Abstract

In this present work, we obtain the solution of the generalized additive functional equation and also establish Hyers–Ulam stability results by using alternative fixed point for a generalized additive functional equation . where is a nonnegative integer with in Banach spaces.

1. Introduction

The problem of Ulam–Hyers stability concerns determining circumstances under which, given an approximate solution of a functional equation, one may locate an exact key that is closer to it in some sense. The investigation of stability problem for functional equations is identified to a question of Ulam [1] about the stability of group homomorphisms and affirmatively answered for Banach space by Hyers [2, 3]. It was further generalized and interesting results were obtained by a number of authors [4–6].

Rassias investigated the Hyers–Ulam stability results for the various functional equations in [7–10] through different spaces. Czerwik [11, 12] examined the stability of the quadratic functional equation involving several variables in the normed spaces. Numerous mathematicians investigated the various stability results in [13–32].

In 2019, Park et al. [33] introduced additive s-functional inequality. Using the fixed-point method and direct method, he established the Hyers–Ulam stability for the abovementioned one in complex Banach spaces. Also, he examined the Hyers–Ulam stability of homomorphism and derivations in complex Banach algebras. In 2018, Almahalebi [34] investigated the quadratic functional equation in Banach spaces. And, he established the hyperstability outcome of the same equation through the fixed-point approach.

Radu [35] investigated various results about the stability problem by using the fixed-point alternative. He applied the fixed-point method to examine the stability of Cauchy functional equation and Jensen’s functional equations. After his work, numerous authors used the fixed-point method to investigate several functional equations [36–41]. The functional equationis called the Cauchy additive functional equation and it is the most famous functional equation. As is the solution of (1), every solution of the additive equation is called an additive function.

In this present work, we derive the solution of the generalized additive functional equation along with established Hyers–Ulam stability results by using direct and fixed-point methods for a generalized additive functional equationwhere is a nonnegative integer in Banach spaces.

2. General Solution of the Functional Equation (2)

In this section, we derive the general solution of the generalized additive functional equation (2).

Here, we consider and be real vector spaces.

Theorem 1. If a mapping satisfies the functional equation (2) for all , then the mapping satisfies the functional equation (1) for all .

Proof. Suppose a mapping satisfies the functional equation (2). Replacing by in the functional equation (2), we have . Replacing by in equation (2), we get for all . Therefore, the function is an odd function. Replacing by in equation (2) and using the property of odd function, we havefor all . Replacing by in (3), we obtainfor all . Again, replacing by in (5) and using (3), we havefor all . We can generalize for any nonnegative integer and we getfor all . Now, replacing by in (2), we obtain our desired result of equation (1).

Remark 1. Let be a linear space and a function satisfies the functional equation (2). Then, the following claims hold:(1) for all , , integers(2) for all if is continuousIn Sections 3 and 4, we take be a normed space and be a Banach space. For our convincing effortlessness, we describe a function asfor every .

3. Hyers–Ulam Stability of the Functional Equation (2): Direct Method

In this section, we investigated the Hyers–Ulam stability of the generalized additive functional equation (2) in Banach space by using the direct method.

Theorem 2. Let , be a mapping such thatconverges in withfor all . If a mapping satisfies the inequalityfor all , then there exists a unique additive mapping satisfying equation (2) andfor all .

Proof. Assume . Replacing by in (10), we havefor all . From equality (12), we getfor all . Exchanging through in (13), we obtainfor all . From (14), we achievefor all . Adding together (13) and (15), we get the following outcome:for all . It follows from (13), (15), and (16), and we can generalize that as follows:for all . In order to establish the convergence of the sequence , switch through and also divide by in (17). We conclude that, for some ,for all . Therefore, the sequence is a Cauchy. As is complete, there exists so that for all . Taking the limit in (17), we obtain that result (11) holds for all . To prove that the function satisfies equation (2), replacing by and also dividing by in (10), we getfor all . Taking the limit in the above inequality and using the definition of , we have for all . Thus, the function satisfies equation (2). To prove that the function is unique, let be another additive mapping satisfying the functional equation (2) and (11). Hence,Hence, is unique. Now, replacing through in (12), we havefor all . The rest of the proof is similar to that when . So for , we can prove the results by a similar manner. Hence, the proof is completed.

Corollary 1. Let and be positive real numbers. If there exists a mapping satisfying the inequalityfor all , then there exists a unique additive mapping such thatfor all .

4. Hyers–Ulam Stability of the Functional Equation (2): Fixed-Point Method

In this section, we examined the Hyers–Ulam stability of the generalized additive functional equation (2) in Banach space by using the fixed-point method.

Theorem 3. Let be a mapping for which there exists a mapping andwhere and such that it satisfies the inequalityfor all . If there exists a Lipschitz constant such thathas the propertyfor all . Then, there exists a unique additive mapping satisfying equation (2) andfor all .

Proof. Consider a set and initiate the generalized metric on , . It is easy to view that is complete. Fix as for all . For and , we haveThat is, . Accordingly, is a strictly contractive mapping on with Lipschitz constant . From inequality (12), we havefor all . It follows from (30) thatfor all . Using inequality (27) when , we getfor all . So we obtainfor all . Replacing by in (31), we havefor all . Using inequality (27) when , we havefor all . Therefore, we obtain the result thatfor all . From inequalities (33) and (36), we conclude thatfor all . Next, using fixed-point alternative theorem [35], there exists a fixed point of in such thatfor all . In order to establish satisfying equation (2), we use an argument similar to that in the proof of Theorem 2. As is a unique fixed point of in the set , is a unique function such thatfor all . This accomplished the proof.