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Abstract and Applied Analysis
Volume 2009, Article ID 417473, 14 pages
http://dx.doi.org/10.1155/2009/417473
Research Article

Solution and Stability of a Mixed Type Cubic and Quartic Functional Equation in Quasi-Banach Spaces

1Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran
2Section of Mathematics and Informatics, Pedagogical Department, National and Capodistrian University of Athens, 4, Agamemnonos Street, Aghia Paraskevi, 15342 Athens, Greece

Received 24 January 2009; Accepted 6 August 2009

Academic Editor: Elena Litsyn

Copyright © 2009 M. Eshaghi Gordji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We obtain the general solution and the generalized Ulam-Hyers stability of the mixed type cubic and quartic functional equation in quasi-Banach spaces.

1. Introduction

We recall some basic facts concerning quasiBanach space. A quasinorm is a real-valued function on satisfying the following.

(1) for all and if and only if (2) for all and all (3)There is a constant such that for all

The pair is called a quasinormed space if is a quasinorm on . A quasiBanach space is a complete quasinormed space. A quasinorm is called a -norm if for all In this case, a quasiBanach space is called a -Banach space. Given a -norm, the formula gives us a translation invariant metric on . By the Aoki-Rolewicz theorem [1] (see also [2]), each quasinorm is equivalent to some -norm. Since it is much easier to work with -norms, henceforth we restrict our attention mainly to -norms. The stability problem of functional equations originated from a question of Ulam [3] in concerning the stability of group homomorphisms. Let be a group and let be a metric group with the metric Given , does there exist a , such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all In the other words, Under what condition does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In Hyers [4] gave the first affirmative answer to the question of Ulam for Banach spaces. Let be a mapping between Banach spaces such that for all and for some Then there exists a unique additive mapping such that for all Moreover if is continuous in for each fixed then is linear. Rassias [5] succeeded in extending the result of Hyers' Theorem by weakening the condition for the Cauchy difference controlled by , to be unbounded. This condition has been assumed further till now, through the complete Hyers direct method, in order to prove linearity for generalized Hyers-Ulam stability problem forms. A number of mathematicians were attracted to the pertinent stability results of Rassias [6], and stimulated to investigate the stability problems of functional equations. The stability phenomenon that was introduced and proved by Rassias is called Hyers-Ulam-Rassias stability. And then the stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [4, 5, 718]).

The following cubic functional equation, which is the oldest cubic functional equation, was introduced by the third author of this paper, Rassias [6] (in 2001): Jun and Kim [19] introduced the following cubic functional equation: and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.5) The function satisfies the functional equation (1.5) which is thus called a cubic functional equation. Every solution of the cubic functional equation is said to be a cubic function. Jun and Kim proved that a function between real vector spaces and is a solution of (1.5) if and only if there exists a unique function such that for all and is symmetric for each fixed one variable and is additive for fixed two variables (see also [20]).

The quartic functional equation (1.6) was introduced by Rassias [21] (in 2000) and then (in 2005) was employed by Park and Bae [22] and others, such that: In fact they proved that a function between real vector spaces and is a solution of (1.6) if and only if there exists a unique symmetric multiadditive function such that for all (see also [2129]). It is easy to show that the function satisfies the functional equation (1.6) which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic function. In this paper we deal with the following functional equation: in quasiBanach spaces. It is easy to see that the function is a solution of the functional equation (1.7) In the present paper we investigate the general solution of functional equation (1.7) when is a mapping between vector spaces, and we establish the generalized Hyers-Ulam-Rassias stability of the functional equation (1.7) whenever is a mapping between two quasiBanach spaces. We only mention here the papers [30, 31] concerning the stability of the mixed type functional equations.

2. General Solution

Throughout this section, and will be real vector spaces. Before proceeding to the proof of Theorem 2.3 which is the main result in this section, we shall need the following two lemmas.

Lemma 2.1. If an even function satisfies (1.7) then is quartic.

Proof. Putting in (1.7) we get . Setting in (1.7) by evenness of we obtain for all Hence (1.7) can be written as This means that is quartic function, which completes the proof of the lemma.

Lemma 2.2. If an odd function satisfies (1.7) then f is a cubic function.

Proof. Setting in (1.7) gives Putting in (1.7) then by oddness of , we have Hence (1.7) can be written as Replacing by in (2.4) we obtain Substituting for in (2.5) gives If we subtract (2.5) from (2.6) we obtain Let us interchange and in (2.7) Then we see that With the substitution in (2.4) we have From the substitution in (2.9) it follows that If we add (2.9) to (2.10) we have Replacing by in (2.7) and using (2.3), we obtain Interchanging with in (2.12) gives the equation If we compare (2.11) and (2.13) and employ (2.4) we conclude that This means that is cubic function. This completes the proof of Lemma.

