Hyers-Ulam-Rassias stability of the additive-quadratic mappings in non-Archimedean Banach spaces

Abstract

Using the fixed point and direct methods, we prove the generalized Hyers-Ulam stability of the following additive-quadratic functional equation in non-Archimedean normed spaces

$$\begin{array}{c}r[f\left(\frac{x+y+z}{s}\right)+f\left(\frac{x-y+z}{s}\right)+f\left(\frac{x+y-z}{s}\right)+f\left(\frac{-x+y+z}{s}\right)]\hfill \\ =\gamma f(x)+\gamma f(y)+\gamma f(z),\hfill \end{array}$$

where r, s, γ are positive real numbers.

MSC:39B55, 46S10.

1 Introduction and preliminaries

A classical question in the theory of functional equations is the following: ‘When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation?’ If the problem accepts a solution, we say that the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [44] in 1940. In the next year, Hyers [23] gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces. In 1978, Rassias [39] proved a generalization of Hyers’ theorem for additive mappings. This new concept is known as generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. Furthermore, in 1994, a generalization of Rassias’ theorem was obtained by Gǎvruta [21] by replacing the bound $ϵ({\parallel x\parallel }^p+{\parallel y\parallel }^p)$ by a general control function $\phi (x,y)$.

In 1983, a generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [43] for mappings $f:X\to Y$, where X is a normed space and Y is a Banach space. In 1984, Cholewa [11] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group and, in 2002, Czerwik [13] proved the generalized Hyers-Ulam stability of the quadratic functional equation. The reader is referred to [1–42] and references therein for detailed information on stability of functional equations.

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 (see [16, 25–27, 33]).

Definition 1.1 By a non-Archimedean field, we mean a field $\mathbb{K}$ equipped with a function (valuation) $|\cdot |:\mathbb{K}\to [0,\mathrm{\infty })$ such that for all $r,s\in \mathbb{K}$, the following conditions hold:

• (1) $|r|=0$ if and only if $r=0$;

• (2) $|rs|=|r||s|$;

• (3) $|r+s|\le max\{|r|,|s|\}$.

Definition 1.2 Let X be a vector space over a scalar field $\mathbb{K}$ with a non-Archimedean non-trivial valuation $|\cdot |$. A function $\parallel \cdot \parallel :X\to R$ is a non-Archimedean norm (valuation) if it satisfies the following conditions:

• (1) $\parallel x\parallel =0$ if and only if $x=0$;

• (2) $\parallel rx\parallel =|r\parallel |x\parallel$ ($r\in \mathbb{K}$, $x\in X$);

• (3) The strong triangle inequality (ultrametric); namely,

  $$\parallel x+y\parallel \le max\{\parallel x\parallel ,\parallel y\parallel \},x,y\in X.$$

Then $(X,\parallel \cdot \parallel )$ is called a non-Archimedean space.

Due to the fact that

$$\parallel x_n-x_m\parallel \le max\{\parallel x_{j+1}-x_j\parallel :m\le j\le n-1\}(n>m).$$

Definition 1.3 A sequence $\{x_n\}$ is Cauchy if and only if $\{x_{n+1}-x_n\}$ converges to zero in a non-Archimedean space. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent.

Definition 1.4 Let X be a set. A function $d:X\times X\to [0,\mathrm{\infty }]$ is called a generalized metric on X if d satisfies

• (1) $d(x,y)=0$ if and only if $x=y$;

• (2) $d(x,y)=d(y,x)$ for all $x,y\in X$;

• (3) $d(x,z)\le d(x,y)+d(y,z)$ for all $x,y,z\in X$.

We recall a fundamental result in fixed point theory.

Theorem 1.5 ([13, 17])

Let $(X,d)$ be a complete generalized metric space and let $J:X\to X$ be a strictly contractive mapping with Lipschitz constant $\alpha <1$. Then for each given element $x\in X$, either

$$d(J^{n}x,J^{n+1}x)=\mathrm{\infty }$$

for all nonnegative integers n or there exists a positive integer $n_0$ such that

• (1) $d(J^{n}x,J^{n+1}x)<\mathrm{\infty }$, $\mathrm{\forall }n\ge n_0$;

• (2) the sequence $\{J^{n}x\}$ converges to a fixed point $y^{\ast }$ of J;

• (3) $y^{\ast }$ is the unique fixed point of J in the set $Y=\{y\in X\mid d(J^{n_0}x,y)<\mathrm{\infty }\}$;

• (4) $d(y,y^{\ast })\le \frac{1}{1-\alpha }d(y,Jy)$ for all $y\in Y$.

In 1996, G. Isac and Th. M. Rassias [24] were the first to provide applications of stability theory of functional equations for the proof of new fixed-point theorems with applications. By using fixed-point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [14, 15, 35, 36, 40]).

This paper is organized as follows: In Section 2, using the fixed-point method, we prove the Hyers-Ulam stability of the following additive-quadratic functional equation:

$$\begin{array}{r}rf\left(\frac{x+y+z}{s}\right)+rf\left(\frac{x-y+z}{s}\right)+rf\left(\frac{x+y-z}{s}\right)+rf\left(\frac{-x+y+z}{s}\right)\\ =\gamma f(x)+\gamma f(y)+\gamma f(z),\end{array}$$

(1.1)

where $x,y,z\in X$, in non-Archimedean normed space. In Section 3, using direct methods, we prove the Hyers-Ulam stability of the additive-quadratic functional equation (1.1) in non-Archimedean normed spaces.

It is easy to see that a mapping f with $f(0)=0$ is a solution of equation (1.1) if and only if f is of the form $f(x)=A(x)+Q(x)$ for all $x\in X$.

2 Stability of functional equation (1.1): a fixed point method

In this section, we deal with the stability problem for the additive-quadratic functional equation (1.1). In the rest of the present article, let $|2|\ne 1$.

