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 [1] in 1940 and affirmatively solved by Hyers [2]. The result of Hyers was generalized by Aoki [3] for approximate additive mappings and by Rassias [4] 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 [5], 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 [6]. 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 [9] 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 $\left|\right|:K\to ℝ$ such that for any $a,b\in K$, we have

• (i) |a| ≥ 0 and equality holds if and only if a = 0,

• (ii) |ab| = |a||b|,

• (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 → ℝ 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 $r\in K$ ℝ and x ∈ X;

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

$$\parallel x+y\parallel \le max\left\{\parallel x\parallel ,\parallel y\parallel \right\}\left(x,y\in X\right).$$

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

Thanks to the inequality

$$\parallel x_m-x_l\parallel \le max\left\{\parallel x_{j+1}-x_j\parallel :l\le j\le m-1\right\}\left(m>l\right)$$

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 [10] 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 [11]. 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 [12–16].

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, 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 [17] as well as for the fuzzy stability of a functional equation related to inner product spaces by Park [18] and Eshaghi Gordji and Khodaei [19]. During the last decades, several stability problems for various functional equations have been investigated by many mathematicians (see [20–49]).

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 : G → X:

$$\Delta f\left(x_1,\dots ,x_n\right)=\sum _{i=1}^{n}f\left(x_i-\frac{1}{n}\sum _{j=1}^n{x}_j\right)-\sum _{i=1}^{n}f\left(x_i\right)+nf\left(\frac{1}{n}\sum _{i=1}^n{x}_i\right)$$

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

Lemma 2.1. [17]. Let V ₁ and V ₂ be real vector spaces. If an odd mapping f : V ₁ → V 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

$$\underset{m\to \infty }{{\displaystyle \lim }}\frac{\phi {(2}^m{x}_1{\mathrm{,2}}^m{x}_2\mathrm{,}\dots {\mathrm{,2}}^m{x}_n)}{{\mid 2\mid }^m}=0=\underset{m\to \infty }{{\displaystyle \lim }}\frac{1}{{\mid 2\mid }^m}\Phi {(2}^{m-1}x)$$

(2.1)

for all x, x ₁, x ₂,⋯, x _n ∈ G, and

$${\tilde{\phi }}_a(x)=\underset{m\to \infty }{{\displaystyle \lim }}\max \left\{\frac{1}{{\mid 2\mid }^k}\Phi {(2}^{k}x):0\le k<m\right\}$$

(2.2)

exists for all x ∈ G, where

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

(2.3)

for all x ∈ G. Suppose that an odd mapping f : G → X 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 : G → X such that

$$\parallel f\left(x\right)-A\left(x\right)\parallel \le \frac{1}{\left|2\right|}{\tilde{\phi }}_a\left(x\right)$$

(2.5)

for all x ∈ G, and if

$$\underset{\ell \to \infty }{{\displaystyle \lim }}\underset{m\to \infty }{{\displaystyle \lim }}\max \left\{\frac{1}{{\mid 2\mid }^k}\Phi {(2}^{k}x):\ell \le k<m+\ell \right\}=0$$

(2.6)

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

Proof. Letting x ₁ = nx ₁, $x_i=n{x}_1^{\prime }\left(i=2,\dots ,n\right)$ in (2.4) and using the oddness of f, we obtain that

$$\begin{array}{c}\parallel nf\left(x_1+\left(n-1\right){x^{\prime }}_1\right)+f\left(\left(n-1\right)\left(x_1-{x^{\prime }}_1\right)\right)-\left(n-1\right)f\left(x_1-{x^{\prime }}_1\right)\\ -f\left(n{x}_1\right)-\left(n-1\right)f\left(n{x^{\prime }}_1\right)\parallel \le \phi \left(n{x}_1,n{x^{\prime }}_1,\dots ,n{x^{\prime }}_1\right)\end{array}$$

(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

$$\begin{array}{c}\parallel nf\left(\left(n-1\right)x_1+{x^{\prime }}_1\right)-f\left(\left(n-1\right)\left(x_1-{x^{\prime }}_1\right)\right)+\left(n-1\right)f\left(x_1-{x^{\prime }}_1\right)\\ -\left(n-1\right)f\left(n{x}_1\right)-f\left(n{x^{\prime }}_1\right)\parallel \le \phi \left(n{x^{\prime }}_1,n{x}_1,\dots ,n{x}_1\right)\end{array}$$

