Hyers-Ulam Stability of Functional Equation Deriving from Quadratic Mapping in Non-Archimedean -Normed Spaces

Abstract

In this work, we have to introduce a generalized quadratic functional equation and derive its solution. The main objective of this work is to investigate the Hyers-Ulam stability of quadratic functional equation in non-Archimedean -normed spaces.

1. Introduction

The theory of functional equation is one of the most interesting topics in the field of Mathematics. It deals with the search of functions which satisfies a given equation. A functional equation is like a regular algebraic equation; though instead of unknown elements in some set, we are interested in finding a function satisfying our equation.

The concept of stability of a functional equation arises when one replaces a functional equation by an inequality which acts as a perturbation of the equation. The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940 and affirmatively solved by Hyers [2]. Aoki was the first author who has generalized the theorem of Hyers (see [3]).

Moreover, Gavruta [4], Rassias [5], and Bourgin [6] have considered the stability problem with unbounded Cauchy difference (see also [7]). On the other hand, Rassias [8–13] considered the Cauchy difference controlled by a product of different powers of norm. This stability phenomenon is called the Ulam-Gavruta-Rassias stability (see also [14]).

The Hyers-Ulam stability issue for the quadratic functional equation was cleared by Skof [15]. In [16], Czerwik demonstrated the Hyers-Ulam-Rassias stability of the quadratic functional equation. Afterward, Jung [17] has summed up the outcomes gotten by Skof and Czerwik.

The Hyers-Ulam stability problem for the quadratic functional equation was solved by Skof [15]. In [16], Czerwik proved the Hyers-Ulam-Rassias stability of quadratic functional equation. Later, Jung [17] has generalized the results obtained by Skof and Czerwik.

The first work on the Hyers-Ulam stability of functional equations in complete non-Archimedean normed spaces (some particular cases were considered earlier; see [18] for details) is [19]. After it, a lot of articles (see, for instance, [20] and the references given there) on the stability of other equations in such spaces were published. In [21], the stability of the additive Cauchy equation in non-Archimedean fuzzy normed spaces under the strongest -norm TM Rassias has been established.

In 1897, Hensel [22] has introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications [23–26].

Initially, Liu [27] introduced the notions of -normed space and non-Archimedean -normed space. Then, they investigated Hyers-Ulam stability of the Cauchy functional equation and the Jensen functional equation in non-Archimedean -normed spaces and that of the pexiderized Cauchy functional equation in -normed spaces. During the most recent many years, a few stability issues of functional equations have been researched by various mathematicians (see [28–40]).

The objective of this work is to introduce that a generalized quadratic functional equation is where and examine the Hyers-Ulam stability of the above mentioned equation in non-Archimedean -normed spaces.

2. Preliminaries

Now, we recall some notions and results which will be used.

Throughout this paper, let denote the set of nonnegative integers and , and let be fixed.

Definition 1 (see [27]). Let be a linear space over with dim , , and , let a mapping satisfies the following conditions: (i) if and only if are linearly dependent(ii) is invariant under permutations of (iii)(iv)for all , and .
Then, is called as -norm on , and is called a linear -normed space or -normed space.
We remark that the notion of a linear -normed space is a summed up of a linear -normed space and of a -normed space .

Definition 2 (see [27]). Let be a real vector space with dim over a scalar field with a non-Archimedean nontrivial valuation , where and a constant with . A real-valued function is called an -norm on if the upcoming conditions true: (a) if and only if are linearly dependent(b) is invariant under permutations of (c)(d)for all and . Then, is called as non-Archimedean -normed space.

Example 3. Let be a prime number. For any nonzero rational number such that and are coprime to the prime number , define the -adic absolute value . Then, is a non-Archimedean norm on . The completion of with respect to is denoted by and is called the -adic number field.

Note that if ; then, in for each integer .

Remark 4 (see [27]). A sequence in a non-Archimedean -normed space is a Cauchy sequence if and only if converges to zero.

Lemma 5 (see [27]). For a convergent sequence in a linear -normed space , for all .

Lemma 6 (see [27]). Let be a linear -normed space, , . If and for all , then .

