Hyers-Ulam Stability of Quadratic Functional Equation Based on Fixed Point Technique in Banach Spaces and Non-Archimedean Banach Spaces

: In this paper, the authors investigate the Hyers–Ulam stability results of the quadratic functional equation in Banach spaces and non-Archimedean Banach spaces by utilizing two different techniques in terms of direct and ﬁxed point techniques.


Introduction and Preliminaries
The study of stability problems for functional equations is one of the essential research areas in mathematics, which originated in issues related to applied mathematics. The first question concerning the stability of homomorphisms was given by Ulam [1] as follows.
Given a group (G, * ), a metric group (G , ·) with the metric d, and a mapping f from G and G , does δ > 0 exist such that d( f (x * y), f (x) · f (y)) ≤ δ for all x, y ∈ G. If such a mapping exists, then does a homomorphism h : G → G exist such that d( f (x), h(x)) ≤ for all x ∈ G? Hyers partially answered affirmatively with respect to the question of Ulam for Banach spaces [2]. By assuming an infinite Cauchy difference, Aoki [3] expanded Hyers' Theorem for additive mappings and Rassias [4] for linear mappings. Gajda [5] discovered an affirmative answer to the issue p > 1 by using the same approach as Rassias [4]. Gajda [5], as well as Rassias and Šemrl [6], showed that a Rassias' type theorem cannot be established for p = 1.
One of the most famous functional equations is the additive functional equation In 1821, it was first solved by A.L. Cauchy in the class of continuous real-valued functions. It is often called the Cauchy additive functional equation in honor of A.L. Cauchy. The theory of additive functional equations is frequently applied to the development of theories of other functional equations. Moreover, the properties of additive functional equations are powerful tools in almost every field of natural and social sciences. Since the function f (x) = x is the solution of (1), every solution of the additive functional Equation (1) is called an additive function.
Gajda's [5], as well as Rassias and Šemrl [6], counterexamples have prompted numerous mathematicians to create alternative definitions of roughly additive or approximately linear mapping. Gȃvruţa [7] explored the Hyers-Ulam stability of functional equations, among other situations (see [8][9][10]). The quadratic functional equation is defined by φ(u + v) + φ(u − v) = 2φ(u) + 2φ(v). Every solution of the quadratic functional equation, in particular, is referred to as a quadratic function. Skof [11] demonstrated the stability of quadratic functional equations for mappings between normed space and Banach space. Cholewa [12] observed that, if the appropriate domain normed space is substituted by an Abelian group, the Skof theorem still holds. More functional equations may be found in [13][14][15][16].
Xiuzhong Yang [17] examined the Hyers-Ulam-Rassias stability of an additivequadratic-cubic-quartic functional equation in non-Archimedean (n, β)-normed spaces. Anurak Thanyacharoen [18,19] proved the generalized Hyers-Ulam-Rassias stability for the following composite functional equation: where f maps from a (β, p)-Banach space into itself by using the fixed point method and the direct method. Moreover, the generalized Hyers-Ulam-Rassias stability for the composite s-functional inequality is discussed via our results and also investigated the generalized Hyers-Ulam stability for the additive-quartic functional equation that associated the mapping from an additive group to a complete non-Archimedean space. Definition 1 ([14]). Let us assume a vector space V over a field K with a non-Archimedean valuation | · |. A mapping · : V → [0, ∞) is said to be a non-Archimedean norm if it satisfies the following conditions: The strong triangle inequality of the following: holds. Then, the pair (V, · ) is called as a non-Archimedean normed space.
In fixed point theory, there is a fundamental result. Theorem 1 ([14]). Suppose that a complete generalized metric space (V, d) and a mapping H : V → V is strictly contractive with Lipschitz constant L < 1. Then, for every v ∈ V, either d(H l v, H l+1 v) = ∞ for all integers l > 0 or there is an integer l 0 > 0 satisfies the following: (2) The sequence {H l v} converges to a fixed point u * of H; In [13], Nazek Alessa et al. introduced a new type of generalized quadratic functional equation as the following: where m ≥ 2, and derived its solution. A non-Archimedean (n, β)-normed space was used to study the stability of the functional Equation (2) in terms of Hyers-Ulam. In this paper, we study the Ulam-Hyers stability results of the generalized additive functional Equation (2) in Banach spaces and non-Archimedean Banach spaces by using different approaches of direct and fixed point techniques. This paper is structured as follows: In Sections 2 and 3, we investigate the Ulam-Hyers stability results in Banach spaces by using direct and fixed point techniques where we consider that V and W are normed spaces and Banach spaces, respectively. In Sections 4 and 5, we examined the Ulam-Hyers stability results in non-Archimedean Banach spaces by using direct and fixed point techniques where we consider that V is a non-Archimedean normed space, W is a non-Archimedean Banach space, and let |2| = 1.
For notational simplicity, we define φ : V → W by the following:

