. Replacing y by x 2 and letting g(x) := f(2x) - 2f(x) in (2.5), we get
P g ( x ) - 8 g x 2 2 r + 9 2 r θ x r

for all x Y.

The rest of the proof is similar to the proof of Theorem 2.1.

Theorem 2.4. Let r be a positive real number with r < 3, and let f : XY be an odd mapping satisfying (2.7). Then there exists a unique cubic mapping C : XY such that
f ( 2 x ) - 2 f ( x ) - C ( x ) 9 + 2 r 8 - 2 r P ( x ) r

for all x X.

Proof. Replacing y by x and letting g(x) := f(2x) - 2f(x) in (2.11), we get
g ( x ) - 1 8 g ( 2 x ) 2 r + 9 8 P ( x ) r

for all x X.

The rest of the proof is similar to the proof of Theorem 2.2.

3. Hyers-Ulam stability of the functional equation (1.3): an even mapping case

In this section, we prove the Hyers-Ulam stability of the functional equation Df(x, y) = 0 in paranormed spaces: an even mapping case.

Note that P(2x) ≤ 2P(x) for all x Y.

Theorem 3.1. Let r, θ be positive real numbers with r > 2, and let f : YX be an even mapping satisfying f(0) = 0 and (2.1). Then there exists a unique quadratic mapping Q2 : YX such that
P ( f ( 2 x ) - 16 f ( x ) - Q 2 ( x ) ) 2 r + 9 2 r - 4 θ x r

for all x Y.

Proof. Letting x = y in (2.1), we get
P ( f ( 3 y ) - 6 f ( 2 y ) + 15 f ( y ) ) 2 θ y r
(3.1)

for all y Y.

Replacing x by 2y in (2.1), we get
P ( f ( 4 y ) - 4 f ( 3 y ) + 4 f ( 2 y ) + 4 f ( y ) ) ( 2 r + 1 ) θ y r
(3.2)

for all y Y.

By (3.1) and (3.2),
P ( f ( 4 y ) - 20 f ( 2 y ) + 64 f ( y ) ) P ( 4 ( f ( 3 y ) - 6 f ( 2 y ) + 15 f ( y ) ) ) + P ( f ( 4 y ) - 4 f ( 3 y ) + 4 f ( 2 y ) + 4 f ( y ) ) 4 P ( f ( 3 y ) - 6 f ( 2 y ) + 15 f ( y ) ) + P ( f ( 4 y ) - 4 f ( 3 y ) + 4 f ( 2 y ) + 4 f ( y ) ) 8 θ y r + ( 2 r + 1 ) θ y r = ( 2 p + 9 ) θ y r
(3.3)
for all y Y. Replacing y by x 2 and letting g(x) := f(2x) - 16f(x) in (3.3), we get
P g ( x ) - 4 g x 2 2 r + 9 2 r θ x r

for all x Y.

The rest of the proof is similar to the proof of Theorem 2.1.

Theorem 3.2. Let r be a positive real number with r < 2, and let f : XY be an even mapping satisfying f(0) = 0 and (2.7). Then there exists a unique quadratic mapping Q2 : XY such that
f ( 2 x ) - 16 f ( x ) - Q 2 ( x ) 9 + 2 r 4 - 2 r P ( x ) r
(3.4)

for all x X.

Proof. Letting x = y in (2.7), we get
f ( 3 y ) - 6 f ( 2 y ) + 15 f ( y ) 2 P ( y ) r
(3.5)

for all y X.

Replacing x by 2y in (2.7), we get
f ( 4 y ) - 4 f ( 3 y ) + 4 f ( 2 y ) + 4 f ( y ) ( 2 r + 1 ) P ( y ) r
(3.6)

for all y X.

By (3.5) and (3.6),
f ( 4 y ) - 20 f ( 2 y ) + 64 f ( y ) 4 ( f ( 3 y ) - 6 f ( 2 y ) + 15 f ( y ) ) + f ( 4 y ) - 4 f ( 3 y ) + 4 f ( 2 y ) + 4 f ( y ) 4 f ( 3 y ) - 6 f ( 2 y ) + 15 f ( y ) + f ( 4 y ) - 4 f ( 3 y ) + 4 f ( 2 y ) + 4 f ( y ) 8 P ( y ) r + ( 2 r + 1 ) P ( y ) r = ( 2 p + 9 ) P ( y ) r
(3.7)
for all y X. Replacing y by x and letting g(x) := f(2x) - 16f(x) in (3.7), we get
g ( x ) - 1 4 g ( 2 x ) 2 r + 9 4 P ( x ) r

for all x X.

The rest of the proof is similar to the proof of Theorem 2.2.

