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

Abstract

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.

1. 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 $(G,\ast )$, a metric group $(G^{{}^{\prime }},·)$ with the metric d, and a mapping f from G and $G^{{}^{\prime }}$, does $\delta >0$ exist such that

$$d\left(f\right(x\ast y),f(x)·f(y\left)\right)\le \delta$$

for all $x,y\in G$. If such a mapping exists, then does a homomorphism $h:G\to G^{{}^{\prime }}$ exist such that

$$d\left(f\right(x),h(x\left)\right)\le ϵ$$

for all $x\in G$?

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

$$f(x+y)=f\left(x\right)+f\left(y\right).$$

(1)

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 $f\left(x\right)=x$ 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. Găvruţa [7] explored the Hyers–Ulam stability of functional equations, among other situations (see [8,9,10]). The quadratic functional equation is defined by $\varphi (u+v)+\varphi (u-v)=2\varphi \left(u\right)+2\varphi \left(v\right)$. 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 additive-quadratic-cubic-quartic functional equation in non-Archimedean $(n,\beta )$-normed spaces. Anurak Thanyacharoen [18,19] proved the generalized Hyers–Ulam–Rassias stability for the following composite functional equation:

$$f\left(f\right(x)-f(y\left)\right)=f(x+y)+f(x-y)-f\left(x\right)-f\left(y\right),$$

where f maps from a $(\beta ,p)$-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 $\mathbb{K}$ with a non-Archimedean valuation $|·|$. A mapping $\parallel ·\parallel :V\to [0,\infty )$ is said to be a non-Archimedean norm if it satisfies the following conditions:

(i) 

$\parallel v\parallel =0$ if and only if $v=0$;

(ii) 

$\parallel rv\parallel =\left|v\right|\parallel v\parallel$ for all $r\in \mathbb{K},v\in V$;

(iii) 

The strong triangle inequality of the following:

$$\parallel u+v\parallel \le max\{\parallel u\parallel ,\parallel v\parallel \},\forall u,v\in V$$

holds. Then, the pair $(V,\parallel ·\parallel )$ 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 $H:V\to V$ is strictly contractive with Lipschitz constant $L<1$. Then, for every $v\in V$, either

$$d(H^{l}v,H^{l+1}v)=\infty$$

for all integers $l>0$ or there is an integer $l_0>0$ satisfies the following:

(1) 

$d(H^{l}v,H^{l+1}v)<\infty$ for all $l\ge l_0$;

(2) 

The sequence $\left\{H^{l}v\right\}$ converges to a fixed point $u^{\ast }$ of H;

(3) 

$u^{\ast }$ is the unique fixed point of H in $W=\{u\in V|d(H^{l_0}v,u)<\infty \}$;

(4) 

$d(u,u^{\ast })\le \frac{1}{1-L}d(u,Hu)$ for all $u\in W$.

In [13], Nazek Alessa et al. introduced a new type of generalized quadratic functional equation as the following:

$$\sum _{1\le i<j\le m}\varphi \left(v_i+v_j\right)+\sum _{1\le i<j\le m}\varphi \left(v_i-v_j\right)=2\left(m-1\right)\sum _{1\le i\le m}\varphi \left(v_i\right)$$

(2)

where $m\ge 2$, and derived its solution. A non-Archimedean $(n,\beta )$-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 Section 2 and Section 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 Section 4 and Section 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 $\left|2\right|\ne 1$.

Lemma 1

([13]). If a mapping $\varphi :V\to W$ satisfies the functional Equation (2), then the mapping $\varphi :V\to W$ is quadratic.

For notational simplicity, we define $\varphi :V\to W$ by the following:

$$\begin{array}{ccc}\hfill \mathsf{\Lambda }\varphi (v_1,v_2,\cdots ,v_m)& =& \sum _{1\le i<j\le m}\varphi \left(v_i+v_j\right)+\sum _{1\le i<j\le m}\varphi \left(v_i-v_j\right)-2\left(m-1\right)\sum _{1\le i\le m}\varphi \left(v_i\right).\hfill \end{array}$$

2. Stability Results in Banach Spaces: Direct Technique

Theorem 2.

Let $\zeta \in \{-1,1\}$ and a mapping $\chi :V^m\to [0,\infty )$ such that

$$\underset{l\to \infty }{lim}\frac{\chi \left(2^{l\zeta }{v}_1,2^{l\zeta }{v}_2,\cdots ,2^{l\zeta }{v}_m\right)}{2^{2l\zeta }}=0$$

(3)

for all $v_1,v_2,\cdots ,v_m\in V$. If a mapping $\varphi :V\to W$ with $\varphi \left(0\right)=0$, and it satisfies the below inequality:

$$\begin{array}{c}\hfill \parallel \mathsf{\Lambda }\varphi (v_1,v_2,\cdots ,v_m)\parallel \le \chi (v_1,v_2,\cdots ,v_m)\end{array}$$

(4)

for all $v_1,v_2,\cdots ,v_m\in V$. Then, there exists a unique quadratic mapping $Q_2:V\to W$ such that

$$\parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \frac{1}{2^2}\sum _{l=\frac{1-\zeta }{2}}^{\infty }\frac{\chi (2^{l}v,2^{l}v,0,\cdots ,0)}{2^{2l}}$$

(5)

for all $v\in V$. Then, the mapping $Q_2\left(v\right)$ is defined by

$$Q_2\left(v\right):=\underset{l\to \infty }{lim}\frac{\varphi \left(2^{l}v\right)}{2^{2l}}$$

for all $v\in V$.

Proof. 

