A functional equation related to inner product spaces in non-Archimedean $\mathcal{L}$-random normed spaces

Abstract

In this paper, we prove the stability of a functional equation related to inner product spaces in non-Archimedean $\mathcal{L}$-random normed spaces.

MSC: 46S10, 39B52, 47S10, 26E30, 12J25.

1 Introduction

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 ThM Rassias [4] for approximate linear mappings by allowing the difference Cauchy equation $\parallel f(x_1+x_2)-f(x_1)-f(x_2)\parallel$ to be controlled by $\epsilon ({\parallel x_1\parallel }^p+{\parallel x_2\parallel }^p)$. In 1994, a generalization of the ThM Rassias‘ theorem was obtained by Gǎvruta [5], who replaced $\epsilon ({\parallel x_1\parallel }^p+{\parallel x_2\parallel }^p)$ by a general control function $\phi (x_1,x_2)$.

Quadratic functional equations were used to characterize inner product spaces [6]. A square norm on an inner product space satisfies the parallelogram equality ${\parallel x_1+x_2\parallel }^2+{\parallel x_1-x_2\parallel }^2=2({\parallel x_1\parallel }^2+{\parallel x_1\parallel }^2)$. The functional equation

$$f(x+y)+f(x-y)=2f(x)+2f(y)$$

(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 ThM 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 $n\ge 2$

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

for all $x_1,\dots ,x_n\in X$.

Let be a field. A non-Archimedean absolute value on is a function such that for any we have

(i) $|a|\ge 0$ and equality holds if and only if $a=0$,

(ii) $|ab|=|a||b|$,

(iii) $|a+b|\le max\{|a|,|b|\}$.

The condition (iii) is called the strict triangle inequality. By (ii), we have $|1|=|-1|=1$. Thus, by induction, it follows from (iii) that $|n|\le 1$ for each integer n. We always assume in addition that $||$ is non-trivial, i.e., that there is an such that $|a_0|\ne 0,1$.

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

(NA1) $\parallel x\parallel =0$ if and only if $x=0$;

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

(NA3) the strong triangle inequality (ultra-metric); 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.

Thanks to the inequality

$$\parallel x_m-x_l\parallel \le max\{\parallel x_{ȷ+1}-x_{ȷ}\parallel :l\le ȷ\le m-1\}(m>l)$$

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, but 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:

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

(1.2)

($n\in \mathbb{N}$, $n\ge 2$) in non-Archimedean normed spaces. Interesting new results concerning functional equations related to inner product spaces have recently been obtained by Najati and ThM Rassias [18] as well as for the fuzzy stability of a functional equation related to inner product spaces by Park [19] and Gordji and Khodaei [20]. During the last decades, several stability problems for various functional equations have been investigated by many mathematicians; [21–56].

2 Preliminaries

The theory of random normed spaces (RN-spaces) is important as a generalization of the deterministic result of linear normed spaces and also in the study of random operator equations. The RN-spaces may also provide us the appropriate tools to study the geometry of nuclear physics and have important applications in quantum particle physics. The Hyers-Ulam stability of different functional equations in RN-spaces and fuzzy normed spaces has been recently studied by Alsina [57], Mirmostafaee, Mirzavaziri, and Moslehian [58, 59], Miheţ and Radu [60], Miheţ, Saadati, and Vaezpour [61, 62], Baktash et al.[63], Najati [64], and Saadati et al.[65].

Let $\mathcal{L}=(L,{\ge }_L)$ be a complete lattice, that is, a partially ordered set in which every non-empty subset admits supremum and infimum and $0_{\mathcal{L}}=infL$, $1_{\mathcal{L}}=supL$. The space of latticetic random distribution functions, denoted by ${\mathrm{\Delta }}_L^{+}$, is defined as the set of all mappings $F:\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}\to L$ such that F is left continuous, non-decreasing on $\mathbb{R}$ and $F(0)=0_{\mathcal{L}}$, $F(+\mathrm{\infty })=1_{\mathcal{L}}$.

The subspace $D_L^{+}\subseteq {\mathrm{\Delta }}_L^{+}$ is defined as $D_L^{+}=\{F\in {\mathrm{\Delta }}_L^{+}:l^{-}F(+\mathrm{\infty })=1_{\mathcal{L}}\}$, where $l^{-}f(x)$ denotes the left limit of the function f at the point x. The space ${\mathrm{\Delta }}_L^{+}$ is partially ordered by the usual point-wise ordering of functions, that is, $F\ge G$ if and only if $F(t){\ge }_{L}G(t)$ for all t in $\mathbb{R}$. The maximal element for ${\mathrm{\Delta }}_L^{+}$ in this order is the distribution function given by

$${\epsilon }_0(t)=\{\begin{array}{cc}{0}_{\mathcal{L}},\hfill & \text{if }t\le 0,\hfill \\ 1_{\mathcal{L}},\hfill & \text{if }t>0.\hfill \end{array}$$

Definition 2.1[66]

A triangular norm (t-norm) on L is a mapping $\mathcal{T}:{(L)}^2⟶L$ satisfying the following conditions:

(1) $(\mathrm{\forall }x\in L)(\mathcal{T}(x,1_{\mathcal{L}})=x)$ (: boundary condition);

(2) $(\mathrm{\forall }(x,y)\in {(L)}^2)(\mathcal{T}(x,y)=\mathcal{T}(y,x))$ (: commutativity);

(3) $(\mathrm{\forall }(x,y,z)\in {(L)}^3)(\mathcal{T}(x,\mathcal{T}(y,z))=\mathcal{T}(\mathcal{T}(x,y),z))$ (: associativity);

(4) $(\mathrm{\forall }(x,x^{\prime },y,y^{\prime })\in {(L)}^4)(x{\le }_L{x}^{\prime }\text{ and }y{\le }_L{y}^{\prime }⟹\mathcal{T}(x,y){\le }_L\mathcal{T}(x^{\prime },y^{\prime }))$ (: monotonicity).