3. General Solution

Here, the authors discussed the general solution of the equation (1). Consider and are real vector spaces.

Theorem 7. If be a mapping satisfies the functional equation (1) for all , then the mapping is quadratic, that is satisfies the equality for all .

Proof. Suppose that the mapping satisfies the functional equation (1). Replacing by in (1), we get . Now, setting by in (1), we get for all . Therefore, the function is an even function. Substituting by and in (1), we obtain , , and so on, for every . In general, for any nonnegative integer , we get for all . Next, replacing by in (1), we get our desired outcome.

4. Stability Results

Here, we consider and examine the Hyers-Ulam stability of the functional equation (1).

Let us assume and are non-Archimedean -normed space and complete non-Archimedean -normed space, respectively, where , .

Define a mapping by for all .

Theorem 8. Let and with , and let be a function. Suppose that be a mapping satisfies for all and . Then, there exists a unique quadratic mapping satisfies for all .

Proof. Replacing by in (5), we obtain for all . Dividing both sides by , we have for all . Replacing by in (8), we attain As and , we obtain that for all . From Remark 4, we conclude that the sequence is a Cauchy sequence in . Since is complete space, we can define by for all . Next, our aim is to prove that the function is quadratic. From (5), (11) and Lemma 5 that for all and . As and , we obtain for all . By Lemma 6, we have for all . Therefore, the function is quadratic. Switching through in (7) and divide by , we attain Thus, by (7) and (15), we reach As and , we obtain for all . By induction on , we can conclude that for all , , and . Replacing by in (18) and dividing both sides by , we have for all , , and . It follows from (7) and (19) that for all , , and . Passing the limit as tends to in inequality (18), we can get (6). Next, we want to prove that the function is unique. Let be an another quadratic mapping which satisfies (6), Taking the limit as tends to in the last inequality, we obtain that for all . By Lemma 6, we conclude that for all . Hence, is the unique quadratic mapping which satisfies (6).

Theorem 9. Let , with . Let be a function. Suppose that a mapping satisfies for all and . Then, there exists a unique quadratic mapping satisfies for all .

Proof. Switching by in (23), we have for all . Interchanging by in (25), we obtain for all . Replacing by in (26), we reach for all and . As and , we have for all and . From Remark 4, we conclude that the sequence is a Cauchy sequence in . As is complete space. We can define by for all . Next, our aim is to prove that the function is quadratic. From (23), (29) and Lemma 5 that for all and . Since and , we have for all . By Lemma 6, we obtain for all . Therefore, the function is quadratic. Switching through in (25) and multiplying by , we reach Thus, by (25) and (33), we obtain Since and , we attain for all . By induction on , we can conclude that for all , and . Replacing by in (36) and multiplying both sides by , we get. for all , . It follows from (25) and (37) that for all , , and . Passing the limit as tends to in (36), we can get (24). Finally, we want to prove that the function is unique. Consider an another quadratic mapping satisfying (24), for all and . Taking the limit as tends to , we obtain that for all . By Lemma 6, we conclude that for all . So that the function is the unique quadratic function. Hence, the proof is completed.

We obtain the following results of theorem with a generalized control function when the domain is a vector space and codomain be a complete non-Archimendean -normed space, where and .

Theorem 10. Let be a function such that for all , and let be a function. Suppose be a mapping which satisfies for all and . Then, there exists a quadratic mapping such that where for all . Moreover, if for all . Then, the unique mapping is quadratic which satisfies the inequality (43).

Proof. Setting by in (42) and dividing both sides by , we have for all and . Replacing by in (46) and dividing both sides by , we obtain for all , , and . Taking the limit as tends to and considering (41), we attain for all and . Utilizing Remark 4, it is clear that the sequence is a Cauchy sequence. As is a complete space. We can define the mapping by for all . Next, we want to prove that the function is quadratic. So, for all and . Taking the limit as and considering (41), we arrive for all . Using Lemma 6, we conclude that is quadratic. Switching through in (46) and dividing by , we obtain for all and . Considering (46), we receive By induction on , we reach