Assume that $\zeta =1$. Replacing $(v_1,v_2,\cdots ,v_m)$ by $(v,v,0,\cdots ,0)$ in (4), we obtain

$$\parallel 2^2\varphi \left(v\right)-\varphi \left(2v\right)\parallel \le \chi \left(v,v,0,\cdots ,0\right)$$

(6)

for all $v\in V$. From inequality (6), we have

$$\Vert \frac{\varphi \left(2v\right)}{2^2}-\varphi \left(v\right)\Vert \frac{\chi (v,v,0,\cdots ,0)}{2^2}$$

(7)

for all $v\in V$. By replacing v by $2v$ and dividing by $2^2$ in (7) and then combining the resultant inequality with (7), we obtain

$$\Vert \frac{\varphi \left(2^{2}v\right)}{2^4}-\varphi \left(v\right)\parallel \le \frac{1}{2^2}\left[\chi (v,v,0,\cdots ,0)+\frac{\chi (2v,2v,0,\cdots ,0)}{2^2}\right]$$

for all $v\in V$. We conclude for any non-negative integer p that one can easy to verify the following:

$$\Vert \frac{\varphi \left(2^{p}v\right)}{2^{2p}}-\varphi \left(v\right)\parallel \le \frac{1}{2^2}\sum _{l=0}^{p-1}\frac{\chi (2^{l}v,2^{l}v,0,\cdots ,0)}{2^{2l}}$$

(8)

for all $v\in V$. To show that the sequence $\left\{\frac{\varphi \left(2^{p}v\right)}{2^{2p}}\right\}$ is converging, replacing v by $2^{l}v$ and dividing by $2^{2l}$ in (8) for $p,l>0$, we obtain

$$\begin{array}{ccc}\hfill \parallel \frac{\varphi \left(2^{p+l}v\right)}{2^{2(p+l)}}-\frac{\varphi \left(2^{l}v\right)}{2^{2l}}\parallel & =& \frac{1}{2^{2l}}\parallel \frac{\varphi \left(2^{p+l}v\right)}{2^{2p}}-\varphi \left(2^{l}v\right)\parallel \hfill \\ & \le & \frac{1}{2^2}\sum _{p=0}^{\infty }\frac{\chi (2^{p+l}v,2^{p+l}v,0,\cdots ,0)}{2^{2(p+l)}}\to 0\mathrm{as}l\to \infty \hfill \end{array}$$

for all $v\in V$. Hence, $\left\{\frac{\varphi \left(2^{p}v\right)}{2^{2p}}\right\}$ is a Cauchy sequence. Since W is complete, there exists a mapping $Q_2:V\to W$ such that

$$Q_2\left(v\right)=\underset{l\to \infty }{lim}\frac{\varphi \left(2^{l}v\right)}{2^{2l}}$$

for all $v\in V$. Taking limit l tending to ∞ in (8), we can observe that (5) holds for all $v\in V$. Next, we want to prove that the function $Q_2$ satisfies the functional Equation (2). By replacing $(v_1,v_2,\cdots ,v_m)$ by $(2^l{v}_1,2^l{v}_2,\cdots ,2^l{v}_m)$ and dividing by $2^{2l}$ in (4), we obtain

$$\frac{1}{2^{2l}}\parallel \mathsf{\Lambda }\varphi (2^l{v}_1,2^l{v}_2,\cdots ,2^l{v}_m)\parallel \le \frac{1}{2^{2l}}\chi \left(2^l{v}_2,2^l{v}_2,\cdots ,2^l{v}_m\right)$$

for all $v_1,v_2,\cdots ,v_m\in V$. Allowing $l\to \infty$ in the above inequality and using the definition of $Q_2\left(v\right)$, we see that $Q_2(v_1,v_2,\cdots ,v_m)=0$. Hence, the function $Q_2$ satisfies the functional Equation (2) for all $v_1,v_2,\cdots ,v_m\in V$. Next, we want to show the uniqueness of $Q_2$. Consider another quadratic function $R_2\left(v\right)$ which satisfies the functional Equation (2) and inequality (5), then

$$\begin{array}{ccc}\hfill \parallel Q_2\left(v\right)-R_2\left(v\right)\parallel & \le & \frac{1}{2^{2l}}\left\{\parallel Q_2\left(2^{l}v\right)-\varphi \left(2^{l}v\right)\parallel +\parallel \varphi \left(2^{l}v\right)-R_2\left(2^{l}v\right)\parallel \right\}\hfill \\ & \le & \frac{2}{2^2}\sum _{p=0}^{\infty }\frac{\chi (2^{p+1}v,2^{p+1}v,0,\cdots ,0)}{2^{2(p+l)}}\to 0\mathrm{as}l\to \infty \hfill \end{array}$$

for all $v\in V$. Hence, the function $Q_2$ is unique. On the other hand, for $\zeta =-1$, 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 $\varphi :V\to W$ with $\varphi \left(0\right)=0$ and it satisfies the following inequality:

$$\begin{array}{c}\hfill \parallel \mathsf{\Lambda }\varphi (v_1,v_2,\cdots ,v_m)\parallel \le \lambda \left(\sum _{j=1}^m{\parallel v_j\parallel }^{\alpha }\right)\end{array}$$

for all $v_1,v_2,\cdots ,v_m\in V$, where λ and α are two non-negative real numbers with $\alpha \ne 2$, then there exists a unique quadratic mapping $Q_2:V\to W$ such that

$$\begin{array}{c}\hfill \parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \frac{{2\lambda \parallel v\parallel }^{\alpha }}{|2^2-2^{\alpha }|}\end{array}$$

(9)

for all $v\in V$.

Proof. 

If we replace $\chi (v_1,v_2,\cdots ,v_m)=\lambda \left({\sum }_{j=1}^m{\parallel v_j\parallel }^{\alpha }\right)$ in Theorem 2, we obtain the result (9). □

Corollary 2.

If a mapping $\varphi :V\to W$ with $\varphi \left(0\right)=0$ satisfies the following inequality:

$$\begin{array}{c}\hfill \parallel \mathsf{\Lambda }\varphi (v_1,v_2,\cdots ,v_m)\parallel \le \lambda \left(\sum _{j=1}^m\parallel v_j{\parallel }^{m\alpha }+\prod _{j=1}^m{\parallel v_j\parallel }^{\alpha }\right)\end{array}$$

for all $v_1,v_2,\cdots ,v_m\in V$, where λ and α are two non-negative real numbers with $m\alpha \ne 2$, then there exists a unique quadratic mapping $Q_2:V\to W$ such that

$$\begin{array}{c}\hfill \parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \frac{{2\lambda \parallel v\parallel }^{m\alpha }}{|2^2-2^{m\alpha }|}\end{array}$$

(10)

for all $v\in V$.