Theorem 2.1 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let $\phi :X^3\to [0,\mathrm{\infty })$ be a function such that there exists an $\alpha <1$ with

$$\phi (2x,2y,2z)\le |2|\alpha \phi (x,y,z)$$

(2.1)

for all $x,y,z\in X$. Let $f:X\to Y$ be an odd mapping satisfying

$$\begin{array}{r}\parallel rf\left(\frac{x+y+z}{s}\right)+rf\left(\frac{x-y+z}{s}\right)+rf\left(\frac{x+y-z}{s}\right)\\ +rf\left(\frac{-x+y+z}{s}\right)-\gamma f(x)-\gamma f(y)-\gamma f(z){\parallel }_Y\le \phi (x,y,z)\end{array}$$

(2.2)

for all $x,y,z\in X$. Then there exists a unique additive mapping $A:X\to Y$ such that

$${\parallel f(x)-A(x)\parallel }_Y\le \frac{max\{\phi (2x,0,0),\phi (x,x,0)\}}{|2\gamma |(1-\alpha )}$$

(2.3)

for all $x\in X$.

Proof Putting $y=z=0$ in (2.2) and replacing x by 2x, we get

$${\parallel rf\left(\frac{2x}{s}\right)-\frac{\gamma }{2}f(2x)\parallel }_Y\le \frac{1}{|2|}\phi (2x,0,0)$$

(2.4)

for all $x\in X$. Putting $y=x$ and $z=0$ in (2.2), we have

$${\parallel rf\left(\frac{2x}{s}\right)-\gamma f(x)\parallel }_Y\le \frac{1}{|2|}\phi (x,x,0)$$

(2.5)

for all $x\in X$. By (2.4) and (2.5), we get

$$\begin{array}{rcl}{\parallel \frac{f(2x)}{2}-f(x)\parallel }_Y& =& \frac{1}{|\gamma |}{\parallel \frac{\gamma }{2}f(2x)\pm rf\left(\frac{2x}{s}\right)-\gamma f(x)\parallel }_Y\\ \le & \frac{1}{|\gamma |}max\{{\parallel rf\left(\frac{2x}{s}\right)-\frac{\gamma }{2}f(2x)\parallel }_Y,{\parallel rf\left(\frac{2x}{s}\right)-\gamma f(x)\parallel }_Y\}\\ \le & \frac{1}{|2\gamma |}max\{\phi (2x,0,0),\phi (x,x,0)\}.\end{array}$$

(2.6)

Consider the set $S:=\{h:X\to Y\}$ and introduce the generalized metric on S:

$$d(g,h)=inf\{\mu \in (0,+\mathrm{\infty }):{\parallel g(x)-h(x)\parallel }_Y\le \mu max\{\phi (2x,0,0),\phi (x,x,0)\},\mathrm{\forall }x\in X\},$$

where, as usual, $inf\varphi =+\mathrm{\infty }$. It is easy to show that $(S,d)$ is complete (see [30]). Now we consider the linear mapping $J:S\to S$ such that $Jg(x):=\frac{1}{2}g(2x)$ for all $x\in X$. Let $g,h\in S$ be given such that $d(g,h)=\epsilon$. Then

$${\parallel g(x)-h(x)\parallel }_Y\le ϵmax\{\phi (2x,0,0),\phi (x,x,0)\}$$

for all $x\in X$. Hence,

$$\begin{array}{rcl}{\parallel Jg(x)-Jh(x)\parallel }_Y& =& {\parallel \frac{1}{2}g(2x)-\frac{1}{2}h(2x)\parallel }_Y\\ =& \frac{{\parallel g(2x)-h(2x)\parallel }_Y}{|2|}\\ \le & \frac{ϵ}{|2|}max\{\phi (4x,0,0),\phi (2x,2x,0)\}\\ \le & \alpha \cdot ϵmax\{\phi (2x,0,0),\phi (x,x,0)\}\end{array}$$

for all $x\in X$. So $d(g,h)=\epsilon$ implies that $d(Jg,Jh)\le \alpha \epsilon$. This means that $d(Jg,Jh)\le \alpha d(g,h)$ for all $g,h\in S$.

It follows from (2.6) that $d(f,Jf)\le \frac{1}{|2\gamma |}$. By Theorem 1.5, there exists a mapping $A:X\to Y$ satisfying the following:

• (1) A is a fixed point of J, i.e.,

  $$2A(x)=A(2x)$$

  (2.7)

for all $x\in X$. The mapping A is a unique fixed point of J in the set $M=\{g\in S:d(h,g)<\mathrm{\infty }\}$. This implies that A is a unique mapping satisfying (2.7) such that there exists a $\mu \in (0,\mathrm{\infty })$ satisfying ${\parallel f(x)-A(x)\parallel }_Y\le \mu max\{\phi (2x,0,0),\phi (x,x,0)\}$ for all $x\in X$;

• (2) $d(J^{n}f,A)\to 0$ as $n\to \mathrm{\infty }$. This implies the equality

  $$\underset{n\to \mathrm{\infty }}{lim}\frac{f(2^{n}x)}{2^n}=A(x)\text{for all }x\in X;$$

• (3) $d(f,A)\le \frac{1}{1-\alpha }d(f,Jf)$, which implies the inequality $d(f,A)\le \frac{1}{|2\gamma |(1-\alpha )}$. This implies that the inequalities (2.3) holds.

It follows from (2.1) and (2.2) that

$$\begin{array}{c}\parallel rA\left(\frac{x+y+z}{s}\right)+rA\left(\frac{x-y+z}{s}\right)+rA\left(\frac{x+y-z}{s}\right)\hfill \\ +rA\left(\frac{-x+y+z}{s}\right)-\gamma A(x)-\gamma A(y)-\gamma A(z){\parallel }_Y\hfill \\ =\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{|2|}^n}\parallel rf\left(\frac{2^n(x+y+z)}{s}\right)+rf\left(\frac{2^n(x-y+z)}{s}\right)+rf\left(\frac{2^n(x+y-z)}{s}\right)\hfill \\ +rf\left(\frac{2^n(-x+y+z)}{s}\right)-\gamma f\left(2^{n}x\right)-\gamma f\left(2^{n}y\right)-\gamma f\left(2^{n}z\right){\parallel }_Y\hfill \\ \le \underset{n\to \mathrm{\infty }}{lim}\frac{1}{{|2|}^n}\phi (2^{n}x,2^{n}y,2^{n}z)\hfill \\ \le \underset{n\to \mathrm{\infty }}{lim}\frac{1}{{|2|}^n}\cdot {|2|}^n{\alpha }^n\phi (x,y,z)\hfill \\ =0\hfill \end{array}$$

for all $x,y,z\in X$. So

$$\begin{array}{c}rA\left(\frac{x+y+z}{s}\right)+rA\left(\frac{x-y+z}{s}\right)+rA\left(\frac{x+y-z}{s}\right)\hfill \\ +rA\left(\frac{-x+y+z}{s}\right)-\gamma A(x)-\gamma A(y)-\gamma A(z)=0\hfill \end{array}$$

for all $x,y,z\in X$. Hence, $A:X\to Y$ satisfying (1.1). This completes the proof. □