(2.8)

for all $x_1\mathrm{,}{x^{\prime }}_1\in G$. It follows from (2.7) and (2.8) that

$$\begin{array}{c}\parallel nf\left(x_1+\left(n-1\right){x^{\prime }}_1\right)-nf\left(\left(n-1\right)x_1+{x^{\prime }}_1\right)+2f\left(\left(n-1\right)\left(x_1-{x^{\prime }}_1\right)\right)\\ -2\left(n-1\right)f\left(x_1-{x^{\prime }}_1\right)+\left(n-2\right)f\left(n{x}_1\right)-\left(n-2\right)f\left(n{x^{\prime }}_1\right)\parallel \\ \le max\left\{\phi \left(n{x}_1,n{x^{\prime }}_1,\dots ,n{x^{\prime }}_1\right),\phi \left(n{x^{\prime }}_1,n{x}_1,\dots ,n{x}_1\right)\right\}\end{array}$$

(2.9)

for all $x_1\mathrm{,}{x^{\prime }}_1\in G$. 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

$$\begin{array}{c}\parallel f\left(\left(n-1\right)x_1+{x^{\prime }}_1\right)-f\left(x_1+\left(n-1\right){x^{\prime }}_1\right)+2f\left(x_1-{x^{\prime }}_1\right)\\ -f\left(n{x}_1\right)+f\left(n{x^{\prime }}_1\right)\parallel \le \phi \left(n{x}_1,-n{x^{\prime }}_1,0,\dots ,0\right)\end{array}$$

(2.10)

for all $x_1\mathrm{,}{x^{\prime }}_1\in G$. It follows from (2.9) and (2.10) that

$$\begin{array}{l}\parallel f((n-1)(x_1-{x^{\prime }}_1))+f(x_1-{x^{\prime }}_1)-f(n{x}_1)+f(n{x^{\prime }}_1)\parallel \\ \le \frac{1}{\mid 2\mid }\max \{n\phi (n{x}_1\mathrm{,}-n{x^{\prime }}_1\mathrm{,0,}\dots ,0),\\ \phi (n{x}_1\mathrm{,}n{x^{\prime }}_1\mathrm{,}\dots \mathrm{,}n{x^{\prime }}_1),\phi (n{x^{\prime }}_1\mathrm{,}n{x}_1\mathrm{,}\dots \mathrm{,}n{x}_1)\}\end{array}$$

(2.11)

for all $x_1\mathrm{,}{x^{\prime }}_1\in G$. 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 $x_1\mathrm{,}{x^{\prime }}_1\in G$. It follows from (2.11) and (2.12) that

$$\begin{array}{c}\parallel f\left(n\left(x_1-{x^{\prime }}_1\right)\right)-f\left(n{x}_1\right)+f\left(n{x^{\prime }}_1\right)\parallel \\ \le max\left\{\phi \left(n\left(x_1-{x^{\prime }}_1\right),0,\dots ,0\right),\frac{n}{\left|2\right|}\phi \left(n{x}_1,-n{x^{\prime }}_1,0,\dots ,0\right),\right.\\ \left.\frac{1}{\left|2\right|}max\left\{\phi \left(n{x}_1,n{x^{\prime }}_1,\dots ,n{x^{\prime }}_1\right),\phi \left(n{x^{\prime }}_1,n{x}_1,\dots ,n{x}_1\right)\right\}\right\}\end{array}$$

(2.13)

for all $x_1\mathrm{,}{x^{\prime }}_1\in G$. Replacing x ₁ and $x_1^{\prime }$ by $\frac{x}{n}$ and $\frac{-x}{n}$ in (2.13), respectively, we obtain

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

for all x ∈ G. Hence,

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

(2.14)

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

$$\Vert \frac{f{(2}^{m-1}x)}{2^{m-1}}-\frac{f{(2}^{m}x)}{2^m}\Vert \le \frac{1}{{\mid 2\mid }^m}\Phi {(2}^{m-1}x)$$

(2.15)

