Stability of general multi-Euler-Lagrange quadratic functional equations in non-Archimedean fuzzy normed spaces

Abstract

In this paper we prove the generalized Hyers-Ulam stability of the system defining general Euler-Lagrange quadratic mappings in non-Archimedean fuzzy normed spaces over a field with valuation using the direct and the fixed point methods.

MSC:39B82, 39B52, 46H25.

1 Introduction

Let $\mathbb{K}$ be a field. A valuation mapping on $\mathbb{K}$ is a function $|\cdot |:\mathbb{K}\to \mathbb{R}$ such that for any $r,s\in \mathbb{K}$ the following conditions are satisfied: (i) $|r|\ge 0$ and equality holds if and only if $r=0$; (ii) $|rs|=|r|\cdot |s|$; (iii) $|r+s|\le |r|+|s|$.

A field endowed with a valuation mapping will be called a valued field. The usual absolute values of $\mathbb{R}$ and $\mathbb{C}$ are examples of valuations. A trivial example of a non-Archimedean valuation is the function $|\cdot |$ taking everything except for 0 into 1 and $|0|=0$. In the following we will assume that $|\cdot |$ is non-trivial, i.e., there is an $r_0\in \mathbb{K}$ such that $|r_0|\ne 0,1$.

If the condition (iii) in the definition of a valuation mapping is replaced with a strong triangle inequality (ultrametric): $|r+s|\le max\{|r|,|s|\}$, then the valuation $|\cdot |$ is said to be non-Archimedean. In any non-Archimedean field we have $|1|=|-1|=1$ and $|n|\le 1$ for all $n\in \mathbb{N}$.

Throughout this paper, we assume that $\mathbb{K}$ is a valued field, $\mathcal{X}$ and $\mathcal{Y}$ are vector spaces over $\mathbb{K}$, $a,b\in \mathbb{K}$ are fixed with $\lambda :=a^2+b^2\ne 0,1$ (${\lambda }_1:=2a\ne 0,1$ if $a=b$) and n is a positive integer. Moreover, $\mathbb{N}$ stands for the set of all positive integers and $\mathbb{R}$ (respectively, $\mathbb{Q}$) denotes the set of all reals (respectively, rationals).

A mapping $f:{\mathcal{X}}^n\to \mathcal{Y}$ is called a general multi-Euler-Lagrange quadratic mapping if it satisfies the general Euler-Lagrange quadratic equations in each of their n arguments:

(1.1)

for all $i=1,\dots ,n$ and all $x_1,\dots ,x_{i-1},x_i,x_i^{\prime },x_{i+1},\dots ,x_n\in \mathcal{X}$. Letting $x_i=x_i^{\prime }=0$ in (1.1), we get $f(x_1,\dots ,x_{i-1},0,x_{i+1},\dots ,x_n)=0$. Putting $x_i^{\prime }=0$ in (1.1), we have

$$f(x_1,\dots ,x_{i-1},a{x}_i,x_{i+1},\dots ,x_n)+f(x_1,\dots ,x_{i-1},b{x}_i,x_{i+1},\dots ,x_n)=\lambda f(x_1,\dots ,x_n).$$

(1.2)

Replacing $x_i$ by $a{x}_i$ and $x_i^{\prime }$ by $b{x}_i$ in (1.1), respectively, we obtain

(1.3)

From (1.2) and (1.3), one gets

$$f(x_1,\dots ,x_{i-1},\lambda x_i,x_{i+1},\dots ,x_n)={\lambda }^{2}f(x_1,\dots ,x_n)$$

(1.4)

for all $i=1,\dots ,n$ and all $x_1,\dots ,x_n\in \mathcal{X}$. If $a=b$ in (1.1), then we have

(1.5)

Letting $x_i^{\prime }=x_i$ in (1.5), we obtain

$$f(x_1,\dots ,x_{i-1},{\lambda }_1{x}_i,x_{i+1},\dots ,x_n)={\lambda }_1^{2}f(x_1,\dots ,x_n)$$

(1.6)

for all $i=1,\dots ,n$ and all $x_1,\dots ,x_n\in \mathcal{X}$.