Corollary 2.2 Let θ be a positive real number and q is a real number with $q>1$. Let $f:X\to Y$ be an odd mapping satisfying

$$\begin{array}{r}\parallel rf\left(\frac{x+y+z}{s}\right)+rf\left(\frac{x-y+z}{s}\right)+rf\left(\frac{x+y-z}{s}\right)\\ +rf\left(\frac{-x+y+z}{s}\right)-\gamma f(x)-\gamma f(y)-\gamma f(z){\parallel }_Y\le \theta ({\parallel x\parallel }^q+{\parallel y\parallel }^q+{\parallel z\parallel }^q)\end{array}$$

(2.8)

for all $x,y,z\in X$. Then there exists a unique additive mapping $A:X\to Y$ such that

$${\parallel f(x)-A(x)\parallel }_Y\le \frac{2|2|\theta {\parallel x\parallel }^q}{|2\gamma |(|2|-{|2|}^q)}$$

for all $x\in X$.

Proof The proof follows from Theorem 2.1 by taking $\phi (x,y,z)=\theta ({\parallel x\parallel }^q+{\parallel y\parallel }^q+{\parallel z\parallel }^q)$ for all $x,y,z\in X$. Then we can choose $\alpha ={|2|}^{q-1}$ and we get the desired result. □

Theorem 2.3 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let $\phi :X^3\to [0,\mathrm{\infty })$ be a function such that there exists an $\alpha <1$ with

$$\phi (\frac{x}{2},\frac{y}{2},\frac{z}{2})\le \frac{\alpha }{|2|}\phi (x,y,z)$$

(2.9)

for all $x,y,z\in X$. Let $f:X\to Y$ be an odd mapping satisfying (2.2). Then there exists a unique additive mapping $A:X\to Y$ such that

$${\parallel f(x)-A(x)\parallel }_Y\le \frac{\alpha max\{\phi (2x,0,0),\phi (x,x,0)\}}{|2\gamma |(1-\alpha )}$$

for all $x\in X$.

Proof Let $(S,d)$ be the generalized metric space defined in the proof of Theorem 2.1.

Now we consider the linear mapping $J:S\to S$ such that

$$Jg(x):=2g\left(\frac{x}{2}\right)$$

for all $x\in X$.

Replacing x by $\frac{x}{2}$ in (2.6) and using (2.9), we have

$$\begin{array}{rcl}{\parallel f(x)-2f\left(\frac{x}{2}\right)\parallel }_Y& \le & \frac{1}{|\gamma |}max\{\phi (x,0,0),\phi (\frac{x}{2},\frac{x}{2},0)\}\\ \le & \frac{\alpha }{|2\gamma |}max\{\phi (2x,0,0),\phi (x,x,0)\}.\end{array}$$

(2.10)

So $d(f,Jf)\le \frac{\alpha }{|2\gamma |}$.

The rest of the proof is similar to the proof of Theorem 2.1. □

Corollary 2.4 Let θ be a positive real number and q is a real number with $0<q<1$. Let $f:X\to Y$ be an odd mapping satisfying (2.8). Then there exists a unique additive mapping $A:X\to Y$ such that

$${\parallel f(x)-A(x)\parallel }_Y\le \frac{2|2|\theta {\parallel x\parallel }^q}{|2\gamma |({|2|}^q-|2|)}$$

for all $x\in X$.

Proof The proof follows from Theorem 2.3 by taking $\phi (x,y,z)=\theta ({\parallel x\parallel }^q+{\parallel y\parallel }^q+{\parallel z\parallel }^q)$ for all $x,y,z\in X$. Then we can choose $\alpha ={|2|}^{1-q}$ and we get the desired result. □

Theorem 2.5 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let $\phi :X^3\to [0,\mathrm{\infty })$ be a function such that there exists an $\alpha <1$ with

$$\phi (2x,2y,2z)\le |4|\alpha \phi (x,y,z)$$

(2.11)

for all $x,y,z\in X$. Let $f:X\to Y$ be an even mapping with $f(0)=0$ and satisfying (2.2). Then there exists a unique quadratic mapping $Q:X\to Y$ such that

$${\parallel f(x)-Q(x)\parallel }_Y\le \frac{max\{\phi (2x,0,0),|2|\phi (x,x,0)\}}{|4\gamma |(1-\alpha )}$$

(2.12)

for all $x\in X$.

Proof Consider the set $S^{\ast }=\{g:X\to Y;g(0)=0\}$ and the generalized metric $d^{\ast }$ in $S^{\ast }$ defined by

$$d^{\ast }(g,h)=inf\{\mu \in (0,+\mathrm{\infty }):{\parallel g(x)-h(x)\parallel }_Y\le \mu max\{\phi (2x,0,0),|2|\phi (x,x,0)\},\mathrm{\forall }x\in X\},$$

where, as usual, $inf\varphi =+\mathrm{\infty }$. It is easy to show that $(S^{\ast },d^{\ast })$ is complete (see [30]). Now we consider the linear mapping $J:(S^{\ast },d^{\ast })\to (S^{\ast },d^{\ast })$ such that

$$Jg(x):=\frac{1}{4}g(2x)$$

for all $x\in X$.

Putting $y=x$ and $z=0$ in (2.2), we have

$${\parallel 2rf\left(\frac{2x}{s}\right)-2\gamma f(x)\parallel }_Y\le \phi (x,x,0)$$

(2.13)

for all $x\in X$.

