Skip to main content

An AQCQ-functional equation in paranormed spaces

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.

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 xp+ ypby xp· yqfor 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.

References

  1. Fast H: Sur la convergence statistique. Colloq Math 1951, 2: 241–244.

    MathSciNet  MATH  Google Scholar 

  2. Steinhaus H: Sur la convergence ordinaire et la convergence asymptotique. Colloq Math 1951, 2: 73–34.

    MathSciNet  Google Scholar 

  3. Fridy JA: On statistical convergence. Analysis 1985, 5: 301–313.

    Article  MathSciNet  MATH  Google Scholar 

  4. Karakus S: Statistical convergence on probabilistic normed spaces. Math Commun 2007, 12: 11–23.

    MathSciNet  MATH  Google Scholar 

  5. Mursaleen M: λ -statistical convergence. Math Slovaca 2000, 50: 111–115.

    MathSciNet  MATH  Google 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.2009.07.005

    Article  MathSciNet  MATH  Google Scholar 

  7. Šalát T: On the statistically convergent sequences of real numbers. Math Slovaca 1980, 30: 139–150.

    MathSciNet  MATH  Google Scholar 

  8. Kolk E: The statistical convergence in Banach spaces. Tartu Ul Toime 1991, 928: 41–52.

    MathSciNet  Google 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.27.4.222

    Article  MathSciNet  Google 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/00210064

    Article  MATH  Google Scholar 

  13. Rassias TM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1

    Article  MATH  Google 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.1994.1211

    Article  MathSciNet  MATH  Google Scholar 

  15. Rassias TM: Problem 16; 2, in Report of the 27th International Symposium on Functional Equations. Aequationes Math 1990, 39: 292–293. 309

    Google Scholar 

  16. Gajda Z: On stability of additive mappings. Int J Math Math Sci 1991, 14: 431–434. 10.1155/S016117129100056X

    Article  MathSciNet  MATH  Google 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/S0002-9939-1992-1059634-1

    Article  MATH  Google 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.

    Chapter  Google 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)90048-9

    Article  MathSciNet  MATH  Google Scholar 

  21. Skof F: Proprietà locali e approssimazione di operatori. Rend Sem Mat Fis Milano 1983, 53: 113–129. 10.1007/BF02924890

    Article  MathSciNet  MATH  Google Scholar 

  22. Cholewa PW: Remarks on the stability of functional equations. Aequationes Math 1984, 27: 76–86. 10.1007/BF02192660

    Article  MathSciNet  MATH  Google 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/BF02941618

    Article  MathSciNet  MATH  Google Scholar 

  24. Aczel J, Dhombres J: Functional Equations in Several Variables. Cambridge University Press, Cambridge; 1989.

    Chapter  Google 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.2010.05.011

    Article  MathSciNet  MATH  Google 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.1993.1010

    Article  MathSciNet  MATH  Google 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.2004.04.009

    Article  MathSciNet  MATH  Google 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-z

    Article  MathSciNet  MATH  Google Scholar 

  30. Rassias JM: Solution of a problem of Ulam. J Approx Theory 1989, 57: 268–273. 10.1016/0021-9045(89)90041-5

    Article  MathSciNet  MATH  Google 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)00415-8

    Article  MathSciNet  MATH  Google Scholar 

  32. Lee S, Im S, Hwang I: Quartic functional equations. J Math Anal Appl 2005, 307: 387–394. 10.1016/j.jmaa.2004.12.062

    Article  MathSciNet  MATH  Google 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, 20

    Google 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, 26

    Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jung Rye Lee.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Park, C., Lee, J.R. An AQCQ-functional equation in paranormed spaces. Adv Differ Equ 2012, 63 (2012). https://doi.org/10.1186/1687-1847-2012-63

Download citation

  • Received:

  • Accepted:

  • Published:

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

Keywords

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