Theory and Modern Applications

# A Functional equation related to inner product spaces in non-archimedean normed spaces

## Abstract

In this paper, we prove the Hyers-Ulam stability of a functional equation related to inner product spaces in non-Archimedean normed spaces.

2010 Mathematics Subject Classification: Primary 46S10; 39B52; 47S10; 26E30; 12J25.

## 1. Introduction and preliminaries

One of the most interesting questions in the theory of functional analysis concerning the Ulam stability problem of functional equations is as follows: When is it true that a mapping satisfying a functional equation approximately must be close to an exact solution of the given functional equation?

The first stability problem concerning group homomorphisms was raised by Ulam  in 1940 and affirmatively solved by Hyers . The result of Hyers was generalized by Aoki  for approximate additive mappings and by Rassias  for approximate linear mappings by allowing the difference Cauchy equation || f(x 1 + x 2) - f(x 1) - f(x 2) || to be controlled by ε (|| x 1 || p + || x 2 || p ). Taking into consideration a lot of influence of Ulam, Hyers and Rassias on the development of stability problems of functional equations, the stability phenomenon that was proved by Rassias is called Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. In 1994, a generalization of the Rassias' theorem was obtained by Gǎvruta , who replaced ε (|| x 1 || p + || x 2 || p ) by a general control function φ(x 1, x 2).

Quadratic functional equations were used to characterize inner product spaces . A square norm on an inner product space satisfies the parallelogram equality ||x 1 + x 2||2 + ||x 1 - x 2||2 = 2(||x 1||2 + ||x 1||2). The functional equation

$f\left(x+y\right)+f\left(x-y\right)=2f\left(x\right)+2f\left(y\right)$
(1.1)

is related to a symmetric bi-additive mapping [7, 8]. It is natural that this equation is called a quadratic functional equation, and every solution of the quadratic equation (1.1) is said to be a quadratic mapping.

It was shown by Rassias  that the norm defined over a real vector space X is induced by an inner product if and only if for a fixed integer n ≥ 2

$\sum _{i=1}^{n}{∥{x}_{i}-\frac{1}{n}\sum _{j=1}^{n}{x}_{j}∥}^{2}=\sum _{i=1}^{n}\parallel {x}_{i}{\parallel }^{2}-n{∥\frac{1}{n}\sum _{i=1}^{n}{x}_{i}∥}^{2}$

for all x 1,, x n X.

Let $K$ be a field. A non-Archimedean absolute value on $K$ is a function $|\phantom{\rule{2.77695pt}{0ex}}|:K\to ℝ$ such that for any $a,b\in K$, we have

1. (i)

|a| ≥ 0 and equality holds if and only if a = 0,

2. (ii)

|ab| = |a||b|,

3. (iii)

|a + b| ≤ max{|a|, |b|}.

The condition (iii) is called the strict triangle inequality. By (ii), we have |1| = | - 1| = 1. Thus, by induction, it follows from (iii) that |n| ≤ 1 for each integer n. We always assume in addition that | | is non-trivial, i.e., that there is an ${a}_{0}\in K$ such that |a 0| ≠ 0, 1.

Let X be a linear space over a scalar field $K$ with a non-Archimedean non-trivial valuation |·|. A function || · || : X is a non-Archimedean norm (valuation) if it satisfies the following conditions:

(NA1) ||x|| = 0 if and only if x = 0;

(NA2) ||rx|| = |r|||x|| for all $r\in K$ and x X;

(NA3) the strong triangle inequality (ultrametric); namely,

$\parallel x+y\parallel \le max\left\{\parallel x\parallel ,\parallel y\parallel \right\}\phantom{\rule{1em}{0ex}}\left(x,y\in X\right).$

Then (X, || · ||) is called a non-Archimedean space.

Thanks to the inequality

$\parallel {x}_{m}-{x}_{l}\parallel \le max\left\{\parallel {x}_{j+1}-{x}_{j}\parallel :l\le j\le m-1\right\}\phantom{\rule{1em}{0ex}}\left(m>l\right)$

a sequence {x m } is Cauchy in X if and only if {x m+1- x m } converges to zero in a non-Archimedean space. By a complete non-Archimedean space, we mean a non-Archimedean space in which every Cauchy sequence is convergent.

In 1897, Hensel  introduced a normed space which does not have the Archimedean property.

During the last three decades, the theory of non-Archimedean spaces has gained the interest of physicists for their research in particular in problems coming from quantum physics, p-adic strings and superstrings . Although many results in the classical normed space theory have a non-Archimedean counterpart, their proofs are essentially different and require an entirely new kind of intuition .

The main objective of this paper is to prove the Hyers-Ulam stability of the following functional equation related to inner product spaces

$\sum _{i=1}^{n}f\left({x}_{i}-\frac{1}{n}\sum _{j=1}^{n}{x}_{j}\right)=\sum _{i=1}^{n}f\left({x}_{i}\right)-nf\left(\frac{1}{n}\sum _{i=1}^{n}{x}_{i}\right)$
(1.2)

(n , n ≥ 2) in non-Archimedean normed spaces. Interesting new results concerning functional equations related to inner product spaces have recently been obtained by Najati and Rassias  as well as for the fuzzy stability of a functional equation related to inner product spaces by Park  and Eshaghi Gordji and Khodaei . During the last decades, several stability problems for various functional equations have been investigated by many mathematicians (see ).

## 2. Hyers-Ulam stability in non-Archimedean spaces

In the rest of this paper, unless otherwise explicitly stated, we will assume that G is an additive group and that X is a complete non-Archimedean space. For convenience, we use the following abbreviation for a given mapping f : GX:

$\Delta f\left({x}_{1},\dots ,{x}_{n}\right)=\sum _{i=1}^{n}f\left({x}_{i}-\frac{1}{n}\sum _{j=1}^{n}{x}_{j}\right)-\sum _{i=1}^{n}f\left({x}_{i}\right)+nf\left(\frac{1}{n}\sum _{i=1}^{n}{x}_{i}\right)$

for all x 1,, x n G, where n ≥ 2 is a fixed integer.