Substituting $y=z=0$ and then replacing x by 2x in (2.2), we obtain

$${\parallel 4rf\left(\frac{2x}{s}\right)-\gamma f(2x)\parallel }_Y\le \phi (2x,0,0).$$

(2.14)

By (2.13) and (2.14), we get

$$\begin{array}{rcl}{\parallel \frac{f(2x)}{4}-f(x)\parallel }_Y& =& \frac{1}{|4\gamma |}{\parallel 2(2rf\left(\frac{2x}{s}\right)-2\gamma f(x))-(4rf\left(\frac{2x}{s}\right)-\gamma f(2x))\parallel }_Y\\ \le & \frac{1}{|4\gamma |}max\{|2|{\parallel 2rf\left(\frac{2x}{s}\right)-2\gamma f(x)\parallel }_Y,{\parallel 4rf\left(\frac{2x}{s}\right)-\gamma f(2x)\parallel }_Y\}\\ \le & \frac{1}{|4\gamma |}max\{\phi (2x,0,0),|2|\phi (x,x,0)\}.\end{array}$$

(2.15)

The rest of the proof is similar to the proof of Theorem 2.1. □

Corollary 2.6 Let θ be a positive real number and q is a real number with $q>2$. Let $f:X\to Y$ be an even mapping with $f(0)=0$ and satisfying (2.8). Then there exists a unique quadratic mapping $Q:X\to Y$ such that

$${\parallel f(x)-Q(x)\parallel }_Y\le \frac{|4|2|2|\theta {\parallel x\parallel }^q}{|4\gamma |(|4|-{|2|}^q)}$$

for all $x\in X$.

Proof The proof follows from Theorem 2.5 by taking $\phi (x,y,z)=\theta ({\parallel x\parallel }^q+{\parallel y\parallel }^q+{\parallel z\parallel }^q)$ for all $x,y,z\in X$. Then we can choose $\alpha ={|2|}^{q-2}$ and we get the desired result. □

Theorem 2.7 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let $\phi :X^3\to [0,\mathrm{\infty })$ be a function such that there exists an $\alpha <1$ with

$$\phi (\frac{x}{2},\frac{y}{2},\frac{z}{2})\le \frac{\alpha }{|4|}\phi (x,y,z)$$

(2.16)

for all $x,y,z\in X$. Let $f:X\to Y$ be an even mapping with $f(0)=0$ and satisfying (2.2). Then there exists a unique quadratic mapping $Q:X\to Y$ such that

$${\parallel f(x)-Q(x)\parallel }_Y\le \frac{\alpha max\{\phi (2x,0,0),|2|\phi (x,x,0)\}}{|4\gamma |(1-\alpha )}$$

(2.17)

for all $x\in X$.

Proof It follows from (2.15) that

$$\begin{array}{rcl}{\parallel f(x)-4f\left(\frac{x}{2}\right)\parallel }_Y& \le & \frac{1}{|\gamma |}max\{\phi (x,0,0),|2|\phi (\frac{x}{2},\frac{x}{2},0)\}\\ \le & \frac{\alpha }{|4\gamma |}max\{\phi (2x,0,0),|2|\phi (x,x,0)\}.\end{array}$$

The rest of the proof is similar to the proof of Theorems 2.1 and 2.5. □

Corollary 2.8 Let θ be a positive real number and q is a real number with $0<q<2$. Let $f:X\to Y$ be an even mapping with $f(0)=0$ and satisfying (2.8). Then there exists a unique quadratic mapping $Q:X\to Y$ such that

$${\parallel f(x)-Q(x)\parallel }_Y\le \frac{|4|2|2|\theta {\parallel x\parallel }^q}{|4\gamma |({|2|}^q-|4|)}$$

for all $x\in X$.

Proof The proof follows from Theorem 2.7 by taking $\phi (x,y,z)=\theta ({\parallel x\parallel }^q+{\parallel y\parallel }^q+{\parallel z\parallel }^q)$ for all $x,y,z\in X$. Then we can choose $\alpha ={|2|}^{2-q}$ and we get the desired result. □

Let $f:X\to Y$ be a mapping satisfying $f(0)=0$ and (1.1). Let $f_e(x):=\frac{f(x)+f(-x)}{2}$ and $f_o(x)=\frac{f(x)-f(-x)}{2}$. Then $f_e$ is an even mapping satisfying (1.1) and $f_o$ is an odd mapping satisfying (1.1) such that $f(x)=f_e(x)+f_o(x)$.

On the other hand

$$\parallel D_{f_o}(x,y,z)\parallel \le \frac{max\{D_f(x,y,z),D_f(-x,-y,-z)\}}{|2|}\le \frac{max\{\phi (x,y,z),\phi (-x,-y,-z)\}}{|2|}$$

and

$$\parallel D_{f_e}(x,y,z)\parallel \le \frac{max\{D_f(x,y,z),D_f(-x,-y,-z)\}}{|2|}\le \frac{max\{\phi (x,y,z),\phi (-x,-y,-z)\}}{|2|}$$

for all $x,y,z\in X$, where $D_f(x,y,z)$ is the difference operator of the functional equation (1.1). So we obtain the following theorem.

Theorem 2.9 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let $\phi :X^3\to [0,\mathrm{\infty })$ be a function such that there exists an $\alpha <1$ with

$$\phi (2x,2y,2z)\le |4|\alpha \phi (x,y,z)$$

for all $x,y,z\in X$. Let $f:X\to Y$ be a mapping with $f(0)=0$ and satisfying (2.2). Then there exist a unique additive mapping $A:X\to Y$ and a unique quadratic mapping $Q:X\to Y$ such that

$$\begin{array}{c}{\parallel f(x)-A(x)-Q(x)\parallel }_Y\hfill \\ \le max\{{\parallel \frac{f(x)-f(-x)}{2}-A(x)\parallel }_Y,{\parallel \frac{f(x)+f(-x)}{2}-Q(x)\parallel }_Y\}\hfill \\ \le max\{\frac{max\{max\{\phi (2x,0,0),\phi (-2x,0,0)\},max\{\phi (x,x,0),\phi (-x,-x,0)\}\}}{|4\gamma |(1-\alpha )},\hfill \\ \frac{max\{max\{\phi (2x,0,0),\phi (-2x,0,0)\},|2|max\{\phi (x,x,0),\phi (-x,-x,0)\}\}}{|8\gamma |(1-\alpha )}\}\hfill \end{array}$$

for all $x\in X$.