Theorem 2.3. A function satisfies (1.7) for all if and only if there exists a unique function and a unique symmetric multiadditive function such that for all and that is symmetric for each fixed one variable and is additive for fixed two variables.

Proof. Let satisfy (1.7) We decompose into the even part and odd part by setting for all By (1.7) we have for all This means that satisfies in (1.7) Similarly we can show that satisfies (1.7) By Lemmas 2.1 and 2.2, and are quartic and cubic, respectively. Thus there exists a unique function and a unique symmetric multiadditive function such that and that for all and is symmetric for each fixed one variable and is additive for fixed two variables. Thus for all The proof of the converse is trivial.

3. Stability

Throughout this section, and will be a uniquely two-divisible abelian group and a quasiBanach spaces respectively, and will be a fixed real number in . We need the following lemma in the main theorems. Now before taking up the main subject, given , we define the difference operator by for all We consider the following functional inequality: for an upper bound

Lemma 3.1. Let be nonnegative real numbers. Then

Theorem 3.2. Let be fixed and let be a function such that for all and for all . Suppose that an even function with satisfies the inequality for all Then the limit exists for all and is a unique quartic function satisfying where for all

Proof. Let By putting in (3.6), we get for all Replacing by in (3.10) yields for all Let for all then by (3.11), we get for all Interchanging with in (3.12), and multiplying by it follows that for all and all nonnegative integers . Since is -Banach space, then by (3.13) we have for all nonnegative integers and with and all Since for all Therefore by (3.5) we have for all Therefore we conclude from (3.14) and (3.15) that the sequence is a Cauchy sequence for all Since is complete, it follows that the sequence converges for all We define the mapping by (3.7) for all Letting and passing the limit in (3.14), we get for all Therefore (3.8) follows from (3.9) and (3.16). Now we show that is quartic. It follows from (3.4), (3.6) and (3.7) for all Therefore the mapping satisfies (1.7). Since then by Lemma 2.1 we get that the mapping is quartic. To prove the uniqueness of let be another quartic mapping satisfies (3.8). Since for all and all then for all It follows from (3.8), (3.19) for all Hence For , we obtain from which one can prove the result by a similar technique.

Corollary 3.3. Let be nonnegative real numbers such that Suppose that an even function with satisfies the inequality for all Then there exists a unique quartic function satisfying for all

Proof. It follows from Theorem 3.2that for all

Theorem 3.4. Let be fixed and let be a function such that for all and for all . Suppose that an odd function satisfies the inequality for all Then the limit exists for all and is a unique cubic function satisfying for all where

Proof. Let Setting in (3.26), we get for all If we replace in (3.30) by and divide both sides of (3.30) by 3, we get for all Let for all then by (3.31), we get for all Multiply (3.32) by and replace by we obtain that for all and all nonnegative integers . Since is a -Banach space, (3.33) follows that for all nonnegative integers and with and all Since for all Therefore it follows from (3.25) that for all therefore we conclude from (3.34) and (3.35) that the sequence is a Cauchy sequence for all Since is complete, the sequence converges for all So one can define the mapping by (3.27) for all Letting and passing the limit in (3.34) we get for all Therefore (3.28) follows from (3.29) and (3.36) Now we show that is cubic. It follows from (3.24) (3.26) and (3.27) for all Therefore the mapping satisfies (1.7) Since is an odd function, then (3.27) implies that the mapping odd. Therefore by Lemma 2.2 we get that the mapping is cubic. The rest of proof is similar to the proof of Theorem 3.2.

Corollary 3.5. Let be a nonnegative real number and be real numbers such that Suppose that an odd function satisfies the inequality for all Then there exists a unique cubic function satisfying for all

Proof. It follows from (3.38) and Theorem 3.4that for all

Theorem 3.6. Let be fixed and let be a function which satisfies for all and for all . Suppose that a function with satisfies the inequality for all Then there exists a unique quartic function and a unique cubic function satisfying (1.7) and for all where and that have been defined in (3.9) and (3.29), respectively.

Proof. Let for all Then and for all Let for all So for all Since for all then for all and all Hence, in view of Theorem 3.2, there exists a unique quartic function satisfying for all where We have for all Therefore it follows from (3.48) that, for all Let for all Then and for all From Theorem 3.4, it follows that there exists a unique cubic function satisfying for all where Since for all it follows from (3.52) that, for all Hence (3.43) follows from (3.51) and (3.55).

Corollary 3.7. Let be nonnegative real numbers such that Suppose that a function with satisfies the inequality for all Then there exists a unique quartic function and a unique cubic function satisfying (1.7) and for all

Proof. It follows from Theorem 3.6that for all

Acknowledgment

The second and fourth authors would like to thank the office of gifted students at Semnan University for its financial support.

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