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

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

(2.16)

for all m ∈ ℕ 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

$$\begin{array}{ll}\hfill \parallel \Delta A\left(x_1,x_2,\dots ,x_n\right)\parallel & =\underset{m\to \infty }{lim}\frac{1}{{\left|2\right|}^m}\parallel \Delta f\left(2^m{x}_1,2^m{x}_2,\dots ,2^m{x}_n\right)\parallel \\ \le \underset{m\to \infty }{lim}\frac{1}{{\left|2\right|}^m}\phi \left(2^m{x}_1,2^m{x}_2,\dots ,2^m{x}_n\right)=0\end{array}$$

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

$$\begin{array}{c}\parallel A(x)-A^{\prime }(x)\parallel =\underset{\ell \to \infty }{{\displaystyle \lim }}\mid 2{\mid }^{-\ell }\parallel A(2^{\ell }x)-A^{\prime }(2^{\ell }x)\parallel \\ \le \underset{\ell \to \infty }{{\displaystyle \lim }}\mid 2{\mid }^{-\ell }\max \{\parallel A(2^{\ell }x)-f(2^{\ell }x)\parallel \mathrm{,}\parallel f(2^{\ell }x)-Q^{\prime }(2^{\ell }x)\parallel \}\\ \le \frac{1}{\mid 2\mid }\underset{\ell \to \infty }{{\displaystyle \lim }}\underset{m\to \infty }{{\displaystyle \lim }}\max \left\{\frac{1}{{\mid 2\mid }^k}\tilde{\phi }{(2}^{k}x):\ell \le k<m+\ell \right\}=0\end{array}$$

for all x ∈ G, Thus A = A'. □

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

• (i) ρ (|2|t) ≤ ρ(|2|)ρ(t) for all t ≥ 0,

• (ii) ρ(|2|) < |2|.

Let ε > 0 and let G be a normed space. Suppose that an odd mapping f : G → X 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 : G → X such that

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

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

$$\underset{m\to \infty }{lim}\frac{1}{{\left|2\right|}^m}\phi \left(2^m{x}_1,\dots ,2^m{x}_n\right)\le \underset{m\to \infty }{lim}{\left(\frac{\rho \left(\left|2\right|\right)}{\left|2\right|}\right)}^m\phi \left(x_1,\dots ,x_n\right)=0$$

for all x ₁,⋯, x _n ∈ G. So we have

$${\tilde{\phi }}_a(x):=\underset{m\to \infty }{{\displaystyle \lim }}\max \left\{\frac{1}{{\mid 2\mid }^k}\Phi {(2}^{k}x):0\le k<m\right\}=\Phi (x)$$

and

$$\underset{\ell \to \infty }{{\displaystyle \lim }}\underset{m\to \infty }{{\displaystyle \lim }}\max \left\{\frac{1}{{\mid 2\mid }^k}\Phi {(2}^{k}x):\ell \le k<m+\ell \right\}=\underset{\ell \to \infty }{{\displaystyle \lim }}\frac{1}{{\mid 2\mid }^{\ell }}\Phi {(2}^{\ell }x)=0$$

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

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

Applying Theorem 2.2, we conclude that

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

for all x ∈ G. □

Lemma 2.4. [17]. Let V ₁ and V ₂ be real vector spaces. If an even mapping f : V ₁ → V 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

$$\underset{m\to \infty }{{\displaystyle \lim }}\frac{\phi {(2}^m{x}_1{\mathrm{,2}}^m{x}_2\mathrm{,}\dots {\mathrm{,2}}^m{x}_n)}{{\mid 2\mid }^{2m}}=0=\underset{m\to \infty }{{\displaystyle \lim }}\frac{1}{{\mid 2\mid }^{2m}}\tilde{\phi }{(2}^{m-1}x)$$

(2.17)

for all x, x ₁, x ₂,⋯, x _n ∈ G, and

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

(2.18)

exists for all x ∈ G, where

$$\begin{array}{ll}\hfill \tilde{\phi }\left(x\right):=& \frac{1}{|n-1|}max\left\{\frac{1}{\left|2\right|}\phi \left(nx,nx,0,\dots ,0\right),\right.\\ \left.\phi \left(nx,0,\dots ,0\right),\phi \left(x,\left(n-1\right)x,0,\dots ,0\right),\Psi \left(x\right)\right\}\end{array}$$

(2.19)

and

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

(2.20)

for all x ∈ G. Suppose that an even mapping f : G → X 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 : G → X such that

$$\parallel f\left(x\right)-Q\left(x\right)\parallel \le \frac{1}{{\left|2\right|}^2}{\tilde{\phi }}_q\left(x\right)$$