Theorem 2.10 Let X is a non-Archimedean normed space and that Y be a complete non-Archimedean space. Let $\phi :X^3\to [0,\mathrm{\infty })$ be a function such that there exists an $\alpha <1$ with

$$\phi (\frac{x}{2},\frac{y}{2},\frac{z}{2})\le \frac{\alpha \phi (x,y,z)}{|2|}$$

for all $x,y,z\in X$. Let $f:X\to Y$ be a mapping with $f(0)=0$ and satisfying (2.2). Then there exist a unique additive mapping $A:X\to Y$ and a unique quadratic mapping $Q:X\to Y$ such that

$$\begin{array}{c}{\parallel f(x)-A(x)-Q(x)\parallel }_Y\hfill \\ \le \alpha max\{\frac{max\{max\{\phi (2x,0,0),\phi (-2x,0,0)\},max\{\phi (x,x,0),\phi (-x,-x,0)\}\}}{|4\gamma |(1-\alpha )},\hfill \\ \frac{max\{max\{\phi (2x,0,0),\phi (-2x,0,0)\},|2|max\{\phi (x,x,0),\phi (-x,-x,0)\}\}}{|8\gamma |(1-\alpha )}\}\hfill \end{array}$$

for all $x\in X$.

3 Stability of functional equation (1.1): a direct method

In this section, using direct method, we prove the generalized Hyers-Ulam stability of the additive-quadratic functional equation (1.1) in non-Archimedean space.

Theorem 3.1 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that $\phi :G^3\to [0,+\mathrm{\infty })$ be a function such that

$$\underset{n\to \mathrm{\infty }}{lim}{|2|}^n\phi (\frac{x}{2^n},\frac{y}{2^n},\frac{z}{2^n})=0$$

(3.1)

for all $x,y,z\in G$. Suppose that, for any $x\in G$, the limit

$$\mathrm{\Omega }(x)=\underset{n\to \mathrm{\infty }}{lim}max\{{|2|}^{k}max\{\phi (\frac{x}{2^k},0,0),\phi (\frac{x}{2^{k+1}},\frac{x}{2^{k+1}},0)\};0\le k<n\}$$

(3.2)

exists and $f:G\to X$ be an odd mapping satisfying

$$\begin{array}{r}\parallel rf\left(\frac{x+y+z}{s}\right)+rf\left(\frac{x-y+z}{s}\right)+rf\left(\frac{x+y-z}{s}\right)\\ +rf\left(\frac{-x+y+z}{s}\right)-\gamma f(x)-\gamma f(y)-\gamma f(z){\parallel }_X\le \phi (x,y,z).\end{array}$$

(3.3)

Then the limit

$$A(x):=\underset{n\to \mathrm{\infty }}{lim}{2}^{n}f\left(\frac{x}{2^n}\right)$$

exists for all $x\in G$ and defines an additive mapping $A:G\to X$ such that

$$\parallel f(x)-A(x)\parallel \le \frac{1}{|\gamma |}\mathrm{\Omega }(x).$$

(3.4)

Moreover, if

$$\underset{j\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim}max\{{|2|}^{k}max\{\phi (\frac{x}{2^k},0,0),\phi (\frac{x}{2^{k+1}},\frac{x}{2^{k+1}},0)\};j\le k<n+j\}=0$$

then A is the unique additive mapping satisfying (3.4).

Proof By (2.10), we know

$${\parallel f(x)-2f\left(\frac{x}{2}\right)\parallel }_X\le \frac{1}{|\gamma |}max\{\phi (x,0,0),\phi (\frac{x}{2},\frac{x}{2},0)\}$$

(3.5)

for all $x\in G$. Replacing x by $\frac{x}{2^n}$ in (3.5), we obtain

$$\begin{array}{r}{\parallel 2^{n}f\left(\frac{x}{2^n}\right)-2^{n+1}f\left(\frac{x}{2^{n+1}}\right)\parallel }_X\\ \le \frac{{|2|}^n}{|\gamma |}max\{\phi (\frac{x}{2^n},0,0),\phi (\frac{x}{2^{n+1}},\frac{x}{2^{n+1}},0)\}.\end{array}$$

(3.6)

Thus, it follows from (3.1) and (3.6) that the sequence ${\{2^{n}f(\frac{x}{2^n})\}}_{n\ge 1}$ is a Cauchy sequence. Since X is complete, it follows that ${\{2^{n}f(\frac{x}{2^n})\}}_{n\ge 1}$ is convergent. Set

$$A(x):=\underset{n\to \mathrm{\infty }}{lim}{2}^{n}f\left(\frac{x}{2^n}\right).$$

By induction on n, one can show that

$$\begin{array}{r}{\parallel 2^{n}f\left(\frac{x}{2^n}\right)-f(x)\parallel }_X\\ \le \frac{1}{|\gamma |}max\{{|2|}^{k}max\{\phi (\frac{x}{2^k},0,0),\phi (\frac{x}{2^{k+1}},\frac{x}{2^{k+1}},0)\};0\le k<n\}\end{array}$$

(3.7)

for all $n\ge 1$ and $x\in G$. By taking $n\to \mathrm{\infty }$ in (3.7) and using (3.2), one obtains (3.4). By (3.1) and (3.3), we get

$$\begin{array}{c}\parallel rA\left(\frac{x+y+z}{s}\right)+rA\left(\frac{x-y+z}{s}\right)+rA\left(\frac{x+y-z}{s}\right)\hfill \\ +rA\left(\frac{-x+y+z}{s}\right)-\gamma A(x)-\gamma A(y)-\gamma A(z){\parallel }_X\hfill \\ =\underset{n\to \mathrm{\infty }}{lim}{|2|}^n\parallel rf\left(\frac{x+y+z}{2^{n}s}\right)+rf\left(\frac{x-y+z}{2^{n}s}\right)+rf\left(\frac{x+y-z}{2^{n}s}\right)\hfill \\ +rf\left(\frac{-x+y+z}{2^{n}s}\right)-\gamma f\left(\frac{x}{2^n}\right)-\gamma f\left(\frac{y}{2^n}\right)-\gamma f\left(\frac{z}{2^n}\right){\parallel }_X\hfill \\ \le \underset{n\to \mathrm{\infty }}{lim}{|2|}^n\phi (\frac{x}{2^n},\frac{y}{2^n},\frac{z}{2^n})\hfill \\ =0\hfill \end{array}$$

for all $x,y,z\in X$. Therefore, the mapping $A:G\to X$ satisfies (1.1).