Proof. 

If we replace $\chi (v_1,v_2,\cdots ,v_m)=\lambda \left({\sum }_{j=1}^m\parallel v_j{\parallel }^{m\alpha }+{\prod }_{j=1}^m{\parallel v_j\parallel }^{\alpha }\right)$ in Theorem 2, we obtain the result (10). □

3. Stability Results in Banach Spaces: Fixed Point Technique

Theorem 3.

Suppose a mapping $\varphi :V\to W$ with $\varphi \left(0\right)=0$ for which there exists a mapping $\chi :V^m\to [0,\infty )$ with the condition

$$\underset{l\to \infty }{lim}\frac{\chi ({\sigma }_j^l{v}_1,{\sigma }_j^l{v}_2,\cdots ,{\sigma }_j^l{v}_m)}{{\sigma }_j^{2l}}=0$$

(11)

where

$$\begin{array}{c}\hfill {\sigma }_j=\left\{\begin{array}{c}2,ifj=0;\hfill \\ \frac{1}{2},ifj=1;\hfill \end{array}\right.\end{array}$$

satisfies the inequality (4). If there exists $L=L\left(j\right)$ that satisfies the following:

$$v\to \beta \left(v\right)=\chi \left(\frac{v}{2},\frac{v}{2},0,\cdots ,0\right)$$

and it has the following property:

$$\frac{\beta \left({\sigma }_{j}v\right)}{{\sigma }_j^2}=L\beta \left(v\right)$$

for all $v\in V$, then there exists a unique quadratic mapping $Q_2:V\to W$ satisfying the functional Equation (2) and such that

$$\parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \frac{L^{1-j}}{1-L}\beta \left(v\right)$$

for all $v\in V$.

Proof. 

Consider the following set:

$$\mathrm{\Psi }:=\{q:V\to W,q(0)=0\}$$

and allow a general metric d on $\mathrm{\Psi }$ such that

$$d(p,q)=inf\{c\in (0,\infty ):\parallel p(v)-q(v)\parallel \le c\beta (v),\mathrm{for}\mathrm{all}v\in V\}.$$

It is clear that $(\mathrm{\Psi },d)$ is complete. Define a mapping $F:\mathrm{\Psi }\to \mathrm{\Psi }$ by

$$Fp\left(v\right)=\frac{\varphi \left({\sigma }_{j}v\right)}{{\sigma }_j^2},v\in V.$$

For all $p,q\in \mathrm{\Psi }$, we obtain

$$\begin{array}{ccc}\hfill d(p,q)& =& c\hfill \\ \hfill \Rightarrow \parallel p\left(v\right)-q\left(v\right)\parallel & \le & c\beta \left(v\right)\hfill \\ \hfill \Rightarrow \parallel \frac{p\left({\sigma }_{j}v\right)}{{\sigma }_j^2}-\frac{q\left({\sigma }_{j}v\right)}{{\sigma }_j^2}\parallel & \le & \frac{1}{{\sigma }_j^2}c\beta \left({\sigma }_{j}v\right)\hfill \\ \hfill \Rightarrow \parallel Fp\left(v\right)-Fq\left(v\right)\parallel & \le & \frac{1}{{\sigma }_j^2}c\beta \left({\sigma }_{j}v\right)\hfill \\ \hfill \Rightarrow \parallel Fp\left(v\right)-Fq\left(v\right)\parallel & \le & Lc\beta \left(v\right)\hfill \\ \hfill \Rightarrow d(Fp,Fq)& \le & cL\hfill \\ \hfill i.e.,d(Fp,Fq)& \le & Ld(p,q).\hfill \end{array}$$

As a result, a strictly contractive function F on $\mathrm{\Psi }$ with L is obtained. It is clear from (6) that

$$\parallel 2^2\varphi \left(v\right)-\varphi \left(2v\right)\parallel \le \chi (v,v,0,\cdots ,0)$$

(12)

for all $v\in V$. We have $j=0$ by using the above inequality and definitions of $\beta \left(v\right)$.

$$\begin{array}{ccc}\hfill \parallel \varphi \left(v\right)-\frac{\varphi \left(2v\right)}{2^2}\parallel & \le & \frac{1}{2^2}\beta \left(v\right)\hfill \\ \hfill \Rightarrow \parallel \varphi \left(v\right)-F\varphi \left(v\right)\parallel & \le & L\beta \left(v\right).\hfill \end{array}$$

Hence, we obtain the following:

$$d(F\varphi ,\varphi )\le L=L^{1-j}$$

(13)

for all $v\in V$. Replacing v by $\frac{v}{2}$ in (12), we obtain

$$\begin{array}{c}\hfill \Vert 2^2\varphi \left(\frac{v}{2}\right)-\varphi \left(v\right)\Vert \le \varphi \left(\frac{v}{2},\frac{v}{2},0,\cdots ,0\right)\end{array}$$