Let $\{x_n\}$ be a sequence in L converging to $x\in L$ (equipped the order topology). The t-norm $\mathcal{T}$ is called a continuous t-norm if

$$\underset{n\to \mathrm{\infty }}{lim}\mathcal{T}(x_n,y)=\mathcal{T}(x,y),$$

for any $y\in L$.

A t-norm $\mathcal{T}$ can be extended (by associativity) in a unique way to an n-array operation taking for $(x_1,\dots ,x_n)\in L^n$ the value $\mathcal{T}(x_1,\dots ,x_n)$ defined by

$${\mathcal{T}}_{i=1}^0{x}_i=1,{\mathcal{T}}_{i=1}^n{x}_i=\mathcal{T}({\mathcal{T}}_{i=1}^{n-1}{x}_i,x_n)=\mathcal{T}(x_1,\dots ,x_n).$$

The t-norm $\mathcal{T}$ can also be extended to a countable operation taking, for any sequence $\{x_n\}$ in L, the value

$${\mathcal{T}}_{i=1}^{\mathrm{\infty }}{x}_i=\underset{n\to \mathrm{\infty }}{lim}{\mathcal{T}}_{i=1}^n{x}_i.$$

(2.1)

The limit on the right side of (2.1) exists since the sequence ${({\mathcal{T}}_{i=1}^n{x}_i)}_{n\in \mathbb{N}}$ is non-increasing and bounded from below.

Note that we put $\mathcal{T}=T$ whenever $L=[0,1]$. If T is a t-norm then, for all $x\in [0,1]$ and $n\in N\cup \{0\}$, $x_T^{(n)}$ is defined by 1 if $n=0$ and $T(x_T^{(n-1)},x)$ if $n\ge 1$. A t-norm T is said to be of Hadžić-type (we denote by $T\in \mathcal{H}$) if the family ${(x_T^{(n)})}_{n\in N}$ is equi-continuous at $x=1$ (see [67]).

Definition 2.2[66]

A continuous t-norm $\mathcal{T}$ on $L={[0,1]}^2$ is said to be continuous t-representable if there exist a continuous t-norm ∗ and a continuous t-co-norm ⋄ on $[0,1]$ such that, for all $x=(x_1,x_2)$, $y=(y_1,y_2)\in L$,

$$\mathcal{T}(x,y)=(x_1\ast y_1,x_2\diamond y_2).$$

For example,

$$\mathcal{T}(a,b)=(a_1{b}_1,min\{a_2+b_2,1\})$$

and

$$\mathbf{M}(a,b)=(min\{a_1,b_1\},max\{a_2,b_2\})$$

for all $a=(a_1,a_2)$, $b=(b_1,b_2)\in {[0,1]}^2$ are continuous t-representable.

Define the mapping ${\mathcal{T}}_{\wedge }$ from $L^2$ to L by

$${\mathcal{T}}_{\wedge }(x,y)=min(x,y)=\{\begin{array}{cc}x,\hfill & \text{if }y{\ge }_{L}x,\hfill \\ y,\hfill & \text{if }x{\ge }_{L}y.\hfill \end{array}$$

Recall (see [67, 68]) that, if $\{x_n\}$ is a given sequence in L, then ${({\mathcal{T}}_{\wedge })}_{i=1}^n{x}_i$ is defined recurrently by ${({\mathcal{T}}_{\wedge })}_{i=1}^1{x}_i=x_1$ and ${({\mathcal{T}}_{\wedge })}_{i=1}^n{x}_i={\mathcal{T}}_{\wedge }({({\mathcal{T}}_{\wedge })}_{i=1}^{n-1}{x}_i,x_n)$ for all $n\ge 2$.

A negation on $\mathcal{L}$ is any decreasing mapping $\mathcal{N}:L\to L$ satisfying $\mathcal{N}(0_{\mathcal{L}})=1_{\mathcal{L}}$ and $\mathcal{N}(1_{\mathcal{L}})=0_{\mathcal{L}}$. If $\mathcal{N}(\mathcal{N}(x))=x$ for all $x\in L$, then $\mathcal{N}$ is called an involutive negation. In the following, $\mathcal{L}$ is endowed with a (fixed) negation $\mathcal{N}$.

Definition 2.3 A latticetic random normed space is a triple $(X,\mu ,{\mathcal{T}}_{\wedge })$, where X is a vector space and μ is a mapping from X into $D_L^{+}$ satisfying the following conditions:

(LRN1) ${\mu }_x(t)={\epsilon }_0(t)$ for all $t>0$ if and only if $x=0$;

(LRN2) ${\mu }_{\alpha x}(t)={\mu }_x(\frac{t}{|\alpha |})$ for all x in X, $\alpha \ne 0$ and $t\ge 0$;

(LRN3) ${\mu }_{x+y}(t+s){\ge }_L{\mathcal{T}}_{\wedge }({\mu }_x(t),{\mu }_y(s))$ for all $x,y\in X$ and $t,s\ge 0$.

We note that, from (LPN2), it follows that ${\mu }_{-x}(t)={\mu }_x(t)$ for all $x\in X$ and $t\ge 0$.