Stability Results in Banach Spaces: Direct Technique
for all v 1 , v 2 , · · · , v m ∈ V. If a mapping φ : V → W with φ(0) = 0, and it satisfies the below inequality: for all v 1 , v 2 , · · · , v m ∈ V. Then, there exists a unique quadratic mapping Q 2 : V → W such that for all v ∈ V. Then, the mapping Q 2 (v) is defined by for all v ∈ V. From inequality (6), we have for all v ∈ V. By replacing v by 2v and dividing by 2 2 in (7) and then combining the resultant inequality with (7), we obtain We conclude for any non-negative integer p that one can easy to verify the following: replacing v by 2 l v and dividing by 2 2l in (8) for p, l > 0, we obtain for all v ∈ V. Taking limit l tending to ∞ in (8), we can observe that (5) holds for all v ∈ V. Next, we want to prove that the function Q 2 satisfies the functional Equation (2). By replacing (v 1 , v 2 , · · · , v m ) by (2 l v 1 , 2 l v 2 , · · · , 2 l v m ) and dividing by 2 2l in (4), we obtain Allowing l → ∞ in the above inequality and using the definition of Q 2 (v), we see that Q 2 (v 1 , v 2 , · · · , v m ) = 0. Hence, the function Q 2 satisfies the functional Equation (2) for all v 1 , v 2 , · · · , v m ∈ V. Next, we want to show the uniqueness of Q 2 . Consider another quadratic function R 2 (v) which satisfies the functional Equation (2) and inequality (5), then Hence, the function Q 2 is unique. On the other hand, for ζ = −1, in the same manner, we can verify a similar sense of stability. The proof of the theorem is now complete.
Corollary 1. If a mapping φ : V → W with φ(0) = 0 and it satisfies the following inequality: where λ and α are two non-negative real numbers with α = 2, then there exists a unique quadratic mapping Q 2 : Corollary 2. If a mapping φ : V → W with φ(0) = 0 satisfies the following inequality: where λ and α are two non-negative real numbers with mα = 2, then there exists a unique quadratic mapping Q 2 : obtain the result (10).

Stability Results in Banach Spaces: Fixed Point Technique
satisfies the inequality (4). If there exists L = L(j) that satisfies the following: and it has the following property: for all v ∈ V, then there exists a unique quadratic mapping Q 2 : V → W satisfying the functional Equation (2) and such that Proof. Consider the following set: and allow a general metric d on Ψ such that It is clear that (Ψ, d) is complete. Define a mapping F : Ψ → Ψ by For all p, q ∈ Ψ, we obtain As a result, a strictly contractive function F on Ψ with L is obtained. It is clear from (6) that for all v ∈ V. We have j = 0 by using the above inequality and definitions of β(v).
Hence, we obtain the following: for all v ∈ V. Replacing v by v 2 in (12), we obtain for all v ∈ V. Using the definition of β(v) in the above inequality (14) for j = 0, we have for all v ∈ V. Hence, we obtain for all v ∈ V. Using (13) and (15), we can conclude that for all v ∈ V. Now, in both cases, the fixed point alternative theorem suggests that exists a fixed point Q 2 of F in Ψ such that In order to prove that Q 2 : V → W satisfies (2), the proof follows a similar manner as Theorem 2. Since the function Q 2 is a unique fixed point of F in the set Θ = {φ ∈ Ψ/d(φ, Q 2 ) < ∞}, thus, the function Q 2 is a unique function such that The proof of the Theorem is now complete.
Corollary 3. If a mapping φ : V → W with φ(0) = 0 and such that for all v 1 , v 2 , · · · , v m ∈ V, where λ and α are two non-negative real numbers, then there exists a unique quadratic mapping Q 2 : V → W which satisfies the following: for all v ∈ V.
Proof. We set In other words, (11) holds. As such, we obtain the following.
for all v ∈ V. Hence, Equation (2) holds for the following.
From the above conditions, we obtain our needed outcomes of (16).
Then, there exists a unique quadratic mapping Q 2 : V → W that satisfies for all v ∈ V.
for all v ∈ V. Thus, for all p > l > 0 and for all v ∈ V. As a result of (20), the sequence 2 2n φ v 2 n is a Cauchy sequence for every v ∈ V. Since W is complete, the sequence {2 2n φ v 2 n } converges. As a result, the mapping Q 2 : V → W may be defined Taking l = 0 and the limit p → ∞ in (20), we obtain (18). As a result of (17) and (4), we have Thus, we obtain From Lemma 1, the mapping Q 2 : V → W is quadratic. Now, consider another quadratic mapping R 2 : V → W that satisfies inequality (18). Then, we obtain Thus, we may infer that Q 2 (v) = R 2 (v) for all v ∈ V. This proves the uniqueness of Q 2 . As a result, the mapping Q 2 : V → W is a unique quadratic mapping that satisfies (18).
for all v 1 , v 2 , · · · , v m ∈ V, then there exists a unique quadratic mapping Q 2 : V → W that satisfies for all v ∈ V, where λ < 2 and α are in R + .