(14)

for all $v\in V$. Using the definition of $\beta \left(v\right)$ in the above inequality (14) for $j=0$, we have

$$\begin{array}{ccc}\hfill \parallel 2^2\varphi \left(2^{-1}v\right)-\varphi \left(v\right)\parallel & \le & \beta \left(v\right)\hfill \\ \hfill \Rightarrow \parallel \varphi \left(v\right)-F\varphi \left(v\right)\parallel & \le & \beta \left(v\right)\hfill \end{array}$$

for all $v\in V$. Hence, we obtain

$$d(\varphi ,F\varphi )\le 1=L^{1-j}$$

(15)

for all $v\in V$. Using (13) and (15), we can conclude that

$$d(\varphi ,F\varphi )\le L^{1-j}<\infty$$

for all $v\in V$. Now, in both cases, the fixed point alternative theorem suggests that exists a fixed point $Q_2$ of F in $\mathrm{\Psi }$ such that

$$Q_2\left(v\right)=\underset{l\to \infty }{lim}\frac{\varphi \left({\sigma }_j^{l}v\right)}{{\sigma }_j^{2l}}$$

for all $v\in V$. In order to prove that $Q_2:V\to W$ 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 $\mathsf{\Theta }=\{\varphi \in \mathrm{\Psi }/d(\varphi ,Q_2)<\infty \}$, thus, the function $Q_2$ is a unique function such that

$$\begin{array}{ccc}\hfill d(\varphi ,Q_2)& \le & \frac{1}{1-L}d(\varphi ,F\varphi )\hfill \\ \hfill i.e.,\parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel & \le & \frac{L^{1-j}}{1-L}\beta \left(v\right),v\in V.\hfill \end{array}$$

The proof of the Theorem is now complete. □

Corollary 3.

If a mapping $\varphi :V\to W$ with $\varphi \left(0\right)=0$ and such that

$$\begin{array}{c}\hfill \parallel \mathsf{\Lambda }\varphi (v_1,v_2,\cdots ,v_m)\parallel \le \left\{\begin{array}{c}\lambda ;\hfill \\ \lambda \left(\sum _{j=1}^m{\parallel v_j\parallel }^{\alpha }\right);\hfill \\ \lambda \left(\sum _{j=1}^m\parallel v_j{\parallel }^{m\alpha }+\prod _{j=1}^m{\parallel v_j\parallel }^{\alpha }\right),\hfill \end{array}\right.\end{array}$$

for all $v_1,v_2,\cdots ,v_m\in V$, where λ and α are two non-negative real numbers, then there exists a unique quadratic mapping $Q_2:V\to W$ which satisfies the following:

$$\begin{array}{c}\hfill \parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \left\{\begin{array}{c}\frac{\lambda }{\left|3\right|};\hfill \\ \frac{{2\lambda \parallel v\parallel }^{\alpha }}{|2^2-2^{\alpha }|},\alpha \ne 2;\hfill \\ \frac{{2\lambda \parallel v\parallel }^{m\alpha }}{|2^2-2^{m\alpha }|},m\alpha \ne 2;\hfill \end{array}\right.\end{array}$$

(16)

for all $v\in V$.

Proof. 

We set

$$\begin{array}{c}\hfill \chi (v_1,v_2,\cdots ,v_m)\le \left\{\begin{array}{c}\lambda ;\hfill \\ \lambda \left(\sum _{j=1}^m{\parallel v_j\parallel }^{\alpha }\right);\hfill \\ \lambda \left(\sum _{j=1}^m\parallel v_j{\parallel }^{m\alpha }+\prod _{j=1}^m{\parallel v_j\parallel }^{\alpha }\right),\hfill \end{array}\right.\end{array}$$

for all $v_1,v_2,\cdots ,v_m\in V$. Now,

$$\begin{array}{ccc}\hfill \frac{\chi ({\sigma }_j^l{v}_1,{\sigma }_j^l{v}_2,\cdots ,{\sigma }_j^l{v}_m)}{{\sigma }_j^{2l}}& =& \left\{\begin{array}{c}\frac{\lambda }{{\sigma }_j^{2l}};\hfill \\ \frac{\lambda }{{\sigma }_j^{2l}}\left(\sum _{1\le i\le m}{\parallel {\sigma }_j^l{v}_i\parallel }^{\alpha }\right);\hfill \\ \frac{\lambda }{{\sigma }_j^{2l}}\left(\sum _{1\le i\le m}\parallel {\sigma }_j^l{v}_i{\parallel }^{m\alpha }+\prod _{1\le i\le m}{\parallel {\sigma }_j^l{v}_i\parallel }^{\alpha }\right);\hfill \end{array}\right.\hfill \\ & =& \left\{\begin{array}{c}\to 0\mathrm{as}l\to \infty ;\hfill \\ \to 0\mathrm{as}l\to \infty ;\hfill \\ \to 0\mathrm{as}l\to \infty ,\hfill \end{array}\right.\hfill \end{array}$$

In other words, (11) holds. As such, we obtain the following.

$$\beta \left(v\right)=\chi \left(\frac{v}{2},\frac{v}{2},0,\cdots ,0\right)=\left\{\begin{array}{c}\frac{\lambda }{4};\hfill \\ \frac{2\lambda }{2^{\alpha }}{\parallel v\parallel }^{\alpha };\hfill \\ \frac{2\lambda }{2^{m\alpha }}{\parallel v\parallel }^{m\alpha }.\hfill \end{array}\right.$$

Moreover,

$$\begin{array}{ccc}\hfill \frac{1}{{\sigma }_j^2}\beta \left({\sigma }_{j}v\right)& =& \left\{\begin{array}{c}\frac{\lambda }{{\sigma }_j^2},\hfill \\ \frac{{2\lambda \parallel v\parallel }^{\alpha }{\sigma }_j^{\alpha }}{2^{\alpha }{\sigma }_j^2}\hfill \\ \frac{{2\lambda \parallel v\parallel }^{m\alpha }{\sigma }_j^{m\alpha }}{2^{m\alpha }{\sigma }_j^2}\hfill \end{array}\right.\hfill \\ & =& \left\{\begin{array}{c}{\sigma }_j^{-2}\beta \left(v\right),\hfill \\ {\sigma }_j^{\alpha -2}\beta \left(v\right),\hfill \\ {\sigma }_j^{m\alpha -2}\beta \left(v\right),\hfill \end{array}\right.\hfill \end{array}$$

for all $v\in V$. Hence, Equation (2) holds for the following.

$$\begin{array}{ccc}\hfill L& =& 2^{-2}\mathrm{if}j=0\mathrm{and}L=\frac{1}{2^{-2}}\mathrm{if}j=1.\hfill \\ \hfill L& =& 2^{\alpha -2}\mathrm{for}\alpha <2\mathrm{if}j=0\mathrm{and}L=\frac{1}{2^{\alpha -2}}\mathrm{for}\alpha >2\mathrm{if}j=1.\hfill \\ \hfill L& =& 2^{m\alpha -2}\mathrm{for}m\alpha <2\mathrm{if}j=0\mathrm{and}L=\frac{1}{2^{m\alpha -2}}\mathrm{for}m\alpha >2\mathrm{if}j=1.\hfill \end{array}$$

From the above conditions, we obtain our needed outcomes of (16). □

4. Stability Results in Non-Archimedean Banach Spaces: Direct Technique

Theorem 4.