Example 2.4 Let $L=[0,1]\times [0,1]$ and an operation ${\le }_L$ be defined by

$$\begin{array}{c}L=\{(a_1,a_2):(a_1,a_2)\in [0,1]\times [0,1]\text{ and }{a}_1+a_2\le 1\},\hfill \\ (a_1,a_2){\le }_L(b_1,b_2)⟺a_1\le b_1,a_2\ge b_2,\mathrm{\forall }a=(a_1,a_2),b=(b_1,b_2)\in L.\hfill \end{array}$$

Then $(L,{\le }_L)$ is a complete lattice (see [66]). In this complete lattice, we denote its units by $0_L=(0,1)$ and $1_L=(1,0)$. Let $(X,\parallel \cdot \parallel )$ be a normed space. Let $\mathcal{T}(a,b)=(min\{a_1,b_1\},max\{a_2,b_2\})$ for all $a=(a_1,a_2)$, $b=(b_1,b_2)\in [0,1]\times [0,1]$ and μ be a mapping defined by

$${\mu }_x(t)=(\frac{t}{t+\parallel x\parallel },\frac{\parallel x\parallel }{t+\parallel x\parallel }),\mathrm{\forall }t\in {\mathbb{R}}^{+}.$$

Then $(X,\mu ,\mathcal{T})$ is a latticetic random normed space.

If $(X,\mu ,{\mathcal{T}}_{\wedge })$ is a latticetic random normed space, then we have

$$\mathcal{V}=\{V(\epsilon ,\lambda ):\epsilon {>}_L{0}_{\mathcal{L}},\lambda \in L\setminus \{0_{\mathcal{L}},1_{\mathcal{L}}\}\}$$

is a complete system of neighborhoods of null vector for a linear topology on X generated by the norm F, where

$$V(\epsilon ,\lambda )=\{x\in X:F_x(\epsilon ){>}_L\mathcal{N}(\lambda )\}.$$

Definition 2.5 Let $(X,\mu ,{\mathcal{T}}_{\wedge })$ be a latticetic random normed space.

(1) A sequence $\{x_n\}$ in X is said to be convergent to a point $x\in X$ if, for any $t>0$ and $\epsilon \in L\setminus \{0_{\mathcal{L}}\}$, there exists a positive integer N such that ${\mu }_{x_n-x}(t){>}_L\mathcal{N}(\epsilon )$ for all $n\ge N$.

(2) A sequence $\{x_n\}$ in X is called a Cauchy sequence if, for any $t>0$ and $\epsilon \in L\setminus \{0_{\mathcal{L}}\}$, there exists a positive integer N such that ${\mu }_{x_n-x_m}(t){>}_L\mathcal{N}(\epsilon )$ for all $n\ge m\ge N$.

(3) A latticetic random normed space $(X,\mu ,{\mathcal{T}}_{\wedge })$ is said to be complete if every Cauchy sequence in X is convergent to a point in X.

Theorem 2.6 If$(X,\mu ,{\mathcal{T}}_{\wedge })$is a latticetic random normed space and$\{x_n\}$is a sequence such that$x_n\to x$, then${lim}_{n\to \mathrm{\infty }}{\mu }_{x_n}(t)={\mu }_x(t)$.

Proof The proof is the same as in classical random normed spaces (see [17]). □

Lemma 2.7 Let$(X,\mu ,{\mathcal{T}}_{\wedge })$be a latticetic random normed space and$x\in X$. If

$${\mu }_x(t)=C,\mathrm{\forall }t>0,$$

then$C=1_{\mathcal{L}}$and$x=0$.

Proof Let ${\mu }_x(t)=C$ for all $t>0$. Since $Ran(\mu )\subseteq D_L^{+}$, we have $C=1_{\mathcal{L}}$ and, by (LRN1), we conclude that $x=0$. □

3 Hyers-Ulam stability in non-Archimedean latticetic random 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 latticetic random space. For convenience, we use the following abbreviation for a given mapping $f:G\to X$:

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

for all $x_1,\dots ,x_n\in G$, where $n\ge 2$ is a fixed integer.

Lemma 3.1[18]

Let$V_1$and$V_2$be real vector spaces. If an odd mapping$f:V_1\to V_2$satisfies the functional equation (1.2), then f is additive.

Let $\mathcal{K}$ be a non-Archimedean field, $\mathcal{X}$ a vector space over $\mathcal{K}$ and $(\mathcal{Y},\mu ,{\mathcal{T}}_{\wedge })$ a non-Archimedean complete LRN-space over $\mathcal{K}$. In the following theorem, we prove the Hyers-Ulam stability of the functional equation (1.2) in non-Archimedean latticetic random spaces for an odd mapping case.

Theorem 3.2 Let$\mathcal{K}$be a non-Archimedean field and$(\mathcal{X},\mu ,{\mathcal{T}}_{\wedge })$a non-Archimedean complete LRN-space over$\mathcal{K}$. Let$\phi :G^n\to D_L^{+}$be a distribution function such that

$$\underset{m\to \mathrm{\infty }}{lim}{\phi }_{2^m{x}_1,2^m{x}_2,\dots ,2^m{x}_n}\left({|2|}^{m}t\right)=1_{\mathcal{L}}=\underset{m\to \mathrm{\infty }}{lim}{\mathrm{\Phi }}_{2^{m-1}x}\left({|2|}^{m}t\right)$$

(3.1)

for all$x,x_1,x_2,\dots ,x_n\in G$, and

$${\tilde{\phi }}_x(t)=\underset{m\to \mathrm{\infty }}{lim}min\{{\mathrm{\Phi }}_{2^{k}x}\left({|2|}^{k}t\right):0\le k<m\}$$