To prove the uniqueness property of A, let L be another mapping satisfying (3.4). Then we have

$$\begin{array}{c}{\parallel A(x)-L(x)\parallel }_X\hfill \\ =\underset{n\to \mathrm{\infty }}{lim}{|2|}^n{\parallel A\left(\frac{x}{2^n}\right)-L\left(\frac{x}{2^n}\right)\parallel }_X\hfill \\ \le \underset{k\to \mathrm{\infty }}{lim}{|2|}^{n}max\{{\parallel A\left(\frac{x}{2^n}\right)-f\left(\frac{x}{2^n}\right)\parallel }_X,{\parallel f\left(\frac{x}{2^n}\right)-L\left(\frac{x}{2^n}\right)\parallel }_X\}\hfill \\ \le \underset{j\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim}max\{{|2|}^{k}max\{\phi (\frac{x}{2^k},0,0),\phi (\frac{x}{2^{k+1}},\frac{x}{2^{k+1}},0)\};j\le k<n+j\}\hfill \\ =0\hfill \end{array}$$

for all $x\in G$. Therefore, $A=L$. This completes the proof. □

Corollary 3.2 Let $\xi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be a function satisfying

$$\xi \left({|2|}^{-1}t\right)\le \xi \left({|2|}^{-1}\right)\xi (t),\xi \left({|2|}^{-1}\right)<{|2|}^{-1}$$

for all $t\ge 0$. Assume that $\kappa >0$ and $f:G\to X$ be a mapping with $f(0)=0$ such that

$$\begin{array}{r}\parallel rf\left(\frac{x+y+z}{s}\right)+rf\left(\frac{x-y+z}{s}\right)+rf\left(\frac{x+y-z}{s}\right)\\ +rf\left(\frac{-x+y+z}{s}\right)-\gamma f(x)-\gamma f(y)-\gamma f(z){\parallel }_X\le \kappa (\xi (\parallel x\parallel )+\xi (\parallel y\parallel )+\xi (\parallel z\parallel ))\end{array}$$

(3.8)

for all $x,y,z\in G$. Then there exists a unique additive mapping $A:G\to X$ such that

$${\parallel f(x)-A(x)\parallel }_X\le \frac{1}{|\gamma |}max\{\kappa \zeta (\parallel x\parallel ),\frac{2}{|2|}\kappa \zeta (\parallel x\parallel )\}.$$

Proof Defining $\phi :G^3\to [0,\mathrm{\infty })$ by $\phi (x,y,z):=\kappa (\xi (\parallel x\parallel )+\xi (\parallel y\parallel )+\xi (\parallel z\parallel ))$, then we have

$$\underset{n\to \mathrm{\infty }}{lim}{|2|}^n\phi (\frac{x}{2^n},\frac{y}{2^n},\frac{z}{2^n})\le \underset{n\to \mathrm{\infty }}{lim}{(|2|\xi \left({|2|}^{-1}\right))}^n\phi (x,y,z)=0$$

for all $x,y,z\in G$. The last equality comes form the fact that $|2|\xi ({|2|}^{-1})<1$. On the other hand, it follows that

$$\begin{array}{rcl}\mathrm{\Omega }(x)& =& \underset{n\to \mathrm{\infty }}{lim}max\{{|2|}^{k}max\{\phi (\frac{x}{2^k},0,0),\phi (\frac{x}{2^{k+1}},\frac{x}{2^{k+1}},0)\};0\le k<n\}\\ \le & max\{\phi (x,0,0),\phi (\frac{x}{2},\frac{x}{2},0)\}\\ =& max\{\kappa \zeta (\parallel x\parallel ),\frac{2}{|2|}\kappa \zeta (\parallel x\parallel )\}\end{array}$$

exists for all $x\in G$. Also, we have

$$\begin{array}{c}\underset{j\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim}max\{{|2|}^{k}max\{\phi (\frac{x}{2^k},0,0),\phi (\frac{x}{2^{k+1}},\frac{x}{2^{k+1}},0)\};j\le k<n+j\}\hfill \\ =\underset{j\to \mathrm{\infty }}{lim}{|2|}^{j}max\{\phi (\frac{x}{2^j},0,0),\phi (\frac{x}{2^{j+1}},\frac{x}{2^{j+1}},0)\}\hfill \\ =0.\hfill \end{array}$$

Thus, applying Theorem 3.1, we have the conclusion. This completes the proof. □

Theorem 3.3 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that $\phi :G^3\to [0,+\mathrm{\infty })$ be a function such that

$$\underset{n\to \mathrm{\infty }}{lim}\frac{\phi (2^{n}x,2^{n}y,2^{n}z)}{{|2|}^n}=0$$

(3.9)

for all $x,y,z\in G$. Suppose that, for any $x\in G$, the limit

$$\mathrm{\Omega }(x)=\underset{n\to \mathrm{\infty }}{lim}max\{\frac{max\{\phi (2^{k+1}x,0,0),\phi (2^{k}x,2^{k}x,0)\}}{{|2|}^k};0\le k<n\}$$

(3.10)

exists and $f:G\to X$ be an odd mapping satisfying (3.3). Then the limit $A(x):={lim}_{n\to \mathrm{\infty }}\frac{f(2^{n}x)}{2^n}$ exists for all $x\in G$ and

$${\parallel f(x)-A(x)\parallel }_X\le \frac{1}{|2\gamma |}\mathrm{\Omega }(x)$$

(3.11)

for all $x\in G$. Moreover, if

$$\underset{j\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim}max\{\frac{max\{\phi (2^{k+1}x,0,0),\phi (2^{k}x,2^{k}x,0)\}}{{|2|}^k};j\le k<n+j\}=0,$$

then A is the unique mapping satisfying (3.11).