Let a mapping $\chi :V^m\to [0,\infty )$ and $\varphi :V\to W$ be a mapping that satisfies $\varphi \left(0\right)=0$ and (4) with

$$\underset{n\to \infty }{lim}{\left|2\right|}^{2n}\chi \left(2^{-n}{v}_1,2^{-n}{v}_2,\cdots ,2^{-n}{v}_m\right)=0.$$

(17)

Then, there exists a unique quadratic mapping $Q_2:V\to W$ that satisfies

$$\begin{array}{c}\hfill \parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \underset{n\in \mathbb{N}}{sup}\left\{{\left|2\right|}^{2(n-1)}\chi \left(\frac{v}{2^n},\frac{v}{2^n},0,\cdots ,0\right)\right\}\end{array}$$

(18)

for all $v\in V$.

Proof. 

Switching $(v_1,v_2,\cdots ,v_m)$ by $(v,v,0,\cdots ,0)$ in (4), we obtain

$$\begin{array}{c}\hfill \parallel \varphi \left(2v\right)-2^2\varphi \left(v\right)\parallel \le \chi (v,v,0,\cdots ,0)\end{array}$$

(19)

for all $v\in V$. Thus,

$$\begin{array}{c}\hfill \parallel \varphi \left(v\right)-2^2\varphi \left(\frac{v}{2}\right)\parallel \le \chi \left(\frac{v}{2},\frac{v}{2},0,\cdots ,0\right)\end{array}$$

for all $v\in V$. Hence, we have

$$\begin{array}{ccc}\hfill & & \parallel 2^{2l}\varphi \left(\frac{v}{2^l}\right)-2^{2p}\varphi \left(\frac{v}{2^p}\right)\parallel \hfill \\ & & \le max\left\{\parallel 2^{2l}\varphi \left(\frac{v}{2^l}\right)-2^{2(l+1)}\varphi \left(\frac{v}{2^{l+1}}\right)\parallel ,\cdots ,\parallel 2^{2(p-1)}\varphi \left(\frac{v}{2^{p-1}}\right)-2^{2p}\varphi \left(\frac{v}{2^p}\right)\parallel \right\}\hfill \\ & & \le \left\{{\left|2\right|}^{2l}\parallel \varphi \left(\frac{v}{2^l}\right)-2^2\varphi \left(\frac{v}{2^{l+1}}\right)\parallel ,\cdots ,{\left|2\right|}^{2(p-1)}\parallel \varphi \left(\frac{v}{2^{p-1}}\right)-2^2\varphi \left(\frac{v}{2^p}\right)\parallel \right\}\hfill \\ & & \le \underset{n\in \{l,l+1,\cdots \}}{sup}\left\{{\left|2\right|}^{2n}\chi \left(\frac{v}{2^{n+1}},\frac{v}{2^{n+1}},0,\cdots ,0\right)\right\}\hfill \end{array}$$

(20)

for all $p>l>0$ and for all $v\in V$. As a result of (20), the sequence $\left\{2^{2n}\varphi \left(\frac{v}{2^n}\right)\right\}$ is a Cauchy sequence for every $v\in V$. Since W is complete, the sequence $\left\{2^{2n}\varphi \left(\frac{v}{2^n}\right)\right\}$ converges. As a result, the mapping $Q_2:V\to W$ may be defined

$$Q_2\left(v\right):=\underset{l\to \infty }{lim}{2}^{2l}\varphi \left(2^{-l}v\right)$$

for all $v\in V$. Taking $l=0$ and the limit $p\to \infty$ in (20), we obtain (18). As a result of (17) and (4), we have

$$\begin{array}{ccc}\hfill \parallel \mathsf{\Lambda }{Q}_2(v_1,v_2,\cdots ,v_m)\parallel & =& \underset{n\to \infty }{lim}{\left|2\right|}^{2n}\parallel \mathsf{\Lambda }\varphi (2^{-n}{v}_1,2^{-n}{v}_2,\cdots ,2^{-n}{v}_m)\parallel \hfill \\ & \le & \underset{n\to \infty }{lim}{\left|2\right|}^{2n}\chi \left(2^{-n}{v}_1,2^{-n}{v}_2,\cdots ,2^{-n}{v}_m\right)=0\hfill \end{array}$$

for all $v_1,v_2,\cdots ,v_m\in V$. Thus, we obtain

$$\mathsf{\Lambda }{Q}_2\left(v_1,v_2,\cdots ,v_m\right)=0.$$

From Lemma 1, the mapping $Q_2:V\to W$ is quadratic. Now, consider another quadratic mapping $R_2:V\to W$ that satisfies inequality (18). Then, we obtain

$$\begin{array}{ccc}\hfill \parallel Q_2\left(v\right)-R_2\left(v\right)\parallel & =& \parallel 2^{2k}{Q}_2\left(\frac{v}{2^k}\right)-2^{2k}{R}_2\left(\frac{v}{2^k}\right)\parallel \hfill \\ & \le & max\left\{\parallel 2^{2k}{Q}_2\left(\frac{v}{2^k}\right)-2^{2k}\varphi \left(\frac{v}{2^k}\right)\parallel ,\parallel 2^{2k}{R}_2\left(\frac{v}{2^k}\right)-2^{2k}\varphi \left(\frac{v}{2^k}\right)\parallel \right\}\hfill \\ & \le & \underset{n\in \mathbb{N}}{sup}\left\{{\left|2\right|}^{2(k+n-1)}\chi \left(\frac{v}{2^{k+n}},\frac{v}{2^{k+n}},0,\cdots ,0\right)\right\},\hfill \\ & & \to 0\mathrm{as}k\to \infty .\hfill \end{array}$$

Thus, we may infer that $Q_2\left(v\right)=R_2\left(v\right)$ for all $v\in V$. This proves the uniqueness of $Q_2$. As a result, the mapping $Q_2:V\to W$ is a unique quadratic mapping that satisfies (18). □

Corollary 4.