(3.2)

exists for all$x\in G$, where

$${\mathrm{\Phi }}_x(t):=min\{{\phi }_{2x,0,\dots ,0}(t),min\{{\phi }_{x,x,0,\dots ,0}\left(\frac{|2|t}{n}\right),{\phi }_{x,-x,\dots ,-x}(|2|t),\phi (-x,x,\dots ,x)\}\}$$

(3.3)

for all$x\in G$. Suppose that an odd mapping$f:G\to X$satisfies the inequality

$${\mu }_{\mathrm{\Delta }f(x_1,\dots ,x_n)}(t){\ge }_L{\phi }_{x_1,x_2,\dots ,x_n}(t)$$

(3.4)

for all$x_1,x_2,\dots ,x_n\in G$and$t>0$. Then there exists an additive mapping$A:G\to X$such that

$${\mu }_{f(x)-A(x)}(t){\ge }_L{\tilde{\phi }}_x(|2|t)$$

(3.5)

for all$x\in G$and$t>0$, and if

$$\underset{\ell \to \mathrm{\infty }}{lim}\underset{m\to \mathrm{\infty }}{lim}min\{{\mathrm{\Phi }}_{2^{k}x}\left({|2|}^{k}t\right):\ell \le k<m+\ell \}=1_{\mathcal{L}}$$

(3.6)

then A is a unique additive mapping satisfying (3.5).

Proof Letting $x_1=n{x}_1$, $x_i=n{x}_1^{\prime }$ ($i=2,\dots ,n$) in (3.4) and using the oddness of f, we obtain that

(3.7)

for all $x_1,x_1^{\prime }\in G$ and $t>0$. Interchanging $x_1$ with $x_1^{\prime }$ in (3.7) and using the oddness of f, we get

(3.8)

for all $x_1,x_1^{\prime }\in G$ and $t>0$. It follows from (3.7) and (3.8) that

(3.9)

for all $x_1,x_1^{\prime }\in G$ and $t>0$. Setting $x_1=n{x}_1$, $x_2=-n{x}_1^{\prime }$, $x_i=0$ ($i=3,\dots ,n$) in (3.4) and using the oddness of f, we get

(3.10)

for all $x_1,x_1^{\prime }\in G$ and $t>0$. It follows from (3.9) and (3.10) that

(3.11)

for all $x_1,x_1^{\prime }\in G$ and $t>0$. Putting $x_1=n(x_1-x_1^{\prime })$, $x_i=0$ ($i=2,\dots ,n$) in (3.4), we obtain

$${\mu }_{f(n(x_1-x_1^{\prime }))-f((n-1)(x_1-x_1^{\prime }))-f((x_1-x_1^{\prime }))}(t){\ge }_L{\phi }_{n(x_1-x_1^{\prime }),0,\dots ,0}(t)$$

(3.12)

for all $x_1,x_1^{\prime }\in G$ and $t>0$. It follows from (3.11) and (3.12) that

$$\begin{array}{rcl}{\mu }_{f(n(x_1-x_1^{\prime }))-f(n{x}_1)+f(n{x}_1^{\prime })}(t)& {\ge }_L& min\{{\phi }_{n(x_1-x_1^{\prime }),0,\dots ,0}(t),{\phi }_{n{x}_1,-n{x}_1^{\prime },0,\dots ,0}\left(\frac{|2|}{n}t\right),\\ min\{{\phi }_{n{x}_1,n{x}_1^{\prime },\dots ,n{x}_1^{\prime }}\left(\frac{|2|}{n}t\right),{\phi }_{n{x}_1^{\prime },n{x}_1,\dots ,n{x}_1}\left(\frac{|2|}{n}t\right)\}\}\end{array}$$

(3.13)

for all $x_1,x_1^{\prime }\in G$ and $t>0$. Replacing $x_1$ and $x_1^{\prime }$ by $\frac{x}{n}$ and $\frac{-x}{n}$ in (3.13), respectively, we obtain

$${\mu }_{f(2x)-2f(x)}(t){\ge }_{L}min\{{\phi }_{2x,0,\dots ,0}(t),min\{{\phi }_{x,x,0,\dots ,0}\left(\frac{|2|}{n}t\right),{\phi }_{x,-x,\dots ,-x}(t),{\phi }_{-x,x,\dots ,x}(t)\}\}$$

for all $x\in G$ and $t>0$. Hence,

$${\mu }_{\frac{f(2x)}{2}-f(x)}(t){\ge }_L{\mathrm{\Phi }}_x(|2|t)$$

(3.14)

for all $x\in G$ and $t>0$. Replacing x by $2^{m-1}x$ in (3.14), we have

$${\mu }_{\frac{f(2^{m-1}x)}{2^{m-1}}-\frac{f(2^{m}x)}{2^m}}(t){\ge }_L{\mathrm{\Phi }}_{2^{m-1}x}\left({|2|}^{m}t\right)$$

(3.15)

for all $x\in G$ and $t>0$. It follows from (3.1) and (3.15) that the sequence $\{\frac{f(2^{m}x)}{2^m}\}$ is Cauchy. Since X is complete, we conclude that $\{\frac{f(2^{m}x)}{2^m}\}$ is convergent. So one can define the mapping $A:G\to X$ by $A(x):={lim}_{m\to \mathrm{\infty }}\frac{f(2^{m}x)}{2^m}$ for all $x\in G$. It follows from (3.14) and (3.15) that

$${\mu }_{f(x)-\frac{f(2^{m}x)}{2^m}}(t){\ge }_{L}min\{{\mathrm{\Phi }}_{2^{k}x}\left({|2|}^{k+1}t\right):0\le k<m\}$$