Proof By (2.6), we get

$${\parallel \frac{f(2x)}{2}-f(x)\parallel }_X\le \frac{max\{\phi (2x,0,0),\phi (x,x,0)\}}{|2\gamma |}$$

(3.12)

for all $x\in G$. Replacing x by $2^{n}x$ in (3.12), we obtain

$${\parallel \frac{f(2^{n+1}x)}{2^{n+1}}-\frac{f(2^{n}x)}{2^n}\parallel }_X\le \frac{max\{\phi (2^{n+1}x,0,0),\phi (2^{n}x,2^{n}x,0)\}}{|2\gamma |{|2|}^n}.$$

(3.13)

Thus, it follows from (3.9) and (3.13) that the sequence ${\{\frac{f(2^{n}x)}{2^n}\}}_{n\ge 1}$ is convergent. Set

$$A(x):=\underset{n\to \mathrm{\infty }}{lim}\frac{f(2^{n}x)}{2^n}.$$

On the other hand, it follows from (3.13) that

for all $x\in G$ and $p,q\ge 0$ with $q>p\ge 0$. Letting $p=0$, taking $q\to \mathrm{\infty }$ in the last inequality and using (3.10), we obtain (3.11).

The rest of the proof is similar to the proof of Theorem 3.1. This completes the proof. □

Theorem 3.4 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that $\phi :G^3\to [0,+\mathrm{\infty })$ be a function such that

$$\underset{n\to \mathrm{\infty }}{lim}{|4|}^n\phi (\frac{x}{2^n},\frac{y}{2^n},\frac{z}{2^n})=0$$

(3.14)

for all $x,y,z\in G$. Suppose that, for any $x\in G$, the limit

$$\mathrm{\Theta }(x)=\underset{n\to \mathrm{\infty }}{lim}max\{{|4|}^{k}max\{\phi (\frac{x}{2^k},0,0),|2|\phi (\frac{x}{2^{k+1}},\frac{x}{2^{k+1}},0)\};0\le k<n\}$$

(3.15)

exists and $f:G\to X$ be an even mapping with $f(0)=0$ and satisfying (3.3). Then the limit $Q(x):={lim}_{n\to \mathrm{\infty }}{4}^{n}f(\frac{x}{2^n})$ exists for all $x\in G$ and defines a quadratic mapping $Q:G\to X$ such that

$${\parallel f(x)-Q(x)\parallel }_X\le \frac{1}{|\gamma |}\mathrm{\Theta }(x).$$

(3.16)

Moreover, if

$$\underset{j\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim}max\{{|4|}^{k}max\{\phi (\frac{x}{2^k},0,0),|2|\phi (\frac{x}{2^{k+1}},\frac{x}{2^{k+1}},0)\};j\le k<n+j\}=0$$

then Q is the unique additive mapping satisfying (3.16).

Proof It follows from (2.15) that

$${\parallel f(x)-4f\left(\frac{x}{2}\right)\parallel }_X\le \frac{1}{|\gamma |}max\{\phi (x,0,0),|2|\phi (\frac{x}{2},\frac{x}{2},0)\}.$$

(3.17)

Replacing x by $\frac{x}{2^n}$ in (3.18), we have

$${\parallel 4^{n}f\left(\frac{x}{2^n}\right)-4^{n+1}f\left(\frac{x}{2^{n+1}}\right)\parallel }_X\le \frac{{|4|}^n}{|\gamma |}max\{\phi (\frac{x}{2^n},0,0),|2|\phi (\frac{x}{2^{n+1}},\frac{x}{2^{n+1}},0)\}.$$

(3.18)

It follows from (3.14) and (3.18) that the sequence ${\{4^{n}f(\frac{x}{2^n})\}}_{n\ge 1}$ is Cauchy sequence. The rest of the proof is similar to the proof of Theorem 3.1. □

Similarly, we can obtain the followings. We will omit the proof.

Theorem 3.5 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that $\phi :G^3\to [0,+\mathrm{\infty })$ be a function such that

$$\underset{n\to \mathrm{\infty }}{lim}\frac{\phi (2^{n}x,2^{n}y,2^{n}z)}{{|4|}^n}=0$$

(3.19)

for all $x,y,z\in G$. Suppose that, for any $x\in G$, the limit

$$\mathrm{\Theta }(x)=\underset{n\to \mathrm{\infty }}{lim}max\{\frac{max\{\phi (2^{k+1}x,0,0),\phi (2^{k}x,2^{k}x,0)\}}{{|4|}^k};0\le k<n\}$$

(3.20)

exists and $f:G\to X$ be an even mapping with $f(0)=0$ and satisfying (3.3). Then the limit $Q(x):={lim}_{n\to \mathrm{\infty }}\frac{f(2^{n}x)}{4^n}$ exists for all $x\in G$ and

$${\parallel f(x)-Q(x)\parallel }_X\le \frac{1}{|4\gamma |}\mathrm{\Theta }(x)$$

(3.21)

for all $x\in G$. Moreover, if

$$\underset{j\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim}max\{\frac{max\{\phi (2^{k+1}x,0,0),\phi (2^{k}x,2^{k}x,0)\}}{{|4|}^k};j\le k<n+j\}=0,$$

then Q is the unique mapping satisfying (3.21).

Let $f:X\to Y$ be a mapping satisfying $f(0)=0$ and (1.1). Let $f_e(x):=\frac{f(x)+f(-x)}{2}$ and $f_o(x)=\frac{f(x)-f(-x)}{2}$. Then $f_e$ is an even mapping satisfying (1.1) and $f_o$ is an odd mapping satisfying (1.1) such that $f(x)=f_e(x)+f_o(x)$. On the other hand,

$$\parallel D_{f_o}(x,y,z)\parallel \le \frac{max\{\phi (x,y,z),\phi (-x,-y,-z)\}}{|2|}$$

and

$$\parallel D_{f_e}(x,y,z)\parallel \le \frac{max\{\phi (x,y,z),\phi (-x,-y,-z)\}}{|2|}$$

for all $x,y,z\in X$, where $D_f(x,y,z)$ is the difference operator of the functional equation (1.1). So we obtain the following theorem.