If a mapping $\varphi :V\to W$ with $\varphi \left(0\right)=0$ and such that

$$\parallel \mathsf{\Lambda }\varphi (v_1,v_2,\cdots ,v_m)\parallel \le \alpha \left(\sum _{j=1}^m{\parallel v_j\parallel }^{\lambda }\right)$$

(21)

for all $v_1,v_2,\cdots ,v_m\in V$, then there exists a unique quadratic mapping $Q_2:V\to W$ that satisfies

$$\begin{array}{c}\hfill \parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \frac{2\alpha }{{\left|2\right|}^{\lambda }}{\parallel v\parallel }^{\lambda }\end{array}$$

for all $v\in V$, where $\lambda <2$ and α are in ${\mathbb{R}}^{+}$.

Corollary 5.

If there is a mapping $\varphi :V\to W$ with $\varphi \left(0\right)=0$ and satisfies

$$\parallel \mathsf{\Lambda }\varphi (v_1,v_2,\cdots ,v_m)\parallel \le \alpha \left(\sum _{j=1}^m\parallel v_j{\parallel }^{m\lambda }+\prod _{j=1}^m{\parallel v_j\parallel }^{\lambda }\right)$$

(22)

for all $v_1,v_2,\cdots ,v_m\in V$, then there exists a unique quadratic mapping $Q_2:V\to W$ such that

$$\begin{array}{c}\hfill \parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \frac{2\alpha }{{\left|2\right|}^{m\lambda }}{\parallel v\parallel }^{m\lambda }\end{array}$$

for all $v\in V$, where $m\lambda <2$ and α are in ${\mathbb{R}}^{+}$.

Theorem 5.

If a mapping $\chi :V^m\to [0,\infty )$ and a mapping $\varphi :V\to W$ with $\varphi \left(0\right)=0$ exists and satisfies (4) and the following

$$\underset{n\to \infty }{lim}\left\{\frac{1}{{\left|2\right|}^{2n}}\chi \left(2^{n-1}{v}_1,2^{n-1}{v}_2,\cdots ,2^{n-1}{v}_m\right)\right\}=0,v_1,v_2,\cdots ,v_m\in V.$$

then there exists a unique quadratic mapping $Q_2:V\to W$ such that

$$\begin{array}{c}\hfill \parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \underset{n\in \mathbb{N}}{sup}\left\{\frac{1}{{\left|2\right|}^{2n}}\chi \left(2^{n-1}v,2^{n-1}v,0,\cdots ,0\right)\right\}\end{array}$$

(23)

for all $v\in V$.

Proof. 

It follows from (19) that

$$\parallel \varphi \left(v\right)-2^{-2}\varphi \left(2v\right)\parallel \le \frac{1}{2^2}\chi (v,v,0,\cdots ,0)$$

for all $v\in V$. Hence,

$$\begin{array}{ccc}\hfill & & \parallel \frac{1}{2^{2l}}\varphi \left(2^{l}v\right)-\frac{1}{2^{2p}}\varphi \left(2^{p}v\right)\parallel \hfill \\ \hfill & & \le max\left\{\parallel \frac{1}{2^{2l}}\varphi \left(2^{l}v\right)-\frac{1}{2^{2(l+1)}}\varphi \left(2^{l+1}v\right)\parallel ,\cdots ,\parallel \frac{1}{2^{2(p-1)}}\varphi \left(2^{p-1}v\right)-\frac{1}{2^{2p}}\varphi \left(2^{p}v\right)\parallel \right\}\hfill \\ \hfill & & \le max\left\{\frac{1}{{\left|2\right|}^{2l}}\parallel \varphi \left(2^{l}v\right)-\frac{1}{2^2}\varphi \left(2^{(l+1)}v\right)\parallel ,\cdots ,\frac{1}{{\left|2\right|}^{2(p-1)}}\parallel \varphi \left(2^{p-1}v\right)-\frac{1}{2^2}\varphi \left(2^{p}v\right)\parallel \right\}\hfill \\ \hfill & & \le \underset{n\in \{l,l+1,\cdots \}}{sup}\left\{\frac{1}{{\left|2\right|}^{2(n+1)}}\chi (2^{n}v,2^{n}v,0,\cdots ,0)\right\}\hfill \end{array}$$

(24)

for all $p>l>0$. As a result of (24), $\left\{\frac{1}{2^{2n}}\varphi \left(2^{n}v\right)\right\}$ is a Cauchy sequence.

Since W is complete, $\left\{\frac{1}{2^{2n}}\varphi \left(2^{n}v\right)\right\}$ converges. Thus, we can define a mapping $Q_2:V\to W$ by

$$Q_2\left(v\right):=\underset{n\to \infty }{lim}\frac{1}{2^{2n}}\varphi \left(2^{n}v\right)$$

for all $v\in V$. Now, taking $l=0$ and the limit $p\to \infty$ 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 $\varphi :V\to W$ with $\varphi \left(0\right)=0$ and it satisfies the inequality (21), then there exists a unique quadratic mapping $Q_2:V\to W$ such that

$$\begin{array}{c}\hfill \parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \frac{2\alpha }{{\left|2\right|}^2}{\parallel v\parallel }^{\lambda }\end{array}$$

for all $v\in V$, where $\lambda >2$ and α are in ${\mathbb{R}}^{+}$.

Corollary 7.

