A Fixed Point Approach to the Stability of an Additive-Quadratic-Cubic-Quartic Type Functional Equation

Abstract

By applying a fixed point approach, we investigate the stability problems for an AQCQ functional equation of the form

1. Introduction

In 1940, Ulam [1] posed the problem concerning the stability of group homomorphisms. In the following year, Hyers [2] gave an affirmative answer to Ulam’s problem for additive mappings between Banach spaces. Thereafter, many mathematicians came to deal with this subject (cf. [3, 4]).

We now consider the following functional equation:The mappings are solutions of this functional equation, where are real constants.

The stability problems of (1) were investigated by Lee et al. [5]. Indeed, they proved the stability of the equation by dividing the relevant mapping into the additive, quadratic, cubic, and quartic parts and applying the fixed point method to each part separately. Unfortunately, they could not prove the uniqueness of the exact solution of (1) which approximates the (given) mapping that is a solution of the perturbed equation of (1).

In this paper, we will show that every solution of functional equation (1) is an AQCQ mapping (see below for the definition of AQCQ mapping) and we introduce a strictly contractive mapping which allows us to use the fixed point theory in the sense of Cădariu and Radu (see [6–8]). In a different way from Lee et al. [5], we do not split the (given) mapping which satisfies the perturbed equation of (1) and we apply the fixed point method to construct the explicit form of the exact solution of (1) near the (given) mapping by or where we denote by and the odd and even parts of , respectively.

Using this method, we can prove the uniqueness of the exact solution of (1) near the (given) mapping.

Throughout this paper, let denote the set of all positive integers and let denote the set of all nonnegative integers.

2. Preliminaries

We recall the fixed point theorem of Diaz and Margolis [9].

Theorem 1. Assume that is a complete generalized metric space and is a strictly contractive mapping with the Lipschitz constant . It then holds that, for each , either for all or there exists such that (i) for all ;(ii)the sequence is convergent to a fixed point of ;(iii) is the unique fixed point of in ;(iv) for all .

Throughout this paper, let and be real vector spaces, let be a real normed space, and let be a real Banach space.

For a given mapping , we use the following abbreviations: for all . Each solution of , , , and is called an additive mapping, a quadratic mapping, a cubic mapping, and a quartic mapping, respectively. Every mapping is called an AQCQ mapping (additive-quadratic-cubic-quartic mapping) if can be expressed by the sum of an additive mapping, a quadratic mapping, a cubic mapping, and a quartic mapping.

The following lemmas have been proved in [10].

Lemma 2 ([10], Theorem 2.3). If is an odd mapping satisfying for all , then there exists a cubic mapping and an additive mapping such that for all .

Lemma 3 ([10], Theorem 2.4). If is an even mapping satisfying the equalities for all and for all , then is a quadratic mapping.

In the following theorem, we investigate the general solution of the functional equation .

Theorem 4. A mapping satisfies for all if and only if is an AQCQ mapping.

Proof. First, we assume that a mapping satisfies for all . Then, the odd part satisfies the equalities for all . It follows from Lemma 2 that is a cubic-additive mapping.
Let and for all . By a direct calculation and using the assumption that for all , we obtain andthat is,for all . Using these equalities, we further get the equalities and which imply the equalities and for all . Hence, by (7) and the fact that , Lemma 3 implies that is quadratic. Moreover, is quartic because of for all . Since for all , is an AQCQ mapping.
Conversely, assume that are mappings such that the equalities , , , , and hold for all . Then the equalities , , , , , , , and hold for all . From the above equalities, we obtain the equalities for all , which imply that for all .

3. Main Results

Throughout this section, let be a real vector space and let be a real Banach space.

In the following theorems, we can prove the generalized Hyers-Ulam stability of the functional equation (1) by using a fixed point approach.

Theorem 5. Let be a mapping, for which there exists a mapping such that the inequalityholds for all , , and let . If there exists a constant such that satisfies the conditionfor all , , then there exists a unique solution of (1) satisfying the inequalityfor all , where . In particular, is represented byfor all .

Proof. Let be the set of all functions with . We introduce a generalized metric on by It is not difficult to see that is a generalized complete metric space (see [11, 12]).
We now consider the mapping , which is defined by for all . We assert that the equalityholds for all and . Equality (17) is true if . We assume that (17) is true for some . Then we havefor all and , which we can obtain from (17) provided we replace with .
Let and let be an arbitrary constant with . From the definition of , we havefor all , which implies that for any . That is, is a strictly contractive self-mapping of with the Lipschitz constant .
Moreover, by a long calculation and (11), we see thatfor all . It means that by the definition of . Therefore, according to Theorem 1 and , the sequence converges to the unique fixed point of in the set , which is represented by (14) for all .
Notice that which implies (13).
By the definition of , together with (11) and (12), we havefor all ; that is, is a solution of the functional equation (1). Notice that if is a solution of the functional equation (1), then the equality implies that is a fixed point of .

In the following theorem, we replace condition (12) for with another (25) and prove the generalized Hyers-Ulam stability of AQCQ equation (1) by applying a fixed point approach.

Theorem 6. Let be a mapping, for which there exists a mapping such that inequality (11) holds for all , and let . If there exists a constant such that satisfies the conditionfor all , then there exists a unique solution of (1) satisfying the inequalityfor all . In particular, is represented byfor all .

Proof. Let be the set defined in the proof of Theorem 5. We now consider the mapping defined by for all . We assert that the equalityholds for all and . Equality (29) is obviously true when . We assume that (29) holds for some . Then we havefor all and , which we get from (29) when we replace by .
Let and let be an arbitrary constant with . From the definition of , we have for all , which implies that for any . That is, is a strictly contractive self-mapping of with the Lipschitz constant .
Moreover, by (11), we see thatfor all , which implies that by the definition of .
Therefore, according to Theorem 1, the sequence converges to the unique fixed point of in the set , which is represented by (27) for all . Notice that which implies (26).
By definition of , together with (11) and (25), we havefor all ; that is, is a solution of the functional equation (1). Notice that if is a solution of the functional equation (1), then the equality implies that is a fixed point of .

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final paper.

Acknowledgments

Soon-Mo Jung was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2005557).