(3.16)

for all and all $x\in G$ and $t>0$. By taking m to approach infinity in (3.16) and using (3.2), one gets (3.5). By (3.1) and (3.4), we obtain

$$\begin{array}{rcl}{\mu }_{\mathrm{\Delta }A(x_1,x_2,\dots ,x_n)}(t)& =& \underset{m\to \mathrm{\infty }}{lim}{\mu }_{\mathrm{\Delta }f(2^m{x}_1,2^m{x}_2,\dots ,2^m{x}_n)}\left({|2|}^{m}t\right)\\ {\ge }_L& \underset{m\to \mathrm{\infty }}{lim}{\phi }_{2^m{x}_1,2^m{x}_2,\dots ,2^m{x}_n}\left({|2|}^{m}t\right)=1_{\mathcal{L}}\end{array}$$

for all $x_1,x_2,\dots ,x_n\in G$ and $t>0$. Thus the mapping A satisfies (1.2). By Lemma 3.1, A is additive.

If $A^{\prime }$ is another additive mapping satisfying (3.5), then

$$\begin{array}{rcl}{\mu }_{A(x)-A^{\prime }(x)}(t)& =& \underset{\ell \to \mathrm{\infty }}{lim}{\mu }_{A(2^{\ell }x)-A^{\prime }(2^{\ell }x)}\left({|2|}^{\ell }t\right)\\ {\ge }_L& \underset{\ell \to \mathrm{\infty }}{lim}min\{{\mu }_{A(2^{\ell }x)-f(2^{\ell }x)}\left({|2|}^{\ell }t\right),{\mu }_{f(2^{\ell }x)-Q^{\prime }(2^{\ell }x)}\left({|2|}^{\ell }t\right)\}\\ {\ge }_L& \underset{\ell \to \mathrm{\infty }}{lim}\underset{m\to \mathrm{\infty }}{lim}min\{{\tilde{\phi }}_{2^{k}x}\left({|2|}^{k+1}\right):\ell \le k<m+\ell \}=0\end{array}$$

for all $x\in G$, thus, $A=A^{\prime }$. □

Corollary 3.3 Let $\rho :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be a function satisfying

(i) $\rho (|2|t)\le \rho (|2|)\rho (t)$for all$t\ge 0$,

(ii) $\rho (|2|)<|2|$.

Let$\epsilon >0$and let$(G,\mu ,{\mathcal{T}}_{\wedge })$be an LRN-space in which$L=D^{+}$. Suppose that an odd mapping$f:G\to X$satisfies the inequality

$${\mu }_{\mathrm{\Delta }f(x_1,\dots ,x_n)}(t){\ge }_L\frac{t}{t+\epsilon {\sum }_{i=1}^n\rho (\parallel x_i\parallel )}$$

for all$x_1,\dots ,x_n\in G$and$t>0$. Then there exists a unique additive mapping$A:G\to X$such that

$${\mu }_{f(x)-A(x)}(t){\ge }_L\frac{t}{t+\frac{2n}{{|2|}^2}\epsilon \rho (\parallel x\parallel )}$$

for all$x\in G$and$t>0$.

Proof Defining $\phi :G^n\to D^{+}$ by ${\phi }_{x_1,\dots ,x_n}(t):=\frac{t}{t+\epsilon {\sum }_{i=1}^n\rho (\parallel x_i\parallel )}$, we have

$$\underset{m\to \mathrm{\infty }}{lim}{\phi }_{2^m{x}_1,\dots ,2^m{x}_n}\left({|2|}^{m}t\right){\ge }_L\underset{m\to \mathrm{\infty }}{lim}{\phi }_{x_1,\dots ,x_n}\left({\left(\frac{|2|}{\rho (|2|)}\right)}^m\right)=1_{\mathcal{L}}$$

for all $x_1,\dots ,x_n\in G$ and $t>0$. So, we have

$${\tilde{\phi }}_x(t):=\underset{m\to \mathrm{\infty }}{lim}min\{{\mathrm{\Phi }}_{2^{k}x}\left({|2|}^k\right):0\le k<m\}={\mathrm{\Phi }}_x(t)$$

and

$$\underset{\ell \to \mathrm{\infty }}{lim}\underset{m\to \mathrm{\infty }}{lim}min\{{\mathrm{\Phi }}_{2^{k}x}\left({|2|}^k\right):\ell \le k<m+\ell \}=\underset{\ell \to \mathrm{\infty }}{lim}{\mathrm{\Phi }}_{2^{\ell }x}\left({|2|}^{\ell }\right)=1_{\mathcal{L}}$$

for all $x\in G$ and $t>0$. It follows from (3.3) that

$${\mathrm{\Phi }}_x(t)=min\{\frac{t}{t+\epsilon \rho (\parallel 2x\parallel )},\frac{t}{t+\frac{1}{|2|}2n\epsilon \rho (\parallel x\parallel )}\}=\frac{|2|t}{|2|t+2n\epsilon \rho (\parallel x\parallel )}.$$

Applying Theorem 3.2, we conclude that

$${\mu }_{f(x)-A(x)}(t){\ge }_L{\tilde{\phi }}_x(|2|t)={\mathrm{\Phi }}_x(|2|t)=\frac{t}{t+\frac{2n}{{|2|}^2}\epsilon \rho (\parallel x\parallel )}$$

for all $x\in G$ and $t>0$. □

Lemma 3.4[18]

Let$V_1$and$V_2$be real vector spaces. If an even mapping$f:V_1\to V_2$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 LRN-spaces for an even mapping case.