Lemma 2.1. . Let V 1 and V 2 be real vector spaces. If an odd mapping f : V 1V 2 satisfies the functional equation (1.2), then f is additive.

In the following theorem, we prove the Hyers-Ulam stability of the functional equation (1.2) in non-Archimedean spaces for an odd case.

Theorem 2.2. Let φ : G n → [0, ∞) be a function such that

$\underset{m\to \infty }{\mathrm{lim}}\frac{\phi {\left(2}^{m}{x}_{1}{,2}^{m}{x}_{2},\dots {,2}^{m}{x}_{n}\right)}{{\mid 2\mid }^{m}}=0=\underset{m\to \infty }{\mathrm{lim}}\frac{1}{{\mid 2\mid }^{m}}\Phi {\left(2}^{m-1}x\right)$
(2.1)

for all x, x 1, x 2,, x n G, and

${\stackrel{˜}{\phi }}_{a}\left(x\right)=\underset{m\to \infty }{\mathrm{lim}}\mathrm{max}\left\{\frac{1}{{\mid 2\mid }^{k}}\Phi {\left(2}^{k}x\right):0\le k
(2.2)

exists for all x G, where

$\begin{array}{c}\Phi \left(x\right):=\mathrm{max}\left\{\phi \left(2x,0,\dots ,0\right),\frac{1}{\mid 2\mid }\mathrm{max}\left\{n\phi \left(x,x,0,\dots ,0\right),\\ \phi \left(x,-x,\dots ,-x\right),\phi \left(-x,x,\dots ,x\right)\right\}\right\}\end{array}$
(2.3)

for all x G. Suppose that an odd mapping f : GX satisfies the inequality

$\parallel \Delta f\left({x}_{1},\dots ,{x}_{n}\right)\parallel \le \phi \left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)$
(2.4)

for all x 1, x 2,, x n G. Then there exists an additive mapping A : GX such that

$\parallel f\left(x\right)-A\left(x\right)\parallel \le \frac{1}{|2|}{\stackrel{̃}{\phi }}_{a}\left(x\right)$
(2.5)

for all x G, and if

$\underset{\ell \to \infty }{\mathrm{lim}}\underset{m\to \infty }{\mathrm{lim}}\mathrm{max}\left\{\frac{1}{{\mid 2\mid }^{k}}\Phi {\left(2}^{k}x\right):\ell \le k
(2.6)

then A is a unique additive mapping satisfying (2.5).

Proof. Letting x 1 = nx 1, ${x}_{i}=n{x}_{1}^{\prime }\phantom{\rule{1em}{0ex}}\left(i=2,\dots ,n\right)$ in (2.4) and using the oddness of f, we obtain that

$\begin{array}{c}\parallel nf\left({x}_{1}+\left(n-1\right){{x}^{\prime }}_{1}\right)+f\left(\left(n-1\right)\left({x}_{1}-{{x}^{\prime }}_{1}\right)\right)-\left(n-1\right)f\left({x}_{1}-{{x}^{\prime }}_{1}\right)\\ \phantom{\rule{1em}{0ex}}-f\left(n{x}_{1}\right)-\left(n-1\right)f\left(n{{x}^{\prime }}_{1}\right)\parallel \phantom{\rule{2.77695pt}{0ex}}\le \phi \left(n{x}_{1},n{{x}^{\prime }}_{1},\dots ,n{{x}^{\prime }}_{1}\right)\end{array}$
(2.7)

for all ${x}_{1},{x}_{1}^{\prime }\in G$. Interchanging x 1 with ${x}_{1}^{\prime }$ in (2.7) and using the oddness of f, we get

$\begin{array}{c}\parallel nf\left(\left(n-1\right){x}_{1}+{{x}^{\prime }}_{1}\right)-f\left(\left(n-1\right)\left({x}_{1}-{{x}^{\prime }}_{1}\right)\right)+\left(n-1\right)f\left({x}_{1}-{{x}^{\prime }}_{1}\right)\\ \phantom{\rule{1em}{0ex}}-\left(n-1\right)f\left(n{x}_{1}\right)-f\left(n{{x}^{\prime }}_{1}\right)\parallel \le \phi \left(n{{x}^{\prime }}_{1},n{x}_{1},\dots ,n{x}_{1}\right)\end{array}$
(2.8)

for all ${x}_{1},{{x}^{\prime }}_{1}\in G$. It follows from (2.7) and (2.8) that

$\begin{array}{c}\parallel nf\left({x}_{1}+\left(n-1\right){{x}^{\prime }}_{1}\right)-nf\left(\left(n-1\right){x}_{1}+{{x}^{\prime }}_{1}\right)+2f\left(\left(n-1\right)\left({x}_{1}-{{x}^{\prime }}_{1}\right)\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-2\left(n-1\right)f\left({x}_{1}-{{x}^{\prime }}_{1}\right)+\left(n-2\right)f\left(n{x}_{1}\right)-\left(n-2\right)f\left(n{{x}^{\prime }}_{1}\right)\parallel \\ \phantom{\rule{1em}{0ex}}\le max\left\{\phi \left(n{x}_{1},n{{x}^{\prime }}_{1},\dots ,n{{x}^{\prime }}_{1}\right),\phi \left(n{{x}^{\prime }}_{1},n{x}_{1},\dots ,n{x}_{1}\right)\right\}\end{array}$
(2.9)

for all ${x}_{1},{{x}^{\prime }}_{1}\in G$. Setting x 1 = nx 1, ${x}_{2}=-n{x}_{1}^{\prime }$, x i = 0 (i = 3,..., n) in (2.4) and using the oddness of f, we get

$\begin{array}{c}\parallel f\left(\left(n-1\right){x}_{1}+{{x}^{\prime }}_{1}\right)-f\left({x}_{1}+\left(n-1\right){{x}^{\prime }}_{1}\right)+2f\left({x}_{1}-{{x}^{\prime }}_{1}\right)\\ \phantom{\rule{1em}{0ex}}-f\left(n{x}_{1}\right)+f\left(n{{x}^{\prime }}_{1}\right)\parallel \le \phi \left(n{x}_{1},-n{{x}^{\prime }}_{1},0,\dots ,0\right)\end{array}$
(2.10)

for all ${x}_{1},{{x}^{\prime }}_{1}\in G$. It follows from (2.9) and (2.10) that