The study of stability problems for functional equations is related to a question of Ulam [30] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [13]. The result of Hyers was generalized by Aoki [2] for approximate additive mappings and by Rassias [27] for approximate linear mappings by allowing the Cauchy difference operator $CDf(x,y)=f(x+y)-[f(x)+f(y)]$ to be controlled by $ϵ({\parallel x\parallel }^p+{\parallel y\parallel }^p)$. In 1994, a further generalization was obtained by Găvruţa [9], who replaced $ϵ({\parallel x\parallel }^p+{\parallel y\parallel }^p)$ by a general control function $\phi (x,y)$. We refer the reader to see, for instance, [1, 4–7, 14–16, 18, 20, 22, 23, 25, 26, 28, 31–37] for more information on different aspects of stability of functional equations. On the other hand, for some outcomes on the stability of multi-quadratic and Euler-Lagrange-type quadratic mappings we refer the reader to [7, 11, 24].

The main purpose of this paper is to prove the generalized Hyers-Ulam stability of multi-Euler-Lagrange quadratic functional equation (1.1) in complete non-Archimedean fuzzy normed spaces over a field with valuation using the direct and the fixed point methods.

2 Preliminaries

We recall the notion of non-Archimedean fuzzy normed spaces over a field with valuation and some preliminary results (see for instance [3, 22, 23, 31, 32]). For more details the reader is referred to [3, 22].

Definition 2.1 Let $\mathcal{X}$ be a linear space over a field $\mathbb{K}$ with a non-Archimedean valuation $|\cdot |$. A function $\parallel \cdot \parallel :\mathcal{X}\to [0,\mathrm{\infty })$ is said to be a non-Archimedean norm if it satisfies the following conditions:

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

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

• (iii) the strong triangle inequality

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

Then $(\mathcal{X},\parallel \cdot \parallel )$ is called a non-Archimedean normed space. By a complete non-Archimedean normed space, we mean one in which every Cauchy sequence is convergent.

In 1897, Hensel discovered the p-adic numbers as a number-theoretical analogue of power series in complex analysis. Let p be a prime number. For any nonzero rational number a, there exists a unique integer r such that $a=p^{r}m/n$, where m and n are integers not divisible by p. Then ${|a|}_p:=p^{-r}$ defines a non-Archimedean norm on $\mathbb{Q}$. The completion of $\mathbb{Q}$ with respect to the metric $d(a,b)={|a-b|}_p$ is denoted by ${\mathbb{Q}}_p$ which is called the p-adic number field. Note that if $p>2$, then ${|2^n|}_p=1$ for each integer n but ${|2|}_2<1$.

During the last three decades, p-adic numbers have gained the interest of physicists for their research, in particular, into problems deriving from quantum physics, p-adic strings, and superstrings (see for instance [21]).

A triangular norm (shorter t-norm, [29]) is a binary operation $T:[0,1]\times [0,1]\to [0,1]$ which satisfies the following conditions: (a) T is commutative and associative; (b) $T(a,1)=a$ for all $a\in [0,1]$; (c) $T(a,b)\le T(c,d)$ whenever $a\le c$ and $b\le d$ for all $a,b,c,d\in [0,1]$. Basic examples of continuous t-norms are the Łukasiewicz t-norm $T_L$, $T_L(a,b)=max\{a+b-1,0\}$, the product t-norm $T_P$, $T_P(a,b)=ab$ and the strongest triangular norm $T_M$, $T_M(a,b)=min\{a,b\}$. A t-norm is called continuous if it is continuous with respect to the product topology on the set $[0,1]\times [0,1]$.