Theorem 3.5 Let $\phi :G^n\to D_L^{+}$ be a function such that

$$\underset{m\to \mathrm{\infty }}{lim}{\phi }_{2^m{x}_1,2^m{x}_2,\dots ,2^m{x}_n}\left({|2|}^{2m}t\right)=1_{\mathcal{L}}=\underset{m\to \mathrm{\infty }}{lim}{\widetilde{{\phi }^{\prime }}}_{2^{m-1}x}\left({|2|}^{2m}t\right)$$

(3.17)

for all$x,x_1,x_2,\dots ,x_n\in G$, $t>0$and

$${\widetilde{{\phi }^{\prime }}}_x(t)=\underset{m\to \mathrm{\infty }}{lim}min\{{\widetilde{{\phi }^{\prime }}}_{2^{k}x}\left({|2|}^{2k}t\right):0\le k<m\}$$

(3.18)

exists for all $x\in G$ and $t>0$ where

$$\begin{array}{rcl}{\widetilde{{\phi }^{\prime }}}_x(t)& :=& min\{{\phi }_{nx,nx,0,\dots ,0}(|2n-2|t),{\phi }_{nx,0,\dots ,0}(|n-1|t),\\ {\phi }_{x,(n-1)x,0,\dots ,0}(|n-1|t),{\mathrm{\Psi }}_x(|n-1|t)\}\end{array}$$

(3.19)

and

$${\mathrm{\Psi }}_x(t):=min\{n{\phi }_{nx,0,\dots ,0}\left(\frac{|2|}{n}t\right),{\phi }_{nx,0,\dots ,0}(|2|t),{\phi }_{0,nx,\dots ,nx}(|2|t)\}$$

(3.20)

for all$x\in G$and$t>0$. Suppose that an even mapping$f:G\to X$with$f(0)=0$satisfies the inequality (3.4) for all$x_1,x_2,\dots ,x_n\in G$and$t>0$. Then there exists a quadratic mapping$Q:G\to X$such that

$${\mu }_{f(x)-Q(x)}(t){\ge }_L{\widetilde{{\phi }^{\prime }}}_x\left({|2|}^{2}t\right)$$

(3.21)

for all$x\in G$, $t>0$and if

$$\underset{\ell \to \mathrm{\infty }}{lim}\underset{m\to \mathrm{\infty }}{lim}min\{{\widetilde{{\phi }^{\prime }}}_{2^{k}x}\left({|2|}^{2k}t\right):\ell \le k<m+\ell \}=1_{\mathcal{L}}$$

(3.22)

then Q is a unique quadratic mapping satisfying (3.21).

Proof Replacing $x_1$ by $n{x}_1$, and $x_i$ by $n{x}_2$ ($i=2,\dots ,n$) in (3.4) and using the evenness of f, we obtain

(3.23)

for all $x_1,x_2\in G$ and $t>0$. Interchanging $x_1$ with $x_2$ in (3.23) and using the evenness of f, we obtain

(3.24)

for all $x_1,x_2\in G$ and $t>0$. It follows from (3.23) and (3.24) that

(3.25)

for all $x_1,x_2\in G$ and $t>0$. Setting $x_1=n{x}_1$, $x_2=-n{x}_2$, $x_i=0$ ($i=3,\dots ,n$) in (3.4) and using the evenness of f, we obtain

(3.26)

for all $x_1,x_2\in G$ and $t>0$. So, it follows from (3.25) and (3.26) that

(3.27)

for all $x_1,x_2\in G$ and $t>0$. Setting $x_1=x$, $x_2=0$ in (3.27), we obtain

(3.28)

for all $x\in G$ and $t>0$. Putting $x_1=nx$, $x_i=0$ ($i=2,\dots ,n$) in (3.4), one obtains

$${\mu }_{f(nx)-f((n-1)x)-(2n-1)f(x)}(t){\ge }_L{\phi }_{nx,0,\dots ,0}(t)$$

(3.29)

for all $x\in G$ and $t>0$. It follows from (3.28) and (3.29) that

$$\begin{array}{rcl}{\mu }_{f(nx)-n^{2}f(x)}(t)& {\ge }_L& min\{{\phi }_{nx,0,\dots ,0}(t),{\phi }_{nx,0,\dots ,0}\left(\frac{|2|}{n}t\right),\\ {\phi }_{nx,0,\dots ,0}(|2|t),{\phi }_{0,nx,\dots ,nx}(|2|t)\}\end{array}$$

(3.30)

for all $x\in G$ and $t>0$. Letting $x_2=-(n-1)x_1$ and replacing $x_1$ by $\frac{x}{n}$ in (3.26), we get

$${\mu }_{f((n-1)x)-f((n-2)x)-(2n-3)f(x)}(t){\ge }_L{\phi }_{x,(n-1)x,0,\dots ,0}(t)$$

(3.31)

for all $x\in G$ and $t>0$. It follows from (3.28) and (3.31) that

$$\begin{array}{rcl}{\mu }_{f((n-2)x)-{(n-2)}^{2}f(x)}(t)& {\ge }_L& min\{{\phi }_{x,(n-1)x,0,\dots ,0}(t),{\phi }_{nx,0,\dots ,0}\left(\frac{|2|}{n}t\right),\\ {\phi }_{nx,0,\dots ,0}(|2|t),{\phi }_{0,nx,\dots ,nx}(|2|t)\}\end{array}$$

(3.32)

for all $x\in G$ and $t>0$. It follows from (3.30) and (3.32) that

$${\mu }_{f(nx)-f((n-2)x)-4(n-1)f(x)}(t){\ge }_{L}min\{{\phi }_{nx,0,\dots ,0}(t),{\phi }_{x,(n-1)x,0,\dots ,0}(t),{\mathrm{\Psi }}_x(t)\}$$