$\begin{array}{l}\parallel f\left(\left(n-1\right)\left({x}_{1}-{{x}^{\prime }}_{1}\right)\right)+f\left({x}_{1}-{{x}^{\prime }}_{1}\right)-f\left(n{x}_{1}\right)+f\left(n{{x}^{\prime }}_{1}\right)\parallel \\ \phantom{\rule{0.1em}{0ex}}\le \frac{1}{\mid 2\mid }\mathrm{max}\left\{n\phi \left(n{x}_{1},-n{{x}^{\prime }}_{1},0,\dots ,0\right),\\ \phantom{\rule{0.1em}{0ex}}\phi \left(n{x}_{1},n{{x}^{\prime }}_{1},\dots ,n{{x}^{\prime }}_{1}\right),\phi \left(n{{x}^{\prime }}_{1},n{x}_{1},\dots ,n{x}_{1}\right)\right\}\end{array}$
(2.11)

for all ${x}_{1},{{x}^{\prime }}_{1}\in G$. Putting ${x}_{1}=n\left({x}_{1}-{x}_{1}^{\prime }\right)$, x i = 0 (i = 2,..., n) in (2.4), we obtain

$\parallel f\left(n\left({x}_{1}-{x}_{1}^{\prime }\right)\right)-f\left(\left(n-1\right)\left({x}_{1}-{x}_{1}^{\prime }\right)\right)-f\left(\left({x}_{1}-{x}_{1}^{\prime }\right)\right)\parallel \le \phi \left(n\left({x}_{1}-{x}_{1}^{\prime }\right),0,\dots ,0\right)$
(2.12)

for all ${x}_{1},{{x}^{\prime }}_{1}\in G$. It follows from (2.11) and (2.12) that

(2.13)

for all ${x}_{1},{{x}^{\prime }}_{1}\in G$. Replacing x 1 and ${x}_{1}^{\prime }$ by $\frac{x}{n}$ and $\frac{-x}{n}$ in (2.13), respectively, we obtain

$\begin{array}{l}\parallel f\left(2x\right)-2f\left(x\right)\parallel \le \mathrm{max}\left\{\phi \left(2x,0,\dots ,0\right),\\ \frac{1}{\mid 2\mid }\mathrm{max}\left\{n\phi \left(x,x,0,\dots ,0\right),\phi \left(x,-x,\dots ,-x\right),\phi \left(-x,x,\dots ,x\right)\right\}\right\}\end{array}$

for all x G. Hence,

$∥\frac{f\left(2x\right)}{2}-f\left(x\right)∥\le \frac{1}{|2|}\Phi \left(x\right)$
(2.14)

for all x G. Replacing x by 2m-1 x in (2.14), we have

$‖\frac{f{\left(2}^{m-1}x\right)}{{2}^{m-1}}-\frac{f{\left(2}^{m}x\right)}{{2}^{m}}‖\le \frac{1}{{\mid 2\mid }^{m}}\Phi {\left(2}^{m-1}x\right)$
(2.15)

for all x G. It follows from (2.1) and (2.15) that the sequence $\left\{\frac{f{\left(2}^{m}x\right)}{{2}^{m}}\right\}$ is Cauchy. Since X is complete, we conclude that $\left\{\frac{f{\left(2}^{m}x\right)}{{2}^{m}}\right\}$ is convergent. So one can define the mapping A : GX by $A\left(x\right):={\mathrm{lim}}_{m\to \infty }\frac{f{\left(2}^{m}x\right)}{{2}^{m}}$ for all x G. It follows from (2.14) and (2.15) that

$‖f\left(x\right)-\frac{f{\left(2}^{m}x\right)}{{2}^{m}}‖\le \frac{1}{\mid 2\mid }\mathrm{max}\left\{\frac{1}{{\mid 2\mid }^{k}}\Phi {\left(2}^{k}x\right):0\le k
(2.16)

for all m and all x G. By taking m to approach infinity in (2.16) and using (2.2), one gets (2.5). By (2.1) and (2.4), we obtain

$\begin{array}{ll}\hfill \parallel \Delta A\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)\parallel & =\underset{m\to \infty }{lim}\frac{1}{{|2|}^{m}}\parallel \Delta f\left({2}^{m}{x}_{1},{2}^{m}{x}_{2},\dots ,{2}^{m}{x}_{n}\right)\parallel \phantom{\rule{2em}{0ex}}\\ \le \underset{m\to \infty }{lim}\frac{1}{{|2|}^{m}}\phi \left({2}^{m}{x}_{1},{2}^{m}{x}_{2},\dots ,{2}^{m}{x}_{n}\right)=0\phantom{\rule{2em}{0ex}}\end{array}$

for all x 1, x 2,, x n G. Thus, the mapping A satisfies (1.2). By Lemma 2.1, A is additive.

If A' is another additive mapping satisfying (2.5), then

$\begin{array}{c}\parallel A\left(x\right)-{A}^{\prime }\left(x\right)\parallel =\underset{\ell \to \infty }{\mathrm{lim}}\mid 2{\mid }^{-\ell }\parallel A\left({2}^{\ell }x\right)-{A}^{\prime }\left({2}^{\ell }x\right)\parallel \\ \le \underset{\ell \to \infty }{\mathrm{lim}}\mid 2{\mid }^{-\ell }\mathrm{max}\left\{\parallel A\left({2}^{\ell }x\right)-f\left({2}^{\ell }x\right)\parallel ,\parallel f\left({2}^{\ell }x\right)-{Q}^{\prime }\left({2}^{\ell }x\right)\parallel \right\}\\ \le \frac{1}{\mid 2\mid }\underset{\ell \to \infty }{\mathrm{lim}}\underset{m\to \infty }{\mathrm{lim}}\mathrm{max}\left\{\frac{1}{{\mid 2\mid }^{k}}\stackrel{˜}{\phi }{\left(2}^{k}x\right):\ell \le k

for all x G, Thus A = A'. □

Corollary 2.3. Let ρ : [0, ∞) → [0, ∞) be a function satisfying

1. (i)

ρ (|2|t) ≤ ρ(|2|)ρ(t) for all t ≥ 0,

2. (ii)

ρ(|2|) < |2|.