A t-norm T can be extended (by associativity) in a unique way to an m-array operation taking for $(x_1,\dots ,x_m)\in {[0,1]}^m$, the value $T(x_1,\dots ,x_m)$ defined recurrently by $T_{i=1}^0{x}_i=1$ and $T_{i=1}^m{x}_i=T(T_{i=1}^{m-1}{x}_i,x_m)$ for $m\in \mathbb{N}$. T can also be extended to a countable operation, taking for any sequence ${\{x_i\}}_{i\in \mathbb{N}}$ in $[0,1]$, the value $T_{i=1}^{\mathrm{\infty }}{x}_i$ is defined as ${lim}_{m\to \mathrm{\infty }}{T}_{i=1}^m{x}_i$. The limit exists since the sequence ${\{T_{i=1}^m{x}_i\}}_{m\in \mathbb{N}}$ is non-increasing and bounded from below. $T_{i=m}^{\mathrm{\infty }}{x}_i$ is defined as $T_{i=1}^{\mathrm{\infty }}{x}_{m+i}$.

Definition 2.2 A t-norm T is said to be of Hadžić-type (H-type, we denote by $T\in \mathcal{H}$) if a family of functions $\{T_{i=1}^m(t)\}$ for all $m\in \mathbb{N}$ is equicontinuous at $t=1$, that is, for all $\epsilon \in (0,1)$ there exists $\delta \in (0,1)$ such that

$$t>1-\delta ⟹T_{i=1}^m(t)>1-\epsilon \text{for all }m\in \mathbb{N}.$$

The t-norm $T_M$ is a t-norm of Hadžić-type. Other important triangular norms we refer the reader to [12].

Proposition 2.3 (see [12])

• (1) If $T=T_P$ or $T=T_L$, then

  $$\underset{m\to \mathrm{\infty }}{lim}{T}_{i=1}^{\mathrm{\infty }}{x}_{m+i}=1⟺\sum _{i=1}^{\mathrm{\infty }}(1-x_i)<\mathrm{\infty }.$$

• (2) If T is of Hadžić-type, then

  $$\underset{m\to \mathrm{\infty }}{lim}{T}_{i=m}^{\mathrm{\infty }}{x}_i=\underset{m\to \mathrm{\infty }}{lim}{T}_{i=1}^{\mathrm{\infty }}{x}_{m+i}=1$$

for every sequence ${\{x_i\}}_{i\in \mathbb{N}}$ in $[0,1]$ such that ${lim}_{i\to \mathrm{\infty }}{x}_i=1$.

Definition 2.4 (see [22])

Let $\mathcal{X}$ be a linear space over a valued field $\mathbb{K}$ and T be a continuous t-norm. A function $N:\mathcal{X}\times \mathbb{R}\to [0,1]$ is said to be a non-Archimedean fuzzy Menger norm on $\mathcal{X}$ if for all $x,y\in \mathcal{X}$ and all $s,t\in \mathbb{R}$:

(N1) $N(x,t)=0$ for all $t\le 0$;

(N2) $x=0$ if and only if $N(x,t)=1$, $t>0$;

(N3) $N(cx,t)=N(x,t/|c|)$ if $c\ne 0$;

(N4) $N(x+y,max\{s,t\})\ge T(N(x,s),N(y,t))$, $s,t>0$;

(N5) ${lim}_{t\to \mathrm{\infty }}N(x,t)=1$.

If N is a non-Archimedean fuzzy Menger norm on $\mathcal{X}$, then the triple $(\mathcal{X},N,T)$ is called a non-Archimedean fuzzy normed space. It should be noticed that from the condition (N4) it follows that

$$N(x,t)\ge T(N(0,t),N(x,s))=N(x,s)$$

for every $t>s>0$ and $x,y\in \mathcal{X}$, that is, $N(x,\cdot )$ is non-decreasing for every x. This implies $N(x,s+t)\ge N(x,max\{s,t\})$. If (N4) holds, then so does

(N6) $N(x+y,s+t)\ge T(N(x,s),N(y,t))$.

We repeatedly use the fact $N(-x,t)=N(x,t)$, $x\in \mathcal{X}$, $t>0$, which is deduced from (N3). We also note that Definition 2.4 is more general than the definition of a non-Archimedean Menger norm in [23, 31], where only fields with a non-Archimedean valuation have been considered.