(2.21)

for all x ∈ G, and if

$$\underset{\ell \to \infty }{lim}\underset{m\to \infty }{lim}max\left\{\frac{1}{{\left|2\right|}^{2k}}\tilde{\phi }(2^{k}x):\ell \le k<m+\ell \right\}=0$$

(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

$$\begin{array}{c}\parallel nf\left(x_1+\left(n-1\right)x_2\right)+f\left(\left(n-1\right)\left(x_1-x_2\right)\right)+\left(n-1\right)f\left(x_1-x_2\right)\\ -f\left(n{x}_1\right)-\left(n-1\right)f\left(n{x}_2\right)\parallel \le \phi \left(n{x}_1,n{x}_2,\dots ,n{x}_2\right)\end{array}$$

(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

$$\begin{array}{c}\parallel nf\left(\left(n-1\right)x_1+x_2\right)+f\left(\left(n-1\right)\left(x_1-x_2\right)\right)+\left(n-1\right)f\left(x_1-x_2\right)\\ -\left(n-1\right)f\left(n{x}_1\right)-f\left(n{x}_2\right)\parallel \le \phi \left(n{x}_2,n{x}_1,\dots ,n{x}_1\right)\end{array}$$

(2.24)

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

$$\begin{array}{c}\parallel nf\left(\left(n-1\right)x_1+x_2\right)+nf\left(x_1+\left(n-1\right)x_2\right)+2f\left(\left(n-1\right)\left(x_1-x_2\right)\right)\\ +2\left(n-1\right)f\left(x_1-x_2\right)-nf\left(n{x}_1\right)-nf\left(n{x}_2\right)\parallel \\ \le max\left\{\phi \left(n{x}_1,n{x}_2,\dots ,n{x}_2\right),\phi \left(n{x}_2,n{x}_1,\dots ,n{x}_1\right)\right\}\end{array}$$

(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

$$\begin{array}{c}\parallel f\left(\left(n-1\right)x_1+x_2\right)+f\left(x_1+\left(n-1\right)x_2\right)+2\left(n-1\right)f\left(x_1-x_2\right)\\ -f\left(n{x}_1\right)-f\left(n{x}_2\right)\parallel \le \phi \left(n{x}_1,-n{x}_2,0,\dots ,0\right)\end{array}$$

(2.26)

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

$$\begin{array}{c}\parallel f\left(\left(n-1\right)\left(x_1-x_2\right)\right)-{\left(n-1\right)}^{2}f\left(x_1-x_2\right)\parallel \\ \le \frac{1}{\left|2\right|}max\{n\phi \left(n{x}_1,-n{x}_2,0,\dots ,0\right),\\ \phi \left(n{x}_1,n{x}_2,\dots ,n{x}_2\right),\phi \left(n{x}_2,n{x}_1,\dots ,n{x}_1\right)\}\end{array}$$

(2.27)

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

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

(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

$$\begin{array}{ll}\hfill \parallel f\left(nx\right)-n^{2}f\left(x\right)\parallel & \le max\left\{\phi \left(nx,0,\dots ,0\right),\frac{n}{\left|2\right|}\phi \left(nx,0,\dots ,0\right),\right.\\ \left.\frac{1}{\left|2\right|}\phi \left(nx,0,\dots ,0\right),\frac{1}{\left|2\right|}\phi \left(0,nx,\dots ,nx\right)\right\}\end{array}$$

(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

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

(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

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

(2.34)

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

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

(2.35)

for all x ∈ G. Thus,

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

(2.36)

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

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

(2.37)

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

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

(2.38)

for all n ∈ 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

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

• (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 : G → X 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 : G → X such that

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

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

$$\underset{m\to \infty }{lim}\frac{1}{{\left|2\right|}^{2m}}\phi \left(2^m{x}_1,\dots ,2^m{x}_n\right)\le \underset{m\to \infty }{lim}{\left(\frac{\eta \left(\left|2\right|\right)}{{\left|2\right|}^2}\right)}^m\phi \left(x_1,\dots ,x_n\right)=0$$

for all x ₁,⋯, x _n ∈ G. We have

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

and

$$\underset{\ell \to \infty }{lim}\underset{m\to \infty }{lim}max\left\{\frac{1}{{\left|2\right|}^{2k}}\tilde{\phi }(2^{k}x):\ell \le k<m+\ell \right\}=\underset{\ell \to \infty }{lim}\frac{1}{{\left|2\right|}^{2\ell }}\tilde{\phi }\left(2^{\ell }x\right)=0$$