Let ε > 0 and let G be a normed space. Suppose that an odd mapping f : GX satisfies the inequality

$\parallel \Delta f\left({x}_{1},\dots ,{x}_{n}\right)\parallel \le \epsilon \sum _{i=1}^{n}\rho \left(\parallel {x}_{i}\parallel \right)$

for all x 1,, x n G. Then there exists a unique additive mapping A : GX such that

$\parallel f\left(x\right)-A\left(x\right)\parallel \le \frac{2n}{{|2|}^{2}}\epsilon \rho \left(\parallel x\parallel \right)$

for all x G.

Proof. Defining φ : G n → [0, ∞) by $\phi \left({x}_{1},\dots ,{x}_{n}\right):=\epsilon {\sum }_{i=1}^{n}\rho \left(\parallel {x}_{i}\parallel \right)$, we have

$\underset{m\to \infty }{lim}\frac{1}{{|2|}^{m}}\phi \left({2}^{m}{x}_{1},\dots ,{2}^{m}{x}_{n}\right)\le \underset{m\to \infty }{lim}{\left(\frac{\rho \left(|2|\right)}{|2|}\right)}^{m}\phi \left({x}_{1},\dots ,{x}_{n}\right)=0$

for all x 1,, x n G. So we have

${\stackrel{˜}{\phi }}_{a}\left(x\right):=\underset{m\to \infty }{\mathrm{lim}}\mathrm{max}\left\{\frac{1}{{\mid 2\mid }^{k}}\Phi {\left(2}^{k}x\right):0\le k

and

$\underset{\ell \to \infty }{\mathrm{lim}}\underset{m\to \infty }{\mathrm{lim}}\mathrm{max}\left\{\frac{1}{{\mid 2\mid }^{k}}\Phi {\left(2}^{k}x\right):\ell \le k

for all x G. It follows from (2.3) that

$\begin{array}{ll}\hfill \Phi \left(x\right)& =max\left\{\epsilon \rho \left(\parallel 2x\parallel \right),\frac{1}{|2|}max\left\{2n\epsilon \rho \left(\parallel x\parallel \right),n\epsilon \rho \left(\parallel x\parallel \right),n\epsilon \rho \left(\parallel x\parallel \right)\right\}\right\}\phantom{\rule{2em}{0ex}}\\ =max\left\{\epsilon \rho \left(\parallel 2x\parallel \right),\frac{1}{|2|}max\left\{2n\epsilon \rho \left(\parallel x\parallel \right),n\epsilon \rho \left(\parallel x\parallel \right)\right\}\right\}\phantom{\rule{2em}{0ex}}\\ =max\left\{\epsilon \rho \left(\parallel 2x\parallel \right),\frac{1}{|2|}2n\epsilon \rho \left(\parallel x\parallel \right)\right\}=\frac{2n}{|2|}\epsilon \rho \left(\parallel x\parallel \right).\phantom{\rule{2em}{0ex}}\end{array}$

Applying Theorem 2.2, we conclude that

$\parallel f\left(x\right)-A\left(x\right)\parallel \le \frac{1}{|2|}{\stackrel{̃}{\phi }}_{a}\left(x\right)=\frac{1}{|2|}\Phi \left(x\right)=\frac{2n}{{|2|}^{2}}\epsilon \rho \left(\parallel x\parallel \right)$

for all x G. □

Lemma 2.4. . Let V 1 and V 2 be real vector spaces. If an even mapping f : V 1V 2 satisfies the functional equation (1.2), then f is quadratic.

In the following theorem, we prove the Hyers-Ulam stability of the functional equation (1.2) in non-Archimedean spaces for an even case.

Theorem 2.5. Let φ : G n → [0, ∞) be a function such that

$\underset{m\to \infty }{\mathrm{lim}}\frac{\phi {\left(2}^{m}{x}_{1}{,2}^{m}{x}_{2},\dots {,2}^{m}{x}_{n}\right)}{{\mid 2\mid }^{2m}}=0=\underset{m\to \infty }{\mathrm{lim}}\frac{1}{{\mid 2\mid }^{2m}}\stackrel{˜}{\phi }{\left(2}^{m-1}x\right)$
(2.17)

for all x, x 1, x 2,, x n G, and

${\stackrel{̃}{\phi }}_{q}\left(x\right)=\underset{m\to \infty }{lim}max\left\{\frac{1}{{|2|}^{2k}}\stackrel{̃}{\phi }\left({2}^{k}x\right):0\le k
(2.18)

exists for all x G, where

(2.19)

and

$\Psi \left(x\right):=\frac{1}{|2|}max\left\{n\phi \left(nx,0,\dots ,0\right),\phi \left(nx,0,\dots ,0\right),\phi \left(0,nx,\dots ,nx\right)\right\}$
(2.20)

for all x G. Suppose that an even mapping f : GX with f(0) = 0 satisfies the inequality (2.4) for all x 1, x 2,, x n G. Then there exists a quadratic mapping Q : GX such that

$\parallel f\left(x\right)-Q\left(x\right)\parallel \le \frac{1}{{|2|}^{2}}{\stackrel{̃}{\phi }}_{q}\left(x\right)$
(2.21)

for all x G, and if

$\underset{\ell \to \infty }{lim}\underset{m\to \infty }{lim}max\left\{\frac{1}{{|2|}^{2k}}\stackrel{̃}{\phi }\left({2}^{k}x\right):\ell \le k
(2.22)

then Q is a unique quadratic mapping satisfying (2.21).