Theorem 3.3. Let r, θ be positive real numbers with r > 4, and let f : YX be an even mapping satisfying f(0) = 0 and (2.1). Then there exists a unique quartic mapping Q4 : YX such that
P ( f ( 2 x ) - 4 f ( x ) - Q 4 ( x ) ) 2 r + 9 2 r - 16 θ x r

for all x Y.

Proof. Replacing y by x 2 and letting g(x) := f(2x) - 4f(x) in (3.3), we get
P g ( x ) - 16 g x 2 2 r + 9 2 r θ x r

for all x Y.

The rest of the proof is similar to the proof of Theorem 2.1.

Theorem 3.4. Let r be a positive real number with r < 4, and let f : XY be an even mapping satisfying f(0) = 0 and (2.7). Then there exists a unique quartic mapping Q4 : XY such that
f ( 2 x ) - 4 f ( x ) - Q 4 ( x ) 9 + 2 r 16 - 2 r P ( x ) r

for all x X.

Proof. Replacing y by x and letting g(x) := f(2x) - 4f(x) in (3.7), we get
g ( x ) - 1 16 g ( 2 x ) 2 r + 9 16 P ( x ) r

for all x X.

The rest of the proof is similar to the proof of Theorem 2.2.

Let f o ( x ) : = f ( x ) - f ( - x ) 2 and f e ( x ) : = f ( x ) + f ( - x ) 2 . Then f o is odd and f e is even. f o , f e satisfy the functional equation (1.3). Let g o (x) := f o (2x) - 2f o (x) and h o (x) := f o (2x) - 8f o (x). Then f o ( x ) = 1 6 g o ( x ) - 1 6 h o ( x ) . Let g e (x) := f e (2x) - 4f e (x) and h e (x) := f e (2x) - 16f e (x). Then f e ( x ) = 1 12 g e ( x ) - 1 12 h e ( x ) . Thus
f ( x ) = 1 6 g o ( x ) - 1 6 h o ( x ) + 1 12 g e ( x ) - 1 12 h e ( x ) .
Theorem 3.5. Let r, θ be positive real numbers with r > 4. Let f : YX be a mapping satisfying f(0) = 0 and (2.1). Then there exist an additive mapping A : YX, a quadratic mapping Q2 : YX, a cubic mapping C : YX and a quartic mapping Q4 : YX such that
P ( 24 f ( x ) - 4 A ( x ) - 2 Q 2 ( x ) - 4 C ( x ) - 2 Q 4 ( x ) ) 4 ( 2 r + 9 ) 2 r - 2 + 2 ( 2 r + 9 ) 2 r - 4 + 4 ( 2 r + 9 ) 2 r - 8 + 2 ( 2 r + 9 ) 2 r - 16 θ x r

for all x Y.

Theorem 3.6. Let r be a positive real number with r < 1. Let f : XY be a mapping satisfying f(0) = 0 and (2.7). Then there exist an additive mapping A : XY, a quadratic mapping Q2 : XY, a cubic mapping C : XY and a quartic mapping Q4 : XY such that
24 f ( x ) - 4 A ( x ) - 2 Q 2 ( x ) - 4 C ( x ) - 2 Q 4 ( x ) 4 ( 2 r + 9 ) 2 - 2 r + 2 ( 2 r + 9 ) 4 - 2 r + 4 ( 2 r + 9 ) 8 - 2 r + 2 ( 2 r + 9 ) 16 - 2 r P ( x ) r

for all x X.

Declarations

Acknowledgements

This study was supported by the Daejin University Research Grants in 2012.

Authors’ Affiliations

(1)
Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea
(2)
Department of Mathematics, Daejin University, Kyeonggi, 487-711, Korea