Theorem 3.6 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that $\phi :G^3\to [0,+\mathrm{\infty })$ be a function such that

$$\underset{n\to \mathrm{\infty }}{lim}\frac{\phi (2^{n}x,2^{n}y,2^{n}z)}{{|4|}^n}=0$$

for all $x,y,z\in G$. Suppose that the limits

$$\begin{array}{rcl}{\mathrm{\Omega }}^{\ast }(x)& =& \underset{n\to \mathrm{\infty }}{lim}\underset{0\le k<n}{max}\{max\{max\{\phi (2^{k+1}x,0,0),\phi (-2^{k+1}x,0,0)\},\\ max\{\phi (2^{k}x,2^{k}x,0),\phi (-2^{k}x,-2^{k}x,0)\}\}/{|2|}^{k+1}\}\end{array}$$

and

$$\begin{array}{rcl}{\mathrm{\Theta }}^{\ast }(x)& =& \underset{n\to \mathrm{\infty }}{lim}\underset{0\le k<n}{max}\{max\{max\{\phi (2^{k+1}x,0,0),\phi (-2^{k+1}x,0,0)\},\\ max\{\phi (2^{k}x,2^{k}x,0),\phi (-2^{k}x,-2^{k}x,0)\}\}/(|2|{|4|}^k)\}\end{array}$$

exist for all $x\in G$ and $f:G\to X$ be a mapping with $f(0)=0$ and satisfying (3.3). Then there exist an additive mapping $A:G\to X$ and a quadratic mapping $Q:G\to X$ such that

$$\begin{array}{r}{\parallel f(x)-A(x)-Q(x)\parallel }_X\\ \le max\{{\parallel \frac{f(x)+f(-x)}{2}-Q(x)\parallel }_X,{\parallel \frac{f(x)-f(-x)}{2}-A(x)\parallel }_X\}\\ \le max\{\frac{1}{|2\gamma |}{\mathrm{\Omega }}^{\ast }(x),\frac{1}{|4\gamma |}{\mathrm{\Theta }}^{\ast }(x)\}\end{array}$$

(3.22)

for all $x\in G$. Moreover, if

$$\begin{array}{c}\underset{j\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim}\underset{j\le k<n+j}{max}\{max\{max\{\phi (2^{k+1}x,0,0),\phi (-2^{k+1}x,0,0)\},\hfill \\ max\{\phi (2^{k}x,2^{k}x,0),\phi (-2^{k}x,-2^{k}x,0)\}\}/{|2|}^{k+1}\}=0\hfill \end{array}$$

and

$$\begin{array}{c}\underset{j\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim}\underset{j\le k<n+j}{max}\{max\{max\{\phi (2^{k+1}x,0,0),\phi (-2^{k+1}x,0,0)\},\hfill \\ max\{\phi (2^{k}x,2^{k}x,0),\phi (-2^{k}x,-2^{k}x,0)\}\}/(|2|{|4|}^k)\}=0\hfill \end{array}$$

then A, Q are the unique mappings satisfying (3.22).

Theorem 3.7 Let G be a vector space and that X is a non-Archimedean Banach space. Assume that $\phi :G^3\to [0,+\mathrm{\infty })$ be a function such that

$$\underset{n\to \mathrm{\infty }}{lim}{|2|}^n\phi (\frac{x}{2^n},\frac{y}{2^n},\frac{z}{2^n})=0$$

for all $x,y,z\in G$. Suppose that the limits

$$\begin{array}{rcl}{\mathrm{\Omega }}^{\ast \ast }(x)& =& \frac{1}{|2|}\underset{n\to \mathrm{\infty }}{lim}\underset{0\le k<n}{max}\{{|2|}^{k}max\{max\{\phi (\frac{x}{2^k},0,0),\phi (\frac{-x}{2^k},0,0)\},\\ max\{\phi (\frac{x}{2^{k+1}},\frac{x}{2^{k+1}},0),\phi (\frac{-x}{2^{k+1}},\frac{-x}{2^{k+1}},0)\}\left\}\right\}\end{array}$$

and

$$\begin{array}{rcl}{\mathrm{\Theta }}^{\ast \ast }(x)& =& \frac{1}{|2|}\underset{n\to \mathrm{\infty }}{lim}\underset{0\le k<n}{max}\{{|4|}^{k}max\{max\{\phi (\frac{x}{2^k},0,0),\phi (\frac{-x}{2^k},0,0)\},\\ |2|max\{\phi (\frac{x}{2^{k+1}},\frac{x}{2^{k+1}},0),\phi (\frac{-x}{2^{k+1}},\frac{-x}{2^{k+1}},0)\}\left\}\right\}\end{array}$$

exist for all $x\in G$ and $f:G\to X$ be a mapping with $f(0)=0$ and satisfying (3.3). Then there exist an additive mapping $A:G\to X$ and a quadratic mapping $Q:G\to X$ such that

$${\parallel f(x)-A(x)-Q(x)\parallel }_X\le \frac{max\{{\mathrm{\Omega }}^{\ast \ast }(x),{\mathrm{\Theta }}^{\ast \ast }(x)\}}{|\gamma |}$$

(3.23)

for all $x\in G$. Moreover, if

$$\begin{array}{c}\underset{j\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim}\underset{j\le k<n+j}{max}\{{|2|}^{k}max\{max\{\phi (\frac{x}{2^k},0,0),\phi (\frac{-x}{2^k},0,0)\},\hfill \\ max\{\phi (\frac{x}{2^{k+1}},\frac{x}{2^{k+1}},0),\phi (\frac{-x}{2^{k+1}},\frac{-x}{2^{k+1}},0)\}\left\}\right\}=0\hfill \end{array}$$

and

$$\begin{array}{c}\underset{j\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim}\underset{j\le k<n+j}{max}\{{|4|}^{k}max\{max\{\phi (\frac{x}{2^k},0,0),\phi (\frac{-x}{2^k},0,0)\},\hfill \\ |2|max\{\phi (\frac{x}{2^{k+1}},\frac{x}{2^{k+1}},0),\phi (\frac{-x}{2^{k+1}},\frac{-x}{2^{k+1}},0)\}\left\}\right\}=0\hfill \end{array}$$

then A, Q are the unique mappings satisfying (3.23).

4 Conclusion

We linked here two different disciplines, namely, the non-Archimedean normed spaces and functional equations. We established the generalized Hyers-Ulam stability of the functional equation (1.1) in non-Archimedean normed spaces.