Proof. Letting x 1 = nx 1, x i = nx 2 (i = 2,, n) in (2.4) and using the evenness of f, we obtain

$\begin{array}{c}\parallel nf\left({x}_{1}+\left(n-1\right){x}_{2}\right)+f\left(\left(n-1\right)\left({x}_{1}-{x}_{2}\right)\right)+\left(n-1\right)f\left({x}_{1}-{x}_{2}\right)\\ \phantom{\rule{1em}{0ex}}-f\left(n{x}_{1}\right)-\left(n-1\right)f\left(n{x}_{2}\right)\parallel \le \phi \left(n{x}_{1},n{x}_{2},\dots ,n{x}_{2}\right)\end{array}$
(2.23)

for all x 1, x 2 G. Interchanging x 1 with x 2 in (2.23) and using the evenness of f, we obtain

$\begin{array}{c}\parallel nf\left(\left(n-1\right){x}_{1}+{x}_{2}\right)+f\left(\left(n-1\right)\left({x}_{1}-{x}_{2}\right)\right)+\left(n-1\right)f\left({x}_{1}-{x}_{2}\right)\\ \phantom{\rule{1em}{0ex}}-\left(n-1\right)f\left(n{x}_{1}\right)-f\left(n{x}_{2}\right)\parallel \le \phi \left(n{x}_{2},n{x}_{1},\dots ,n{x}_{1}\right)\end{array}$
(2.24)

for all x 1, x 2 G. It follows from (2.23) and (2.24) that

$\begin{array}{c}\parallel nf\left(\left(n-1\right){x}_{1}+{x}_{2}\right)+nf\left({x}_{1}+\left(n-1\right){x}_{2}\right)+2f\left(\left(n-1\right)\left({x}_{1}-{x}_{2}\right)\right)\\ \phantom{\rule{1em}{0ex}}+2\left(n-1\right)f\left({x}_{1}-{x}_{2}\right)-nf\left(n{x}_{1}\right)-nf\left(n{x}_{2}\right)\parallel \\ \le max\left\{\phi \left(n{x}_{1},n{x}_{2},\dots ,n{x}_{2}\right),\phi \left(n{x}_{2},n{x}_{1},\dots ,n{x}_{1}\right)\right\}\end{array}$
(2.25)

for all x 1, x 2 G. Setting x 1 = nx 1, x 2 = -nx 2, x i = 0 (i = 3,, n) in (2.4) and using the evenness of f, we obtain

$\begin{array}{c}\parallel f\left(\left(n-1\right){x}_{1}+{x}_{2}\right)+f\left({x}_{1}+\left(n-1\right){x}_{2}\right)+2\left(n-1\right)f\left({x}_{1}-{x}_{2}\right)\\ \phantom{\rule{1em}{0ex}}-f\left(n{x}_{1}\right)-f\left(n{x}_{2}\right)\parallel \le \phi \left(n{x}_{1},-n{x}_{2},0,\dots ,0\right)\end{array}$
(2.26)

for all x 1, x 2 G. So we obtain from (2.25) and (2.26) that

$\begin{array}{c}\parallel f\left(\left(n-1\right)\left({x}_{1}-{x}_{2}\right)\right)-{\left(n-1\right)}^{2}f\left({x}_{1}-{x}_{2}\right)\parallel \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{|2|}max\left\{n\phi \left(n{x}_{1},-n{x}_{2},0,\dots ,0\right),\\ \phantom{\rule{1em}{0ex}}\phi \left(n{x}_{1},n{x}_{2},\dots ,n{x}_{2}\right),\phi \left(n{x}_{2},n{x}_{1},\dots ,n{x}_{1}\right)\right\}\end{array}$
(2.27)

for all x 1, x 2 G. Setting x 1 = x, x 2 = 0 in (2.27), we obtain

$\begin{array}{c}\parallel f\left(\left(n-1\right)x\right)-{\left(n-1\right)}^{2}f\left(x\right)\parallel \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{|2|}max\left\{n\phi \left(nx,0,\dots ,0\right),\phi \left(nx,0,\dots ,0\right),\phi \left(0,nx,\dots ,nx\right)\right\}\end{array}$
(2.28)

for all x G. Putting x 1 = nx, x i = 0 (i = 2,, n) in (2.4), one obtains

$\parallel f\left(nx\right)-f\left(\left(n-1\right)x\right)-\left(2n-1\right)f\left(x\right)\parallel \le \phi \left(nx,0,\dots ,0\right)$
(2.29)

for all x G. It follows from (2.28) and (2.29) that

(2.30)

for all x G. Letting x 2 = - (n - 1) x 1 and replacing x 1 by $\frac{x}{n}$ in (2.26), we get

$\parallel f\left(\left(n-1\right)x\right)-f\left(\left(n-2\right)x\right)-\left(2n-3\right)f\left(x\right)\parallel \le \phi \left(x,\left(n-1\right)x,0,\dots ,0\right)$
(2.31)

for all x G. It follows from (2.28) and (2.31) that

(2.32)

for all x G. It follows from (2.30) and (2.32) that

$\begin{array}{c}\parallel f\left(nx\right)-f\left(\left(n-2\right)x\right)-4\left(n-1\right)f\left(x\right)\parallel \\ \le max\left\{\phi \left(nx,0,\dots ,0\right),\phi \left(x,\left(n-1\right)x,0,\dots ,0\right),\Psi \left(x\right)\right\}\end{array}$
(2.33)

for all x G. Setting x 1 = x 2 = n x , x i = 0 (i = 3,, n) in (2.4), we obtain

$\parallel f\left(\left(n-2\right)x\right)+\left(n-1\right)f\left(2x\right)-f\left(nx\right)\parallel \le \frac{1}{|2|}\phi \left(nx,nx,0,\dots ,0\right)$
(2.34)

for all x G. It follows from (2.33) and (2.34) that

(2.35)

for all x G. Thus,

$∥f\left(x\right)-\frac{f\left(2x\right)}{{2}^{2}}∥\le \frac{1}{{|2|}^{2}}\stackrel{̃}{\phi }\left(x\right)$
(2.36)

for all x G. Replacing x by 2 m - 1 x in (2.36), we have

$∥\frac{f\left({2}^{m-1}x\right)}{{2}^{2\left(m-1\right)}}-\frac{f\left({2}^{m}x\right)}{{2}^{2m}}∥\le \frac{1}{{|2|}^{2m}}\stackrel{̃}{\phi }\left({2}^{m-1}x\right)$
(2.37)

for all x G. It follows from (2.17) and (2.37) that the sequence $\left\{\frac{f\left({2}^{m}x\right)}{{2}^{2m}}\right\}$ is Cauchy. Since X is complete, we conclude that $\left\{\frac{f\left({2}^{m}x\right)}{{2}^{2m}}\right\}$ is convergent. So one can define the mapping Q : GX by $Q\left(x\right):={lim}_{m\to \infty }\frac{f\left({2}^{m}x\right)}{{2}^{2m}}$ for all x G. By using induction, it follows from (2.36) and (2.37) that