Corollary 5.
If there is a mapping φ : V → W with φ(0) = 0 and satisfies for all v 1 , v 2 , · · · , v m ∈ V, then there exists a unique quadratic mapping Q 2 : V → W such that for all v ∈ V, where mλ < 2 and α are in R + .
Theorem 5. If a mapping χ : V m → [0, ∞) and a mapping φ : V → W with φ(0) = 0 exists and satisfies (4) and the following then there exists a unique quadratic mapping Q 2 : V → W such that Proof. It follows from (19) that for all v ∈ V. Hence, for all p > l > 0. As a result of (24), { 1 2 2n φ(2 n v)} is a Cauchy sequence. Since W is complete, { 1 2 2n φ(2 n v)} converges. Thus, we can define a mapping Q 2 : for all v ∈ V. Now, taking l = 0 and the limit p → ∞ in (24), we obtain (23). The remaining part of the proof is similar to that of Theorem 4.

Corollary 6.
If there exists a mapping φ : V → W with φ(0) = 0 and it satisfies the inequality (21), then there exists a unique quadratic mapping Q 2 : V → W such that for all v ∈ V, where λ > 2 and α are in R + .

Corollary 7.
If there exists a mapping φ : V → W with φ(0) = 0 and it satisfies (22), then there exists a unique quadratic mapping Q 2 : V → W such that for all v ∈ V, where mλ > 2 and α are in R + .

Stability Results in Non-Archimedean Banach Spaces: Fixed Point Technique
Theorem 6. Let a mapping χ : Let a mapping φ : V → W which satisfies φ(0) = 0 and (4). Then, there exists a unique quadratic mapping Q 2 : V → W such that for all v ∈ V. Let us consider the set as well as the generalised metric d on M: where, as is typical, inf ∅ = +∞. It is simple to demonstrate that (M, d) is complete (see [20]). Now, we examine the linear mapping F : M → M, which has the following property: for all v ∈ V. Let p, q ∈ M be given such that d(p, q) = . Then, we have This means that d(Fp, Fq) ≤ Ld(p, q) for all p, q ∈ M. It follows from (26) that From Theorem 1, there exists a quadratic mapping Q 2 : V → W satisfying the following: The function Q 2 is a unique fixed point of M in the set This yields that Q 2 is a unique function satisfying (27) such that there exists θ ∈ (0, ∞) satisfying (2) d(F n φ, Q 2 ) → 0 as n → ∞. This indicates the below equality Fφ), and it implies the following: for all v ∈ V. It follows from (25) and (4) that Thus, By Lemma 1, the mapping Q 2 : V → W is quadratic.
Corollary 8. If a mapping φ : V → W satisfies φ(0) = 0 and the following: for all v 1 , v 2 , · · · , v m ∈ V, where λ < 2 and α are two non-negative real numbers, then there exists a unique quadratic mapping Q 2 : V → W such that Proof. The proof is based on Theorem 6 by allowing the following: After that, we may use L = |2| 2−λ to obtain our desired result.
Corollary 9. If a mapping φ : V → W satisfies φ(0) = 0 and the following: for all v 1 , v 2 , · · · , v m ∈ V, where mλ < 2 and α are two non-negative real numbers, then there exists a unique quadratic mapping Q 2 : V → W such that Proof. The proof is based on Theorem 6 by allowing the following: After that, we may use L = |2| 2−mλ to obtain our desired result.
Theorem 7. Let a mapping χ : V m → [0, ∞) such that there is L < 1 with the following: If a mapping φ : V → W satisfies φ(0) = 0 and (4), then there exists a unique quadratic mapping Q 2 : V → W such that Proof. It follows from (26) that for all v ∈ V. The remaining part of the proof is similar to that of Theorem 6.
Corollary 10. If a mapping φ : V → W satisfies φ(0) = 0 and (28), then there exists a unique quadratic mapping Q 2 : V → W such that for all v ∈ V, where λ > 2 and α are two positive real numbers.
Proof. The proof is based on Theorem 7 by allowing the following: Then, we can take L = |2| λ−2 , and we obtain our result.
Corollary 11. If a mapping φ : V → W satisfies φ(0) = 0 and (29), then there exists a unique quadratic mapping Q 2 : V → W such that for all v ∈ V, where mλ > 2 and α are two non-negative real numbers.
Proof. The proof is based on Theorem 7 by allowing the following: Then, we can take L = |2| mλ−2 and we obtain our result.

Conclusions
In this work, we studied the Ulam-Hyers stability results of the generalized additive functional Equation (2) in Banach spaces and non-Archimedean Banach spaces by using different approaches of direct and fixed point methods. In future works, the researcher can obtain the Ulam-Hyers stability results of this generalized additive functional equation in various normed spaces such as matrix paranormed spaces, quasi-β-normed spaces, fuzzy normed spaces, etc.
The results obtained and the methods adopted in this study would be useful for other researchers for carrying out further investigations. Since there are lot of applications of functions in various fields including physics, economics, business, medicine, digital image processing, chemistry, etc., the study of this type of equation has a lot of scope for other researchers.