(3.33)

for all $x\in G$ and $t>0$. Setting $x_1=x_2=nx$, $x_i=0$ ($i=3,\dots ,n$) in (3.4), we obtain

$${\mu }_{f((n-2)x)+(n-1)f(2x)-f(nx)}(t){\ge }_L{\phi }_{nx,nx,0,\dots ,0}(|2|t)$$

(3.34)

for all $x\in G$ and $t>0$. It follows from (3.33) and (3.34) that

(3.35)

for all $x\in G$ and $t>0$. Thus,

$${\mu }_{f(x)-\frac{f(2x)}{{}^{22}}}(t){\ge }_L{\widetilde{{\phi }^{\prime }}}_x\left({|2|}^{2}t\right)$$

(3.36)

for all $x\in G$ and $t>0$. Replacing x by $2^{m-1}x$ in (3.36), we have

$${\mu }_{\frac{f(2^{m-1}x)}{2^{2(m-1)}}-\frac{f(2^{m}x)}{2^{2m}}}(t){\ge }_L{\widetilde{{\phi }^{\prime }}}_{2^{m-1}x}\left({|2|}^{2m}t\right)$$

(3.37)

for all $x\in G$ and $t>0$. It follows from (3.17) and (3.37) that the sequence $\{\frac{f(2^{m}x)}{2^{2m}}\}$ is Cauchy. Since X is complete, we conclude that $\{\frac{f(2^{m}x)}{2^{2m}}\}$ is convergent. So, one can define the mapping $Q:G\to X$ by $Q(x):={lim}_{m\to \mathrm{\infty }}\frac{f(2^{m}x)}{2^{2m}}$ for all $x\in G$. By using induction, it follows from (3.36) and (3.37) that

$${\mu }_{f(x)-\frac{f(2^{m}x)}{2^{2m}}}(t){\ge }_{L}min\{{\widetilde{{\phi }^{\prime }}}_{2^{k}x}\left({|2|}^{2k+2}t\right):0\le k<m\}$$

(3.38)

for all and all $x\in G$ and $t>0$. By taking m to approach infinity in (3.38) and using (3.18), one gets (3.21).

The rest of proof is similar to the proof of Theorem 3.2. □

Corollary 3.6 Let $\eta :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be a function satisfying

(i) $\eta (|l|t)\le \eta (|l|)\eta (t)$for all$t\ge 0$,

(ii) $\eta (|l|)<{|l|}^2$for$l\in \{2,n-1,n\}$.

Let$\epsilon >0$and let$(G,\mu ,{\mathcal{T}}_{\wedge })$be a LRN-space in which$L=D^{+}$. Suppose that an even mapping$f:G\to X$with$f(0)=0$satisfies the inequality

$${\mu }_{\mathrm{\Delta }f(x_1,\dots ,x_n)}(t)\ge \frac{t}{t+\epsilon {\sum }_{i=1}^n\eta (\parallel x_i\parallel )}$$

for all$x_1,\dots ,x_n\in G$and$t>0$. Then there exists a unique quadratic mapping$Q:G\to X$such that

$${\mu }_{f(x)-Q(x)}(t)\ge \{\begin{array}{cc}\frac{t}{t+\frac{2}{{|2|}^2}\epsilon \eta (\parallel x\parallel )},\hfill & \mathit{\text{if }}n=2;\hfill \\ \frac{t}{t+\frac{n}{{|2|}^3|n-1|}\epsilon \eta (\parallel nx\parallel )},\hfill & \mathit{\text{if }}n>2,\hfill \end{array}$$

for all$x\in G$and$t>0$.

Proof Defining $\phi :G^n\to D^{+}$ by ${\phi }_{x_1,\dots ,x_n}(t):=\frac{t}{t+\epsilon {\sum }_{i=1}^n\eta (\parallel x_i\parallel )}$, we have

$$\underset{m\to \mathrm{\infty }}{lim}{\phi }_{2^m{x}_1,\dots ,2^m{x}_n}\left({|2|}^{2m}t\right)\ge \underset{m\to \mathrm{\infty }}{lim}{\phi }_{x_1,\dots ,x_n}\left({\left(\frac{{|2|}^2}{\eta (|2|)}\right)}^m\right)=1_{\mathcal{L}}$$

for all $x_1,\dots ,x_n\in G$ and $t>0$. We have

$${\widetilde{{\phi }^{\prime }}}_x(t):=\underset{m\to \mathrm{\infty }}{lim}min\{{\widetilde{{\phi }^{\prime }}}_{2^{k}x}\left({|2|}^{2k}t\right):0\le k<m\}$$

and

$$\underset{\ell \to \mathrm{\infty }}{lim}\underset{m\to \mathrm{\infty }}{lim}min\{{\widetilde{{\phi }^{\prime }}}_{2^{k}x}\left({|2|}^{2k}t\right):\ell \le k<m+\ell \}=\underset{\ell \to \mathrm{\infty }}{lim}{\widetilde{{\phi }^{\prime }}}_{2^{\ell }x}\left({|2|}^{2\ell }t\right)=0$$

for all $x\in G$ and $t>0$. It follows from (3.20) that

$$\begin{array}{rcl}{\mathrm{\Psi }}_x(t)& =& min\{\frac{|2|t}{|2|t+2n\epsilon \eta (\parallel nx\parallel )},\frac{|2|t}{|2|t+2\epsilon \eta (\parallel nx\parallel )},\frac{|2|t}{|2|t+2(n-1)\epsilon \eta (\parallel nx\parallel )}\}\\ =& \frac{|2|t}{|2|t+n\epsilon \eta (\parallel nx\parallel )}.\end{array}$$

