Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 912718, 4 pages

http://dx.doi.org/10.1155/2013/912718

Research Article

Hyperstability of the Drygas Functional Equation

Magdalena Piszczek and Joanna Szczawińska

Institute of Mathematics, Pedagogical University, Podchorążych 2, 30-084 Kraków, Poland

Received 16 May 2013; Accepted 7 July 2013

Academic Editor: Alberto Fiorenza

Copyright © 2013 Magdalena Piszczek and Joanna Szczawińska. 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 use the fixed point theorem for functional spaces to obtain the hyperstability result for the Drygas functional equation on a restricted domain. Namely, we show that a function satisfying the Drygas equation approximately must be exactly the solution of it.

1. Introduction

We say that a function satisfies the Drygas equation if

The above equation was introduced in [1] in order to obtain a characterization of the quasi-inner-product spaces. The general solution of (1), obtained by Ebanks et al. in [2] (see also [3]), is of the following form:

where is an additive function and is a quadratic function; that is, satisfies the quadratic functional equation

A set-valued version of (1) was considered by Smajdor in [4].

The stability in the Hyers-Ulam sense of the Drygas equation has been investigated, for example, in [5–8]. For more information and numerous references on the stability of functional equations in general, we refer to [9, 10].

The term hyperstability was used for the first time probably in [11]; however, it seems that the first hyperstability result was published in [12] and concerned the ring homomorphisms. The hyperstability results for the Cauchy equation were investigated by Brzdȩk in [13, 14]. Gselmann in [15] studied the hyperstability of the parametric fundamental equation of information.

In the paper, we prove the hyperstability of the Drygas equation on a restricted domain. Let be a nonempty subset of an Abelian group and a semigroup. We say that a function satisfies the Drygas functional equation on if

for all such that . We will show that (4) is hyperstable; that is, each function (under some additional assumptions on ) satisfying the inequality

for all such that with must satisfy the Drygas equation (4).

The method of the proof of the main theorem is motivated by an idea used by Brzdȩk in [13] and further by Piszczek in [16]. It is based on a fixed point theorem for functional spaces obtained by Brzdęk et al. (see [17, Theorem  1]). Some generalizations of their result were proved by Cădariu et al. in [18]. The case of fixed point theorem for non-Archimedean metric spaces was also studied by Brzdęk and Ciepliński in [19]. It is worth mentioning that using fixed point theorems is now one of the most popular methods of investigating the stability of functional equations in single as well as in several variables. Let us recall a few recent approaches of Jung in [20], Lee and Jung in [21], or Cădariu and Radu in [22, 23]. More information on the application of the fixed point method was collected by Ciepliński in [24].

First, we take the following three hypotheses (all notations come from [17]).

(H1) is a nonempty set, is a Banach space, and and are given.(H2) is an operator satisfying the inequality

for all , . (H3) is defined by

The mentioned fixed point theorem is stated as follows.

Theorem 1. Let hypotheses (H1)–(H3) be valid and functions and let fulfil the following two conditions:

Then, there exists a unique fixed point of with

Moreover,

Throughout the paper, , , and denote the set of all positive integers, the set of all nonnegative integers, and the set of all integers greater than or equal to , respectively.

2. The Main Result

Theorem 2. Assume that is a nonempty, symmetric with respect to subset of a normed space such that and there exists with for , . Let be a Banach space, , and . If satisfies

for all such that , then satisfies the Drygas equation on .

Proof. First observe that there exists such that

Let us fix . Replacing with and with in (11), we get

Further put

Inequality (13) now takes the form

The operator

has the form described in (H3) with and , , , , , and for . Moreover, for every ,  ,

As , we have

Thus, according to Theorem 1 there exists a unique solution of the equation

such that

Moreover,

To prove that satisfies the Drygas equation on , observe that

for every ,  , and .

Indeed, if , then (22) is simply (11). So, fix and suppose that (22) holds for and such that . Then

By induction, we have shown that (22) holds for all ,  , and . Letting in (22), we obtain

Thus, we have proved that for every there exists a function such that is a solution of the Drygas equation on and

Since , the sequence tends to zero. Consequently, satisfies the Drygas equation on as the pointwise limit of .

Remark 3. If is a normed space and satisfies (11) for , with , then by Theorem 2 we know that satisfies the Drygas equation on . It is easy to see that if , then satisfies the Drygas equation on the whole .

We end with two examples. First notice that the assumption of unboundedness of the set is necessary.

Example 4. Let and let be defined by , . Then for all such that ,

with , but does not satisfy the Drygas equation on .

In the case , the considered Drygas equation is not hyperstable.

Example 5. Let and let be a constant function , for some . Then satisfies the inequality

for all such that , with , but is not a solution of the Drygas equation on .

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