$∥f\left(x\right)-\frac{f\left({2}^{m}x\right)}{{2}^{2m}}∥\le \frac{1}{{|2|}^{2}}max\left\{\frac{1}{{|2|}^{2k}}\stackrel{̃}{\phi }\left({2}^{k}x\right):0\le k
(2.38)

for all n and all x G. By taking m to approach infinity in (2.38) and using (2.18), one gets (2.21).

The rest of proof is similar to proof of Theorem 2.2. □

Corollary 2.6. Let η : [0, ∞) → [0, ∞) be a function satisfying

1. (i)

η(|l|t) ≤ η(|l|)η(t) for all t ≥ 0,

2. (ii)

η(|l|) < |l|2 for l {2, n - 1, n}.

Let ε > 0 and let G be a normed space. Suppose that an even mapping f : GX with f(0) = 0 satisfies the inequality

$\parallel \Delta f\left({x}_{1},\dots ,{x}_{n}\right)\parallel \le \epsilon \sum _{i=1}^{n}\eta \left(\parallel {x}_{i}\parallel \right)$

for all x 1,, x n G. Then there exists a unique quadratic mapping Q : GX such that

$\parallel f\left(x\right)-Q\left(x\right)\parallel \le \left\{\begin{array}{cc}\hfill \frac{2}{{|2|}^{2}}\epsilon \eta \left(\parallel x\parallel \right),\hfill & \hfill if\phantom{\rule{1em}{0ex}}n=2;\hfill \\ \hfill \frac{n}{{|2|}^{3}|n-1|}\epsilon \eta \left(\parallel nx\parallel \right),\hfill & \hfill if\phantom{\rule{1em}{0ex}}n>2,\hfill \end{array}\right\$

for all x G.

Proof. Defining φ : G n → [0, ∞) by $\phi \left({x}_{1},\dots ,{x}_{n}\right):=\epsilon {\sum }_{i=1}^{n}\eta \left(\parallel {x}_{i}\parallel \right)$, we have

$\underset{m\to \infty }{lim}\frac{1}{{|2|}^{2m}}\phi \left({2}^{m}{x}_{1},\dots ,{2}^{m}{x}_{n}\right)\le \underset{m\to \infty }{lim}{\left(\frac{\eta \left(|2|\right)}{{|2|}^{2}}\right)}^{m}\phi \left({x}_{1},\dots ,{x}_{n}\right)=0$

for all x 1,, x n G. We have

${\stackrel{̃}{\phi }}_{q}\left(x\right):=\underset{m\to \infty }{lim}max\left\{\frac{1}{{|2|}^{2k}}\stackrel{̃}{\phi }\left({2}^{k}x\right):0\le k

and

$\underset{\ell \to \infty }{lim}\underset{m\to \infty }{lim}max\left\{\frac{1}{{|2|}^{2k}}\stackrel{̃}{\phi }\left({2}^{k}x\right):\ell \le k

for all x G. It follows from (2.20) that

$\begin{array}{ll}\hfill \Psi \left(x\right)& =\frac{1}{|2|}max\left\{n\epsilon \eta \left(\parallel nx\parallel \right),\epsilon \eta \left(\parallel nx\parallel \right),\left(n-1\right)\epsilon \eta \left(\parallel nx\parallel \right)\right\}\phantom{\rule{2em}{0ex}}\\ =\frac{1}{|2|}max\left\{n\epsilon \eta \left(\parallel nx\parallel \right),\left(n-1\right)\epsilon \eta \left(\parallel nx\parallel \right)\right\}\phantom{\rule{2em}{0ex}}\\ =\frac{n}{|2|}\epsilon \eta \left(\parallel nx\parallel \right)\phantom{\rule{2em}{0ex}}\end{array}$

Hence, by using (2.19), we obtain

for all x G.

Applying Theorem 2.5, we conclude the required result. □

Lemma 2.7. . Let V 1 and V 2 be real vector spaces. A mapping f : V 1V 2 satisfies (1.2) if and only if there exist a symmetric bi-additive mapping B : V 1 × V 1V 2 and an additive mapping A : V 1V 2 such that f(x) = B(x, x) + A(x) for all x V 1.

Now, we prove the main theorem concerning the Hyers-Ulam stability problem for the functional equation (1.2) in non-Archimedean spaces.

Theorem 2.8. Let φ : G n → [0, ∞) be a function satisfying (2.1) and (2.17) for all x, x 1, x 2,, x n G, and ${\stackrel{̃}{\phi }}_{a}\left(x\right)$ and ${\stackrel{̃}{\phi }}_{q}\left(x\right)$ exist for all x G, where ${\stackrel{̃}{\phi }}_{a}\left(x\right)$ and ${\stackrel{̃}{\phi }}_{q}\left(x\right)$ are defined as in Theorems 2.2 and 2.5. Suppose that a mapping f : GX with f(0) = 0 satisfies the inequality (2.4) for all x 1, x 2,, x n G. Then there exist an additive mapping A : GX and a quadratic mapping Q : GX such that

$\begin{array}{c}\parallel f\left(x\right)-A\left(x\right)-Q\left(x\right)\parallel \\ \le \frac{1}{{|2|}^{2}}max\left\{{\stackrel{̃}{\phi }}_{a}\left(x\right),{\stackrel{̃}{\phi }}_{a}\left(-x\right),\frac{1}{|2|}{\stackrel{̃}{\phi }}_{q}\left(x\right),\frac{1}{|2|}{\stackrel{̃}{\phi }}_{q}\left(-x\right)\right\}\end{array}$
(2.39)

for all x G. If

$\begin{array}{c}\underset{\ell \to \infty }{lim}\underset{m\to \infty }{lim}max\left\{\frac{1}{{|2|}^{k}}\Phi \left({2}^{k}x\right):\ell \le k

then A is a unique additive mapping and Q is a unique quadratic mapping satisfying (2.39).