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

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

Hence, by using (2.19), we obtain

$$\begin{array}{ll}\hfill \tilde{\phi }\left(x\right)& =\frac{1}{|n-1|}max\left\{\frac{2}{\left|2\right|}\epsilon \eta \left(\parallel nx\parallel \right),\right.\\ \left.\epsilon \eta \left(\parallel nx\parallel \right),\frac{n}{\left|2\right|}\epsilon \eta \left(\parallel nx\parallel \right),\epsilon \left(\eta \left(\parallel x\parallel \right)+\eta \left(\parallel \left(n-1\right)x\parallel \right)\right)\right\}\\ =\left\{\begin{array}{cc}\hfill 2\epsilon \eta \left(\parallel x\parallel \right),\hfill & \hfill ifn=2;\hfill \\ \hfill \frac{n}{\left|2\right||n-1|}\epsilon \eta \left(\parallel nx\parallel \right),\hfill & \hfill ifn>2,\hfill \end{array}\right.\end{array}$$

for all x ∈ G.

Applying Theorem 2.5, we conclude the required result. □

Lemma 2.7. [17]. Let V ₁ and V ₂ be real vector spaces. A mapping f : V ₁ → V ₂ satisfies (1.2) if and only if there exist a symmetric bi-additive mapping B : V ₁ × V ₁ → V ₂ and an additive mapping A : V ₁ → V 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 ${\tilde{\phi }}_a\left(x\right)$ and ${\tilde{\phi }}_q\left(x\right)$ exist for all x ∈ G, where ${\tilde{\phi }}_a\left(x\right)$ and ${\tilde{\phi }}_q\left(x\right)$ are defined as in Theorems 2.2 and 2.5. Suppose that a mapping f : G → X 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 : G → X and a quadratic mapping Q : G → X such that

$$\begin{array}{c}\parallel f\left(x\right)-A\left(x\right)-Q\left(x\right)\parallel \\ \le \frac{1}{{\left|2\right|}^2}max\left\{{\tilde{\phi }}_a\left(x\right),{\tilde{\phi }}_a\left(-x\right),\frac{1}{\left|2\right|}{\tilde{\phi }}_q\left(x\right),\frac{1}{\left|2\right|}{\tilde{\phi }}_q\left(-x\right)\right\}\end{array}$$

(2.39)

for all x ∈ G. If

$$\begin{array}{c}\underset{\ell \to \infty }{lim}\underset{m\to \infty }{lim}max\left\{\frac{1}{{\left|2\right|}^k}\Phi (2^{k}x):\ell \le k<m+\ell \right\}=0\\ =\underset{\ell \to \infty }{lim}\underset{m\to \infty }{lim}max\left\{\frac{1}{{\left|2\right|}^{2k}}\tilde{\phi }(2^{k}x):\ell \le k<m+\ell \right\}\end{array}$$

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

$$\begin{array}{ll}\hfill \parallel \Delta f_e\left(x_1,\dots ,x_n\right)\parallel & =∥\frac{1}{2}\left(\Delta f\left(x_1,\dots ,x_n\right)+\Delta f\left(-x_1,\dots ,-x_n\right)\right)∥\\ \le \frac{1}{\left|2\right|}max\left\{\phi \left(x_1,\dots ,x_n\right),\phi \left(-x_1,\dots ,-x_n\right)\right\}\end{array}$$

for all x ₁, x ₂,⋯, x _n ∈ G. By Theorem 2.5, there exists a quadratic mapping Q : G → X such that

$$\parallel f_e\left(x\right)-Q\left(x\right)\parallel \le \frac{1}{{\left|2\right|}^3}max\left\{{\tilde{\phi }}_q\left(x\right),{\tilde{\phi }}_q\left(-x\right)\right\}$$

(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 : G → X such that

$$\parallel f_o\left(x\right)-A\left(x\right)\parallel \le \frac{1}{{\left|2\right|}^2}max\left\{{\tilde{\phi }}_a\left(x\right),{\tilde{\phi }}_a\left(-x\right)\right\}$$

(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 : G → X 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 : G → X and a unique quadratic mapping Q : G → X such that

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

for all x ∈ G.

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