Definition 2.5 Let $(\mathcal{X},N,T)$ be a non-Archimedean fuzzy normed space. Let ${\{x_m\}}_{m\in \mathbb{N}}$ be a sequence in $\mathcal{X}$. Then ${\{x_m\}}_{m\in \mathbb{N}}$ is said to be convergent if there exists $x\in \mathcal{X}$ such that ${lim}_{m\to \mathrm{\infty }}N(x_m-x,t)=1$ for all $t>0$. In that case, x is called the limit of the sequence ${\{x_m\}}_{m\in \mathbb{N}}$ and we denote it by ${lim}_{m\to \mathrm{\infty }}{x}_m=x$. The sequence ${\{x_m\}}_{m\in \mathbb{N}}$ in $\mathcal{X}$ is said to be a Cauchy sequence if ${lim}_{m\to \mathrm{\infty }}N(x_{m+p}-x_m,t)=1$ for all $t>0$ and $p=1,2,\dots$ . If every Cauchy sequence in $\mathcal{X}$ is convergent, then the space is called a complete non-Archimedean fuzzy normed space.

Example 2.6 Let $(\mathcal{X},\parallel \cdot \parallel )$ be a real (or non-Archimedean) normed space. For each $k>0$, consider

$$N_k(x,t)=\{\begin{array}{cc}\frac{t}{t+k\parallel x\parallel },\hfill & t>0,\hfill \\ 0,\hfill & t\le 0.\hfill \end{array}$$

Then $(\mathcal{X},N_k,T_M)$ is a non-Archimedean fuzzy normed space.

Example 2.7 (see [22])

Let $(\mathcal{X},\parallel \cdot \parallel )$ be a real normed space. Then the triple $(\mathcal{X},N,T_P)$, where

$$N(x,t)=\{\begin{array}{cc}{e}^{-\parallel x\parallel /t},\hfill & t>0,\hfill \\ 0,\hfill & t\le 0\hfill \end{array}$$

is a non-Archimedean fuzzy normed space. Moreover, if $(\mathcal{X},\parallel \cdot \parallel )$ is complete, then $(\mathcal{X},N,T_P)$ is complete and therefore it is a complete non-Archimedean fuzzy normed space over an Archimedean valued field.

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

• (1) $d(x,y)=0$ if and only if $x=y$; (2) $d(x,y)=d(y,x)$, $x,y\in \mathrm{\Omega }$; (3) $d(x,y)\le d(x,z)+d(y,z)$, $x,y,z\in \mathrm{\Omega }$.

For explicitly later use, we recall the following result by Diaz and Margolis [8].

Theorem 2.8 Let $(\mathrm{\Omega },d)$ be a complete generalized metric space and $J:\mathrm{\Omega }\to \mathrm{\Omega }$ be a strictly contractive mapping with Lipschitz constant $0<L<1$, that is

$$d(Jx,Jy)\le Ld(x,y),x,y\in \mathrm{\Omega }.$$

If there exists a nonnegative integer $m_0$ such that $d(J^{m_0}x,J^{m_0+1}x)<\mathrm{\infty }$ for an $x\in \mathrm{\Omega }$, then

• (1) the sequence ${\{J^{m}x\}}_{m\in \mathbb{N}}$ converges to a fixed point $x^{\ast }$ of J;

• (2) $x^{\ast }$ is the unique fixed point of J in the set ${\mathrm{\Omega }}^{\ast }$,

  $${\mathrm{\Omega }}^{\ast }:=\{y\in \mathrm{\Omega }|d(J^{m_0}x,y)<\mathrm{\infty }\};$$

• (3) if $y\in {\mathrm{\Omega }}^{\ast }$, then

  $$d(y,x^{\ast })\le \frac{1}{1-L}d(y,Jy).$$

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

Throughout this section, using a direct method, we prove the stability of Eq. (1.1) in complete non-Archimedean fuzzy normed spaces.