If there exists a mapping $\varphi :V\to W$ with $\varphi \left(0\right)=0$ and it satisfies (22), then there exists a unique quadratic mapping $Q_2:V\to W$ such that

$$\begin{array}{c}\hfill \parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \frac{2\alpha }{{\left|2\right|}^2}{\parallel v\parallel }^{m\lambda }\end{array}$$

for all $v\in V$, where $m\lambda >2$ and α are in ${\mathbb{R}}^{+}$.

5. Stability Results in Non-Archimedean Banach Spaces: Fixed Point Technique

Theorem 6.

Let a mapping $\chi :V^m\to [0,\infty )$ such that there is $L<1$ with

$$\chi \left(\frac{v_1}{2},\frac{v_2}{2},\cdots ,\frac{v_m}{2}\right)\le \frac{L}{{\left|2\right|}^2}\chi \left(v_1,v_2,\cdots ,v_m\right),v_1,v_2,\cdots ,v_m\in V.$$

(25)

Let a mapping $\varphi :V\to W$ which satisfies $\varphi \left(0\right)=0$ and (4). Then, there exists a unique quadratic mapping $Q_2:V\to W$ such that

$$\parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \frac{L}{{\left|2\right|}^2(1-L)}\chi (v,v,0,·,0)$$

for all $v\in V$.

Proof. 

Replacing $(v_1,v_2,\cdots ,v_m)$ by $(v,v,0,\cdots ,0)$ in (4), we obtain

$$\parallel \varphi \left(2v\right)-2^2\varphi \left(v\right)\parallel \le \chi (v,v,0,\cdots ,0)$$

(26)

for all $v\in V$. Let us consider the set

$$M:=\{q:V\to W,q(0)=0\}$$

as well as the generalised metric d on M:

$$d(p,q)=inf\left\{\theta \in {\mathbb{R}}_{+}:\parallel p\left(v\right)-q\left(v\right)\parallel \le \theta \chi (v,v,0,\cdots ,0),\mathrm{for}\mathrm{all}v\in V\right\},$$

where, as is typical, $inf\varnothing =+\infty$. It is simple to demonstrate that $(M,d)$ is complete (see [20]). Now, we examine the linear mapping $F:M\to M$, which has the following property:

$$Fp\left(v\right):=2^{2}p\left(\frac{v}{2}\right)$$

for all $v\in V$. Let $p,q\in M$ be given such that $d(p,q)=ϵ$. Then, we have

$$\parallel p\left(v\right)-q\left(v\right)\parallel \le ϵ\chi (v,v,0,\cdots ,0)$$

for all $v\in V$. Hence,

$$\begin{array}{ccc}\hfill \parallel Fp\left(v\right)-Fq\left(v\right)\parallel & =& \parallel 2^{2}p\left(\frac{v}{2}\right)-2^{2}q\left(\frac{v}{2}\right)\parallel \hfill \\ & \le & {\left|2\right|}^2ϵ\chi \left(\frac{v}{2},\frac{v}{2},0,\cdots ,0\right)\hfill \\ & \le & {\left|2\right|}^2ϵ\frac{L}{{\left|2\right|}^2}\chi (v,v,0,\cdots ,0)\hfill \\ & \le & Lϵ\chi (v,v,0,\cdots ,0)\hfill \end{array}$$

for all $v\in V$. Thus, $d(p,q)=ϵ$ implies that

$$d(Fp,Fq)\le ϵL.$$

This means that

$$\begin{array}{c}\hfill d(Fp,Fq)\le Ld(p,q)\end{array}$$

for all $p,q\in M$. It follows from (26) that

$$\begin{array}{ccc}\hfill \parallel \varphi \left(v\right)-2^2\varphi \left(\frac{v}{2}\right)\parallel & \le & \chi \left(\frac{v}{2},\frac{v}{2},0,\cdots ,0\right)\hfill \\ & \le & \frac{L}{{\left|2\right|}^2}\chi (v,v,0\cdots ,0)\hfill \end{array}$$

for all $v\in V$. Thus, $d(\varphi ,F\varphi )\le \frac{L}{{\left|2\right|}^2}$. From Theorem 1, there exists a quadratic mapping $Q_2:V\to W$ satisfying the following:

(1) $Q_2$ is a fixed point of F

$$i.e.,Q_2\left(v\right)=2^{2}Q\left(\frac{v}{2}\right),v\in V.$$

(27)

The function $Q_2$ is a unique fixed point of M in the set

$$T=\{p\in M:d(\varphi ,p)<\infty \}.$$

This yields that $Q_2$ is a unique function satisfying (27) such that there exists $\theta \in (0,\infty )$ satisfying

$$\begin{array}{c}\hfill \parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \theta \chi (v,v,0,\cdots ,0),v\in V.\end{array}$$

(2) $d(F^n\varphi ,Q_2)\to 0$ as $n\to \infty$. This indicates the below equality

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{2}^{2n}\varphi \left(2^{-n}v)\right)=Q_2\left(v\right),v\in V.\end{array}$$

(3) $d(\varphi ,Q_2)\le \frac{1}{1-L}d\left(\varphi ,F\varphi \right)$, and it implies the following:

$$\begin{array}{c}\hfill \parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \frac{L}{{\left|2\right|}^2(1-L)}\chi (v,v,0,\cdots ,0)\end{array}$$

for all $v\in V$. It follows from (25) and (4) that

$$\begin{array}{ccc}\hfill \parallel \mathsf{\Lambda }{Q}_2(v_1,v_2,\cdots ,v_m)\parallel & =& \underset{n\to \infty }{lim}{\left|2\right|}^{2n}\parallel \mathsf{\Lambda }\varphi (2^{-n}{v}_1,2^{-n}{v}_2,\cdots ,2^{-n}{v}_m)\parallel \hfill \\ & \le & \underset{n\to \infty }{lim}{\left|2\right|}^{2n}\chi \left(2^{-n}{v}_1,2^{-n}{v}_2,\cdots ,2^{-n}{v}_m\right)=0.\hfill \end{array}$$

Thus,

$$\mathsf{\Lambda }{Q}_2(v_1,v_2,\cdots ,v_m)=0,v_1,v_2,\cdots ,v_m\in V.$$

By Lemma 1, the mapping $Q_2:V\to W$ is quadratic. □

Corollary 8.

