Hyperstability of Some Functional Equations on Restricted Domain

Abstract

The paper concerns functions which approximately satisfy, not necessarily on the whole linear space, a generalization of linear functional equation. A Hyers-Ulam stability result is proved and next applied to give conditions implying the hyperstability of the equation. The results may be used as tools in stability studies on restricted domains for various functional equations. We use the main theorem to obtain a few hyperstability results of Fréchet equation on restricted domain for different control functions.

1. Introduction

The famous Cauchy equationhas a very long history, which started with A. M. Legendre in 1791 and C. F. Gauss in 1809, but the first important result about the solutions of (1) was proved by Cauchy (see [1]) and due to this outcome (1) is today named after him, as is mentioned in [2]. In the meantime, many authors have dealt with this equation, and next Fréchet in [3] studied an important generalization of Cauchy equation, characterizing the polynomials among the continuous mappings, namely, the equationwhere denotes the difference operator defined as usual by for . From the result of Fréchet it follows that a continuous function satisfies (2) for all and if and only if is a polynomial of degree smaller than or equal to (cf. [2, 4]). As a generalization we can consider the functional equationwhere and are the linear spaces over a field , , and .

The stability and hyperstability of the above functional equation and its particular cases were studied by many mathematicians (cf., e.g., [5–16]). Let us recall (see [9]) that the theory of Hyers-Ulam stability, in general, was motivated by a problem raised in 1940 by Ulam [17] and a partial answer to it was provided by Hyers [18]. The term hyperstability, in the meaning used here, was introduced presumably in [19] (cf. also [20]).

Our basic assumptions are the following. is a normed space over a field , are the nonempty subsets of such that , and is a nonempty subset of such that . Denote and , for , and a sum of numbers over an empty set is defined to be zero. In the sequel, for nonempty set .

We say that the equation is -hyperstable in the class of functions (with a control function ), if satisfies the inequalityfor all , and fulfills (5).

The above notion of hyperstability for functions defined on subsets of a linear space (not necessarily on the whole space but on the restricted domain) is a counterpart of hyperstability considered in [8]. In this paper we obtain, applying the fixed point theorem, some results concerning stability and hyperstability for (4) on restricted domain. Our outcomes allow studying the behavior of functions approximately fulfilling functional equations of this linear type on different subsets of the linear space, and this also applies to domain restrictions for control functions. This way we give a tool for the further Hyers-Ulam stability research. Its application is presented in the last section where some new as well as generalizations of known results on hyperstability of Cauchy and Fréchet equation are obtained.

2. Auxiliary Result

First we establish a counterpart of stability result from [7] on restricted domain, and to do this we use the fixed point theorem. This result will be needed to show our main theorem which provides criteria for hyperstability of many functional equations of the linear type on a restricted domain.

Lemma 1. Assume that is a Banach space, or “ and ,” and . Suppose that there exist and such that Let satisfy the inequalityIf fulfills the estimationthen there exists a unique function satisfying (5) such that

Proof. Take and such that assumptions (7) and (8) hold.
First consider the case . Taking and substituting , , in (9) we haveDefineThen (11) may be rewritten as follows:By induction, for every and Since , we see thatNotice that all assumptions of Theorem in [21] are satisfied for operators (12) and (13) and . According to this fixed point theorem, the function given by for is a fixed point of fulfilling (10).
It is sufficient to show that satisfies (5) (with ). First we prove that for every and every such that Obviously, the case is just (9). Next, fix , , and and assume that (18) holds. Then Thus by induction hypothesis and (8) we have which finishes the proof of (18). Letting in (18), we prove that satisfies (5), which completes the proof in the first case.
In the case where and , the function , , satisfiesfor , and . Consequently, we conclude that there exists a unique function such thatObviously and is the desired function.

Denote

Remark 2. It is easily seen that functions (i);(ii);(iii);(iv);(v);with some , , , and fulfill condition (8) for some function . For instance, in case (iv) we can use for , and in case (v) we may take where . If there exists at least one such that or then

3. The Main Result

We can now state a counterpart (on restricted domain) of Theorem in [8], concerning -hyperstability of (5).

Theorem 3. Let be a normed space over , or “ and ,” , , and satisfy (9). If there exist and a sequence of elements of such thatthen satisfies (5).

Proof. We can certainly assume that is a Banach space, for if not we replace it by its completion.
Take and the sequence of the elements of such that conditions (25), (26), and (27) hold. Hence we can find and such thatTherefore, according to Lemma 1, for each there exists a unique function satisfying equation (5) such thatIn this way we obtain the sequence of functions satisfying (5). Letting in (30) , we get, using (28), that satisfies (5).

4. Applications

Using the above outcome we can get the results for the particular form of (5), as, for instance, for Fréchet functional equation, presented in this section.

Corollary 4. Assume that is a normed space over , , is a nonempty subset of such that there exists a positive integer withIf fulfills the inequalitywith and , then is a solution of the equation

Proof. Put , and for and consider two cases: (a),(b).In the case (a) we define the sequence as follows: It follows easily that the assumptions of Theorem 3 are satisfied with and .
In case (b) we can find such that Define There exists such that and for We will verify that we can apply Theorem 3 for . Namely, Therefore Thus conditions (26) and (28) hold, which finishes the proof.

Corollary 5. Assume that is a normed space over , , and is a nonempty subset of such that there exists a positive integer with (31).
If fulfills the inequalitywith and , for , then is a solution of (33).

Proof. Putting , , and for , where , , andwe get that and therefore for , and consequently condition (26) is satisfied. Also, condition (28) holds, because which with Theorem 3 completes the proof.

Denote

Corollary 6. Assume that is a normed space over , , and is a nonempty subset of such that there exists a positive integer withIf fulfills the inequalitywith and , then is a solution of the equation

Proof. Put , , and for There exist and , such that , , and for . Define the sequence as follows: The assumptions of Theorem 3 are fulfilled with and , as is easy to check, which finishes the proof.

In the same way as Corollary 5 we can prove the following corollary.

Corollary 7. Assume that is a normed space over , , and is a nonempty subset of such that there exists a positive integer with (31).
If fulfills the inequalitywith and for , then is a solution of (45).

Remark 8. Observe that putting in the above two corollaries we get hyperstability results for Cauchy equation. Corollary 7 generalizes the result from [9] (see Theorem ), where and (in our considerations may be different). Corollaries 6 and 7 generalize the outcomes from [6].

We end the paper with an example, which shows that the additional assumption on , appearing in the above corollaries, is essential.

Example 9. Let , , and be defined by , . Then satisfies the estimations for , and but satisfies neither (33) nor (45).

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.