Proof. Let ${f}_{e}\left(x\right)=\frac{1}{2}\left(f\left(x\right)+f\left(-x\right)\right)$ for all x G. Then

$\begin{array}{ll}\hfill \parallel \Delta {f}_{e}\left({x}_{1},\dots ,{x}_{n}\right)\parallel & =∥\frac{1}{2}\left(\Delta f\left({x}_{1},\dots ,{x}_{n}\right)+\Delta f\left(-{x}_{1},\dots ,-{x}_{n}\right)\right)∥\phantom{\rule{2em}{0ex}}\\ \le \frac{1}{|2|}max\left\{\phi \left({x}_{1},\dots ,{x}_{n}\right),\phi \left(-{x}_{1},\dots ,-{x}_{n}\right)\right\}\phantom{\rule{2em}{0ex}}\end{array}$

for all x 1, x 2,, x n G. By Theorem 2.5, there exists a quadratic mapping Q : GX such that

$\parallel {f}_{e}\left(x\right)-Q\left(x\right)\parallel \le \frac{1}{{|2|}^{3}}max\left\{{\stackrel{̃}{\phi }}_{q}\left(x\right),{\stackrel{̃}{\phi }}_{q}\left(-x\right)\right\}$
(2.40)

for all x G. Also, let ${f}_{o}\left(x\right)=\frac{1}{2}\left(f\left(x\right)-f\left(-x\right)\right)$ for all x G. By Theorem 2.2, there exists an additive mapping A : GX such that

$\parallel {f}_{o}\left(x\right)-A\left(x\right)\parallel \le \frac{1}{{|2|}^{2}}max\left\{{\stackrel{̃}{\phi }}_{a}\left(x\right),{\stackrel{̃}{\phi }}_{a}\left(-x\right)\right\}$
(2.41)

for all x G. Hence (2.39) follows from (2.40) and (2.41).

The rest of proof is trivial. □

Corollary 2.9. Let γ : [0, ∞) → [0, ∞) be a function satisfying

(i) γ(|l|t) ≤ γ(|l|) γ(t) for all t ≥ 0,

(ii) γ(|l|) < |l|2 for l {2, n - 1, n}.

Let ε > 0, G a normed space and let f : GX satisfy

$\parallel \Delta f\left({x}_{1},\dots ,{x}_{n}\right)\parallel \le \epsilon \sum _{i=1}^{n}\gamma \left(\parallel {x}_{i}\parallel \right)$

for all x 1,, x n G and f (0) = 0. Then there exist a unique additive mapping A : GX and a unique quadratic mapping Q : GX such that

$\parallel f\left(x\right)-A\left(x\right)-Q\left(x\right)\parallel \le \frac{2n}{{|2|}^{3}}\epsilon \gamma \left(\parallel x\parallel \right)$

for all x G.

Proof. The result follows by Corollaries 2.6 and 2.3. □

## References

1. Ulam SM: A Collection of the Mathematical Problems. Interscience Publication, New York; 1960.

2. Hyers DH: On the stability of the linear functional equation. Proc Natl Acad Sci USA 1941, 27: 222-224. 10.1073/pnas.27.4.222

3. Aoki T: On the stability of the linear transformation in Banach spaces. J Math Soc Jpn 1950, 2: 64-66. 10.2969/jmsj/00210064

4. Rassias ThM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc 1978, 72: 297-300. 10.1090/S0002-9939-1978-0507327-1

5. Găvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl 1994, 184: 431-436. 10.1006/jmaa.1994.1211

6. Amir D: Characterizations of Inner Product Spaces. Birkhäuser, Basel; 1986.

7. Aczel J, Dhombres J: Functional Equations in Several Variables. Cambridge University Press, Cambridge; 1989.

8. Kannappan Pl: Quadratic functional equation and inner product spaces. Results Math 1995, 27: 368-372.

9. Rassias ThM: New characterization of inner product spaces. Bull Sci Math 1984, 108: 95-99.

10. Hensel K: Uber eine neue Begrundung der Theorie der algebraischen Zahlen. Jahresber Deutsch Math Verein 1897, 6: 83-88.

11. Khrennikov A: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic Publishers, Dordrecht; 1997.

12. Moslehian MS, Rassias ThM: Stability of functional equations in non-Archimedean spaces. Appl Anal Discrete Math 2007, 1: 325-334. 10.2298/AADM0702325M

13. Narici L, Beckenstein E: Strange terrain-non-Archimedean spaces. Am Math Monthly 1981, 88: 667-676. 10.2307/2320670

14. Eshaghi Gordji M, Savadkouhi MB: Stability of cubic and quartic functional equations in non-Archimedean spaces. Acta Appl Math 2010, 110: 1321-1329. 10.1007/s10440-009-9512-7

15. Eshaghi Gordji M, Savadkouhi MB: Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces. Appl Math Lett 2010, 23: 1198-1202. 10.1016/j.aml.2010.05.011

16. Eshaghi Gordji M, Khodaei H, Khodabakhsh R: General quartic-cubic-quadratic functional equation in non-Archimedean normed spaces. U.P.B Sci Bull Ser A 2010, 72: 69-84.

17. Najati A, Rassias ThM: Stability of a mixed functional equation in several variables on Banach modules. Nonlinear Anal TMA 2010, 72: 1755-1767. 10.1016/j.na.2009.09.017

18. Park C: Fuzzy stability of a functional equation associated with inner product spaces. Fuzzy Sets Syst 2009, 160: 1632-1642. 10.1016/j.fss.2008.11.027

19. Eshaghi Gordji M, Khodaei H: The fixed point method for fuzzy approximation of a functional equation associated with inner product spaces. Discrete Dyn Nat Soc 2010, 2010: 1-15. Article ID 140767

20. Adam M, Czerwik S: On the stability of the quadratic functional equation in topological spaces. Banach J Math Anal 2007, 1: 245-251.