Hence, by using (3.19), we obtain

$$\begin{array}{rcl}{\widetilde{{\phi }^{\prime }}}_x(t)& =& min\{\frac{|2n-2|t}{|2n-2|t+2\epsilon \eta (\parallel nx\parallel )},\frac{|n-1|t}{|n-1|t+\epsilon \eta (\parallel nx\parallel )},\\ \phantom{min\{}\frac{|2n-2|t}{|2n-2|t+n\epsilon \eta (\parallel nx\parallel )},\frac{|n-1|t}{|n-1|t+\epsilon (\eta (\parallel x\parallel )+\eta (\parallel (n-1)x\parallel ))}\}\\ =& \{\begin{array}{cc}\frac{t}{t+2\epsilon \eta (\parallel x\parallel )},\hfill & \text{if }n=2;\hfill \\ \frac{|2||n-1|t}{|2||n-1|t+n\epsilon \eta (\parallel nx\parallel )},\hfill & \text{if }n>2,\hfill \end{array}\end{array}$$

for all $x\in G$ and $t>0$. Applying Theorem 3.5, we conclude the required result. □

Lemma 3.7[18]

Let$V_1$and$V_2$be real vector spaces. A mapping$f:V_1\to V_2$satisfies (1.2) if and only if there exist a symmetric bi-additive mapping$B:V_1\times V_1\to V_2$and an additive mapping$A:V_1\to V_2$such that$f(x)=B(x,x)+A(x)$for all$x\in V_1$.

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

Theorem 3.8 Let$\phi :G^n\to D_L^{+}$be a function satisfying (3.1) for all$x,x_1,x_2,\dots ,x_n\in G$, and${\tilde{\phi }}_x(t)$and${\widetilde{{\phi }^{\prime }}}_x(t)$exist for all$x\in G$and$t>0$, where${\tilde{\phi }}_x(t)$and${\widetilde{{\phi }^{\prime }}}_x(t)$are defined as in Theorems  3.2 and  3.5. Suppose that a mapping$f:G\to X$with$f(0)=0$satisfies the inequality (3.4) for all$x_1,x_2,\dots ,x_n\in G$. Then there exist an additive mapping$A:G\to X$and a quadratic mapping$Q:G\to X$such that

$${\mu }_{f(x)-A(x)-Q(x)}(t){\ge }_{L}min\{{\tilde{\phi }}_x\left({|2|}^{2}t\right),{\tilde{\phi }}_{-x}\left({|2|}^{2}t\right),{\widetilde{{\phi }^{\prime }}}_x(|2|t),\frac{1}{|2|}{\widetilde{{\phi }^{\prime }}}_{-x}(|2|t)\}$$

(3.39)

for all$x\in G$and$t>0$. If

then A is a unique additive mapping and Q is a unique quadratic mapping satisfying (3.39).

Proof Let $f_e(x)=\frac{1}{2}(f(x)+f(-x))$ for all $x\in G$. Then

$$\begin{array}{rcl}\parallel \mathrm{\Delta }{f}_e(x_1,\dots ,x_n)\parallel & =& \parallel \frac{1}{2}(\mathrm{\Delta }f(x_1,\dots ,x_n)+\mathrm{\Delta }f(-x_1,\dots ,-x_n))\parallel \\ \le & \frac{1}{|2|}max\{\phi (x_1,\dots ,x_n),\phi (-x_1,\dots ,-x_n)\}\end{array}$$

for all $x_1,x_2,\dots ,x_n\in G$ and $t>0$. By Theorem 3.5, there exists a quadratic mapping $Q:G\to X$ such that

$${\mu }_{f_e(x)-Q(x)}(t){\ge }_{L}min\{{\widetilde{{\phi }^{\prime }}}_x\left({|2|}^{3}t\right),{\widetilde{{\phi }^{\prime }}}_{-x}\left({|2|}^{3}t\right)\}$$

(3.40)

for all $x\in G$ and $t>0$. Also, let $f_o(x)=\frac{1}{2}(f(x)-f(-x))$ for all $x\in G$. By Theorem 3.2, there exists an additive mapping $A:G\to X$ such that

$${\mu }_{f_o(x)-A(x)}(t){\ge }_{L}min\{{\tilde{\phi }}_x\left({|2|}^{2}t\right),{\tilde{\phi }}_{-x}\left({|2|}^{2}t\right)\}$$

(3.41)

for all $x\in G$ and $t>0$. Hence (3.39) follows from (3.40) and (3.41).

The rest of proof is trivial. □

Corollary 3.9 Let $\gamma :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be a function satisfying

(i) $\gamma (|l|t)\le \gamma (|l|)\gamma (t)$for all$t\ge 0$,

(ii) $\gamma (|l|)<{|l|}^2$for$l\in \{2,n-1,n\}$.

Let$\epsilon >0$, $(G,\mu ,{\mathcal{T}}_{\wedge })$be an LRN-space in which$L=D^{+}$and let$f:G\to X$satisfy

$${\mu }_{\mathrm{\Delta }f(x_1,\dots ,x_n)}(t)\ge \frac{t}{t+\epsilon {\sum }_{i=1}^n\gamma (\parallel x_i\parallel )}$$

for all$x_1,\dots ,x_n\in G$, $t>0$and$f(0)=0$. Then there exist a unique additive mapping$A:G\to X$and a unique quadratic mapping$Q:G\to X$such that

$${\mu }_{f(x)-A(x)-Q(x)}(t)\ge \frac{{|2|}^{3}t}{{|2|}^{3}t+2n\epsilon \gamma (\parallel x\parallel )}$$

for all$x\in G$and$t>0$.

Proof The result follows from Corollaries 3.6 and 3.3. □

Authors‘ contributions

All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.