Hyperstability of the -Cubic Functional Equation in Non-Archimedean Banach Spaces

Abstract

Using the fixed point approach, we investigate a general hyperstability results for the following -cubic functional equations where is a fixed positive integer , in ultrametric Banach spaces.

1. Introduction

The starting point of studying the stability of functional equations seems to be the famous talk of Ulam [1] in 1940, in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms.

Ulam’s problem: let be a group and let be a metric group with a metric . Given , there exists such that if a mapping satisfies the inequalityfor all , then there exists a homomorphism withfor all .

The first partial answer, in the case of Cauchy equation in Banach spaces, to Ulam question was given by Hyers [2]. Later, the result of Hyers was first generalized by Aoki [3], and only much later by Rassias [4] and Găvruţa [5]. Since then, the stability problems of several functional equations have been extensively investigated [6–10].

We say a functional equation is hyperstable if any function satisfying the equation approximately (in some sense) must be actually a solution to it. It seems that the first hyperstability result was published in [11] and concerned the ring homomorphisms. However, the term hyperstability has been used for the first time in [12]. Quite often, hyperstability is confused with superstability, which also admits bounded functions. Numerous papers on this subject have been published, and we refer, for example, to [3, 12–29].

Throughout this paper, stands for the set of all positive integers and , the set of integers greater than or equal , , and we use the notation for the set .

Let us recall (see, for instance, [30]) some basic definitions and facts concerning non-Archimedean normed spaces.

Definition 1. By a non-Archimedean field, we mean a field equipped with a function (valuation) such that, for all , the following conditions hold:(1) if and only if (2)(3)The pair is called a valued field.
In any non-Archimedean field, we have and , for . In any field , the function given byis a valuation which is called trivial, but the most important examples of non-Archimedean fields are -adic numbers which have gained the interest of physicists for their research in some problems coming from quantum physics, -adic strings, and superstrings.

Definition 2. Let be a vector space over a scalar field with a non-Archimedean nontrivial valuation . A function is a non-Archimedean norm (valuation) if it satisfies the following conditions:(1) if and only if ,(2),(3)The strong triangle inequality (ultrametric), namely,Then, is called a non-Archimedean normed space or an ultrametric normed space.

Definition 3. Let be a sequence in a non-Archimedean normed space .(1)A sequence in a non-Archimedean space is a Cauchy sequence iff the sequence converges to zero.(2)The sequence is said to be convergent if there exists such that, for any , there is a positive integer such that , for all . Then, the point is called the limit of the sequence , which is denoted by .(3)If every Cauchy sequence in converges, then the non-Archimedean normed space is called a non-Archimedean Banach space or an ultrametric Banach space.Let and be normed spaces. A function is called a cubic function provided it satisfies the functional equation:and we can say that is cubic on if it satisfies (5) for all .
In 2013, Bahyrycz et al. [31] used the fixed point theorem from Theorem 1 in [24] to prove the stability results for the generalization of -Wright affine equation in ultrametric spaces. Recently, corresponding results for more general functional equations (in classical spaces) have been proved in [32–35].
In this paper, by using the fixed point method derived from [20, 21, 36], we present some hyperstability results for equation (5) in ultrametric Banach spaces. Before proceeding to the main results, we state Theorem 1 which is useful for our purpose. To present it, we introduce the following three hypotheses: (H1): is a nonempty set, is an ultrametric Banach space over a non-Archimedean field, , and are given. (H2): is an operator satisfying the inequality (H3): is a linear operator defined byThanks to a result due to Brzdçk and Ciepliński ([25], Remark 2), we state a slightly modified version of the fixed point theorem ([24], Theorem 1) in ultrametric spaces. We use it to assert the existence of a unique fixed point of operator .

Theorem 1. Let hypotheses ()–() be valid, and functions and fulfill the following two conditions:

Then, there exists a unique fixed point of with

Moreover,

2. Main Results

In this section, we use Theorem 1 as a basic tool to prove the hyperstability results of the -cubic functional equation (5) in ultrametric Banach spaces.

Theorem 2. Let and be normed space and ultrametric Banach space, respectively, , , and , and let satisfiesfor all such that , , , and . Then, is -cubic on .

Proof. Take such thatSince , one of must be negative. Assume that and replacing by and by in (11), we obtainDefine operators and byand writeIt is easily seen that has the form described in (H3) with , , , , , and . Furthermore, (11) can be written in the following way:Moreover, for every , ,So, (H2) is valid.
By using mathematical induction, we will show that, for each , we havewhere . We obtain that (18) holds for . Next, we will assume that (18) holds for , where . Then, we haveThis shows that (18) holds for . Now, we can conclude that inequality (18) holds for all . From (18), we obtainfor all .
Hence, according to Theorem 1, there exists a unique solution of the equation:such thatMoreover,for all .
Now, we show thatfor every such that , . Since the case is just (11), take , and suppose the last inequality holds for and every such that , . Then,for all such that and . Thus, by induction, we have shown that suppose the last inequality holds for every . Letting , we obtain thatfor all such that , . In this way, we obtain a sequence of cubic functions on such thatThis implies thatIt follows, with , that is cubic on . In a similar way, we can prove the following theorem.

Theorem 3. Let and be normed space and ultrametric Banach space, respectively, , , and , and let satisfiesfor all such that and . Then, is cubic on .

Proof. Take such thatSince , one of must be positive, and let and replace by and by in (29). Thus,Writing(31) takes formThe rest of the proof is similar to the proof of the last theorem. It easy to show the hyperstability of cubic equation on the set containing 0. The above theorems imply, in particular, the following corollary, which shows their simple application.

Corollary 1. Let and be normed space and ultrametric Banach space, respectively, and for some , andwhere , . Assume that the numbers satisfy one of the following conditions:(1), and (11) holds for all (2), and (36) holds for all Then, the functional equationhas no solution in the class of functions .

In the following theorem, we present a general hyperstability for the cubic equation where the control function is , which corresponds to the approach introduced in [36].

Theorem 4. Let be a normed space, be an ultrametric Banach space over a field , and be a function such thatis a nonempty set, where for all . Suppose thatand satisfies the inequalityfor all , such that and . Then, is cubic on .

Proof. Replacing by and by for in (39), we getfor all . For each , we define the operator byFurthermore, we putThen, inequality (40) takes the formFor each , the operator which is defined by has the form described in (H3) with andfor all . Moreover, for every , ,So, (H2) is valid. By using mathematical induction, we will show that, for each , we haveFrom (42), we obtain that inequality (46) holds for . Next, we will assume that (46) holds for , where . Then, we haveThis shows that (46) holds for . Now, we can conclude that inequality (46) holds for all . From (46), we obtain , for all and all . Hence, according to Theorem 1, there exists, for each , a unique solution of the equation:, such thatMoreover, , for all . Now, we show thatfor every such that , , and . Since the case is just (39), take and assume that (50) holds for , where and every such that . Then,Thus, by induction, we have shown that (50) holds for every . Letting in (50), we obtain thatfor all such that and . In this way, we obtain a sequence of cubic functions on such thatThis implies thatbecause the precedent inequality holds for over and .
It follows, with , that is cubic on . The following corollary is a particular case of Theorem 4 where with and .

Corollary 2. Let and be normed space and ultrametric Banach space, respectively, , and , and let satisfiesfor all . Then, is cubic on .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.