If a mapping $\varphi :V\to W$ satisfies $\varphi \left(0\right)=0$ and the following:

$$\parallel \mathsf{\Lambda }\varphi (v_1,v_2,\cdots ,v_m)\parallel \le \alpha \left(\sum _{j=1}^m{\parallel v_j\parallel }^{\lambda }\right)$$

(28)

for all $v_1,v_2,\cdots ,v_m\in V$, where $\lambda <2$ and α are two non-negative real numbers, then there exists a unique quadratic mapping $Q_2:V\to W$ such that

$$\parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \frac{{2\alpha \parallel v\parallel }^{\lambda }}{{\left|2\right|}^{\lambda }-{\left|2\right|}^2}$$

for all $v\in V$.

Proof. 

The proof is based on Theorem 6 by allowing the following:

$$\chi (v_1,v_2,\cdots ,v_m)=\alpha \left(\sum _{j=1}^m{\parallel v_j\parallel }^{\lambda }\right)$$

for all $v_1,v_2,\cdots ,v_m\in V$. After that, we may use $L={\left|2\right|}^{2-\lambda }$ to obtain our desired result. □

Corollary 9.

If a mapping $\varphi :V\to W$ satisfies $\varphi \left(0\right)=0$ and the following:

$$\parallel \mathsf{\Lambda }\varphi (v_1,v_2,\cdots ,v_m)\parallel \le \alpha \left(\sum _{j=1}^m\parallel v_j{\parallel }^{m\lambda }+\prod _{j=1}^m{\parallel v_j\parallel }^{\lambda }\right)$$

(29)

for all $v_1,v_2,\cdots ,v_m\in V$, where $m\lambda <2$ and α are two non-negative real numbers, then there exists a unique quadratic mapping $Q_2:V\to W$ such that

$$\parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \frac{{2\alpha \parallel v\parallel }^{m\lambda }}{{\left|2\right|}^{m\lambda }-{\left|2\right|}^2}$$

for all $v\in V$.

Proof. 

The proof is based on Theorem 6 by allowing the following:

$$\chi (v_1,v_2,\cdots ,v_m)=\alpha \left(\sum _{j=1}^m\parallel v_j{\parallel }^{m\lambda }+\prod _{j=1}^m{\parallel v_j\parallel }^{\lambda }\right)$$

for all $v_1,v_2,\cdots ,v_m\in V$. After that, we may use $L={\left|2\right|}^{2-m\lambda }$ to obtain our desired result. □

Theorem 7.

Let a mapping $\chi :V^m\to [0,\infty )$ such that there is $L<1$ with the following:

$$\chi \left(v_1,v_2,\cdots ,v_m\right)\le \left|2^2\right|L\chi \left(2^{-1}{v}_1,2^{-1}{v}_2,\cdots ,2^{-1}{v}_m\right)$$

for all $v_1,v_2,\cdots ,v_m\in V$. If a mapping $\varphi :V\to W$ satisfies $\varphi \left(0\right)=0$ and (4), then there exists a unique quadratic mapping $Q_2:V\to W$ such that

$$\parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \frac{1}{|2^2|(1-L)}\chi (v,v,0,\cdots ,0)$$

for all $v\in V$.

Proof. 

It follows from (26) that

$$\parallel \varphi \left(v\right)-\frac{1}{2^2}\varphi \left(2v\right)\parallel \le \frac{1}{{\left|2\right|}^2}\chi (v,v,0,\cdots ,0)$$

(30)

for all $v\in V$. Let $(M,d)$ denote the generalised metric space noted in Theorem 25. Now, let us consider the linear mapping $F:M\to M$ that satisfies the following:

$$Fp\left(v\right):=\frac{1}{2^2}p\left(2v\right)$$

for all $v\in V$. This comes from (30) that

$$d(\varphi ,F\varphi )\le \frac{1}{{\left|2\right|}^2}.$$

Thus,

$$\parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \frac{1}{{\left|2\right|}^2(1-L)}\chi (v,v,0,\cdots ,0)$$

for all $v\in V$. The remaining part of the proof is similar to that of Theorem 6. □

Corollary 10.

If a mapping $\varphi :V\to W$ satisfies $\varphi \left(0\right)=0$ and (28), then there exists a unique quadratic mapping $Q_2:V\to W$ such that

$$\parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \frac{{2\alpha \parallel v\parallel }^{\lambda }}{{\left|2\right|}^2-{\left|2\right|}^{\lambda }}$$

for all $v\in V$, where $\lambda >2$ and α are two positive real numbers.

Proof. 

The proof is based on Theorem 7 by allowing the following:

$$\chi (v_1,v_2,\cdots ,v_m)=\alpha \left(\sum _{j=1}^m{\parallel v_j\parallel }^{\lambda }\right).$$

Then, we can take $L={\left|2\right|}^{\lambda -2}$, and we obtain our result. □

Corollary 11.

If a mapping $\varphi :V\to W$ satisfies $\varphi \left(0\right)=0$ and (29), then there exists a unique quadratic mapping $Q_2:V\to W$ such that

$$\parallel \varphi \left(v\right)-Q_2\left(v\right)\parallel \le \frac{{2\alpha \parallel v\parallel }^{m\lambda }}{{\left|2\right|}^2-{\left|2\right|}^{m\lambda }}$$

for all $v\in V$, where $m\lambda >2$ and α are two non-negative real numbers.

Proof. 

The proof is based on Theorem 7 by allowing the following:

$$\chi (v_1,v_2,\cdots ,v_m)=\alpha \left(\sum _{j=1}^m\parallel v_j{\parallel }^{m\lambda }+\prod _{j=1}^m{\parallel v_j\parallel }^{\lambda }\right).$$

Then, we can take $L={\left|2\right|}^{m\lambda -2}$ and we obtain our result. □

6. 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-$\beta$-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.