References

  1. Fast H: Sur la convergence statistique. Colloq Math 1951, 2: 241–244.MathSciNetMATHGoogle Scholar
  2. Steinhaus H: Sur la convergence ordinaire et la convergence asymptotique. Colloq Math 1951, 2: 73–34.MathSciNetGoogle Scholar
  3. Fridy JA: On statistical convergence. Analysis 1985, 5: 301–313.MathSciNetView ArticleMATHGoogle Scholar
  4. Karakus S: Statistical convergence on probabilistic normed spaces. Math Commun 2007, 12: 11–23.MathSciNetMATHGoogle Scholar
  5. Mursaleen M: λ -statistical convergence. Math Slovaca 2000, 50: 111–115.MathSciNetMATHGoogle Scholar
  6. Mursaleen M, Mohiuddine SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J Comput Anal Math 2009, 233: 142–149. 10.1016/j.cam.2009.07.005MathSciNetView ArticleMATHGoogle Scholar
  7. Šalát T: On the statistically convergent sequences of real numbers. Math Slovaca 1980, 30: 139–150.MathSciNetMATHGoogle Scholar
  8. Kolk E: The statistical convergence in Banach spaces. Tartu Ul Toime 1991, 928: 41–52.MathSciNetGoogle Scholar
  9. Wilansky A: Modern Methods in Topological Vector Space. McGraw-Hill International Book Co., New York; 1978.Google Scholar
  10. Ulam SM: A Collection of the Mathematical Problems. Interscience Publication, New York; 1960.Google Scholar
  11. Hyers DH: On the stability of the linear functional equation. Proc Natl Acad Sci USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar
  12. Aoki T: On the stability of the linear transformation in Banach spaces. J Math Soc Jpn 1950, 2: 64–66. 10.2969/jmsj/00210064View ArticleMATHGoogle Scholar
  13. Rassias TM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1View ArticleMATHGoogle Scholar
  14. Găvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl 1994, 184: 431–436. 10.1006/jmaa.1994.1211MathSciNetView ArticleMATHGoogle Scholar
  15. Rassias TM: Problem 16; 2, in Report of the 27th International Symposium on Functional Equations. Aequationes Math 1990, 39: 292–293. 309Google Scholar
  16. Gajda Z: On stability of additive mappings. Int J Math Math Sci 1991, 14: 431–434. 10.1155/S016117129100056XMathSciNetView ArticleMATHGoogle Scholar
  17. Rassias TM, Šemrl P: On the behaviour of mappings which do not satisfy Hyers-Ulam stability. Proc Am Math Soc 1992, 114: 989–993. 10.1090/S0002-9939-1992-1059634-1View ArticleMATHGoogle Scholar
  18. Czerwik P: Functional Equations and Inequalities in Several Variables. World Scientific Publishing Company, New Jersey; 2002.Google Scholar
  19. Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.View ArticleGoogle Scholar
  20. Rassias JM: On approximation of approximately linear mappings by linear mappings. J Funct Anal 1982, 46: 126–130. 10.1016/0022-1236(82)90048-9MathSciNetView ArticleMATHGoogle Scholar
  21. Skof F: Proprietà locali e approssimazione di operatori. Rend Sem Mat Fis Milano 1983, 53: 113–129. 10.1007/BF02924890MathSciNetView ArticleMATHGoogle Scholar
  22. Cholewa PW: Remarks on the stability of functional equations. Aequationes Math 1984, 27: 76–86. 10.1007/BF02192660MathSciNetView ArticleMATHGoogle Scholar
  23. Czerwik S: On the stability of the quadratic mapping in normed spaces. Abh Math Sem Univ Hamburg 1992, 62: 59–64. 10.1007/BF02941618MathSciNetView ArticleMATHGoogle Scholar
  24. Aczel J, Dhombres J: Functional Equations in Several Variables. Cambridge University Press, Cambridge; 1989.View ArticleGoogle Scholar
  25. Eshaghi Gordji M, Savadkouhi MB: Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces. Appl Math Lett 2010, 23: 1198–1202. 10.1016/j.aml.2010.05.011MathSciNetView ArticleMATHGoogle Scholar
  26. Isac G, Rassias TM: On the Hyers-Ulam stability of ψ -additive mappings. J Approx Theory 1993, 72: 131–137. 10.1006/jath.1993.1010MathSciNetView ArticleMATHGoogle Scholar
  27. Jun K, Lee Y: A generalization of the Hyers-Ulam-Rassias stability of the pexiderized quadratic equations. J Math Anal Appl 2004, 297: 70–86. 10.1016/j.jmaa.2004.04.009MathSciNetView ArticleMATHGoogle Scholar
  28. Jung S: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press lnc., Palm Harbor, FL; 2001.Google Scholar
  29. Park C: Homomorphisms between Poisson JC *-algebras. Bull Braz Math Soc 2005, 36: 79–97. 10.1007/s00574-005-0029-zMathSciNetView ArticleMATHGoogle Scholar
  30. Rassias JM: Solution of a problem of Ulam. J Approx Theory 1989, 57: 268–273. 10.1016/0021-9045(89)90041-5MathSciNetView ArticleMATHGoogle Scholar
  31. Jun K, Kim H: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. J Math Anal Appl 2002, 274: 867–878. 10.1016/S0022-247X(02)00415-8MathSciNetView ArticleMATHGoogle Scholar
  32. Lee S, Im S, Hwang I: Quartic functional equations. J Math Anal Appl 2005, 307: 387–394. 10.1016/j.jmaa.2004.12.062MathSciNetView ArticleMATHGoogle Scholar
  33. Eshaghi-Gordji M, Kaboli-Gharetapeh S, Park C, Zolfaghari S: Stability of an additive-cubic-quartic functional equation. Adv Diff Equ 2009., 2009: Article ID 395693, 20Google Scholar
  34. Eshaghi-Gordji M, Abbaszadeh S, Park C: On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces. J Inequal Appl 2009., 2009: Article ID 153084, 26Google Scholar

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