21. Ebadian A, Ghobadipour N, Eshaghi Gordji M: A fixed point method for perturbation of bimultipliers and Jordan bimultipliers in C* -ternary algebras. J Math Phys 2010, 51: 1-10. Article ID 103508

22. Ebadian A, Najati A, Eshaghi Gordji M: On approximate additive-quartic and quadratic-cubic functional equations in two variables on abelian groups. Results Math 2010, 58: 39-53. 10.1007/s00025-010-0018-4

23. Eshaghi Gordji M: Stability of a functional equation deriving from quartic and additive functions. Bull Korean Math Soc 2010, 47: 491-502. 10.4134/BKMS.2010.47.3.491

24. Eshaghi Gordji M: Stability of an additive-quadratic functional equation of two variables in F -spaces. J Nonlinear Sci Appl 2009, 2: 251-259.

25. Eshaghi Gordji M, Ghaemi MB, Kaboli Gharetapeh S, Shams S, Ebadian A: On the stability of J * -derivations. J Geom Phys 2010, 60: 454-459. 10.1016/j.geomphys.2009.11.004

26. Eshaghi Gordji M, Najati A: Approximately J * -homomorphisms: a fixed point approach. J Geom Phys 2010, 60: 809-814. 10.1016/j.geomphys.2010.01.012

27. Eshaghi Gordji M, Savadkouhi MB: On approximate cubic homomorphisms. Adv Difference Equ 2009, 2009: 1-11. Article ID 618463

28. Eshaghi Gordji M, Ghaemi MB, Majani H, Park C: Generalized Ulam-Hyers stability of Jensen functional equation in Erstnev PN-spaces. J Inequal Appl 2010, 2010: 14. Article ID 868193

29. Farokhzad R, Hosseinioun SAR: Perturbations of Jordan higher derivations in Banach ternary algebras: an alternative fixed point approach. Int J Nonlinear Anal Appl 2010, 1: 42-53.

30. Gajda Z: On stability of additive mappings. Int J Math Sci 1991, 14: 431-434. 10.1155/S016117129100056X

31. Gǎvruta P, Gǎvruta L: A new method for the generalized Hyers-Ulam-Rassias stability. Int J Nonlinear Anal Appl 2010, 1: 11-18.

32. Eshaghi Gordji M, Kaboli Gharetapeh S, Park C, Zolfaghri S: Stability of an additive-cubic-quartic functional equation. Adv Difference Equ 2009, 2009: 20. Article ID 395693

33. Eshaghi Gordji M, Kaboli Gharetapeh S, Rassias JM, Zolfaghari S: Solution and stability of a mixed type additive, quadratic and cubic functional equation. Adv Difference Equ 2009, 2009: 17. Article ID 826130

34. Eshaghi Gordji M, Savadkouhi MB: Stability of a mixed type cubic and quartic functional equations in random normed spaces. J Inequal Appl 2009, 2009: 9. Article ID 527462

35. Eshaghi Gordji M, Abbaszadeh S, Park C: On the stability of generalized mixed type quadratic and quartic functional equation in quasi-Banach spaces. J Inequal Appl 2009, 2009: 26. Article ID 153084

36. Eshaghi Gordji M, Savadkouhi MB, Park C: Quadratic-quartic functional equations in RN-spaces. J Inequal Appl 2009, 2009: 14. Article ID 868423

37. Eshaghi Gordji M, Savadkouhi MB: Approximation of generalized homomorphisms in quasi-Banach algebras. Analele Univ Ovidius Constata, Math Series 2009, 17: 203-214.

38. Eshaghi Gordji M, Savadkouhi MB, Bidkham M: Stability of a mixed type additive and quadratic functional equation in non-Archimedean spaces. J Comput Anal Appl 2010, 12: 454-462.

39. Jung S: A fixed point approach to the stability of an equation of the square spiral. Banach J Math Anal 2007, 1: 148-153.

40. Khodaei H, Kamyar M: Fuzzy approximately additive mappings. Int J Nonlinear Anal Appl 2010, 1: 44-53.

41. Khodaei H, Rassias ThM: Approximately generalized additive functions in several variables. Int J Nonlinear Anal Appl 2010, 1: 22-41.

42. Park C: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras. Banach J Math Anal 2007, 1: 23-32.

43. Park C, Najati A: Generalized additive functional inequalities in Banach algebras. Int J Nonlinear Anal Appl 2010, 1: 54-62.

44. Park C, Rassias ThM: Isomorphisms in unital C * -algebras. Int J Nonlinear Anal Appl 2010, 1: 1-10.

45. Park C, Rassias ThM: Isometric additive mappings in generalized quasi-Banach spaces. Banach J Math Anal 2008, 2: 59-69.

46. Saadati R, Park C:$\mathcal{L}$ -fuzzy normed spaces and stability of functional equations Non-Archimedean. Comput Math Appl 2010,60(8):2488-2496. 10.1016/j.camwa.2010.08.055

47. Cho Y, Park C, Saadati R: Functional inequalities in Non-Archimedean Banach spaces. Appl Math Lett 2010 ,23(10):1238-1242. 10.1016/j.aml.2010.06.005

48. Park C, Eshaghi Gordji M, Najati A: Generalized Hyers-Ulam stability of an AQCQ-functional equation in non-Archimedean Banach spaces. J Nonlinear Sci Appl 2010,3(4):272-281.

49. Shakeri S, Saadati R, Park C:Stability of the quadratic functional equation in non-Archimedean $\mathcal{L}$ -fuzzy normed spaces. Int J Nonlinear Anal Appl 2010, 1: 72-83.

## Acknowledgements

Dong Yun Shin was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).

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### Authors' contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Gordji, M.E., Khodabakhsh, R., Khodaei, H. et al. A Functional equation related to inner product spaces in non-archimedean normed spaces. Adv Differ Equ 2011, 37 (2011). https://doi.org/10.1186/1687-1847-2011-37 