Theorem 3.1 Let $\mathbb{K}$ be a valued field, $\mathcal{X}$ be a vector space over $\mathbb{K}$ and $(\mathcal{Y},N,T)$ be a complete non-Archimedean fuzzy normed space over $\mathbb{K}$. Assume also that, for every $i\in \{1,2,\dots ,n\}$, ${\mathrm{\Psi }}_i:{\mathcal{X}}^{n+1}\times [0,\mathrm{\infty })\to [0,1]$ is a mapping such that

(3.1)

and

(3.2)

for all $x_1,\dots ,x_i,x_i^{\prime },x_{i+1},\dots ,x_n\in \mathcal{X}$ and $t>0$. If $f:{\mathcal{X}}^n\to \mathcal{Y}$ is a mapping satisfying

$$f(x_1,\dots ,x_{i-1},0,x_{i+1},\dots ,x_n)=0,$$

(3.3)

and

(3.4)

for all $x_1,\dots ,x_i,x_i^{\prime },x_{i+1},\dots ,x_n\in \mathcal{X}$, $i\in \{1,2,\dots ,n\}$ and $t\in [0,\mathrm{\infty })$, then for every $i\in \{1,2,\dots ,n\}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q_i:{\mathcal{X}}^n\to \mathcal{Y}$ such that

(3.5)

for all $x_1,\dots ,x_n\in \mathcal{X}$ and $t>0$.

Proof Fix $x_1,\dots ,x_n\in \mathcal{X}$, $j\in \mathbb{N}\cup \{0\}$, $i\in \{1,2,\dots ,n\}$ and $t>0$. Putting $x_i^{\prime }=0$ in (3.4), we get

(3.6)

Replacing $x_i$ by $a{x}_i$ and $x_i^{\prime }$ by $b{x}_i$ in (3.4), respectively, we have

(3.7)

From (3.6) and (3.7), one gets

(3.8)

Therefore one can get

and thus from (3.2) it follows that ${\{\frac{1}{{\lambda }^{2j}}f(x_1,\dots ,x_{i-1},{\lambda }^j{x}_i,x_{i+1},\dots ,x_n)\}}_{j\in \mathbb{N}}$ is a Cauchy sequence in a complete non-Archimedean fuzzy normed space. Hence, we can define a mapping $Q_i:{\mathcal{X}}^n\to \mathcal{Y}$ such that

$$\underset{j\to \mathrm{\infty }}{lim}N(\frac{1}{{\lambda }^{2j}}f(x_1,\dots ,x_{i-1},{\lambda }^j{x}_i,x_{i+1},\dots ,x_n)-Q_i(x_1,\dots ,x_n),t)=1.$$

Next, for each $k\in \mathbb{N}$ with $k\ge 1$, we have

Therefore,

Letting $k\to \mathrm{\infty }$ in this inequality, we obtain (3.5). Now, fix also $x_i^{\prime }\in \mathcal{X}$, from (3.1) and (3.4) it follows that

Next, fix $k\in \{1,\dots ,n\}\mathrm{\setminus }\{i\}$, $x_k^{\prime }\in \mathcal{X}$, and assume, without loss of generality, that $k<i$ (the same arguments apply to the case where $k>i$). From (3.1) and (3.4), it follows that

Hence the mapping $Q_i$ is a general multi-Euler-Lagrange quadratic mapping. Let us finally assume that $Q_i^{\prime }:{\mathcal{X}}^n\to \mathcal{Y}$ is another multi-Euler-Lagrange quadratic mapping satisfying (3.5). Then, by (1.4), (3.5) and (3.2), it follows that

and therefore $Q_i=Q_i^{\prime }$. □

For $a=b$, we get the following result.

Theorem 3.2 Let $\mathbb{K}$ be a valued field, $\mathcal{X}$ be a vector space over $\mathbb{K}$ and $(\mathcal{Y},N,T)$ be a complete non-Archimedean fuzzy normed space over $\mathbb{K}$. Assume also that, for every $i\in \{1,2,\dots ,n\}$, ${\mathrm{\Psi }}_i:{\mathcal{X}}^{n+1}\times [0,\mathrm{\infty })\to [0,1]$ is a mapping such that

(3.9)

and

(3.10)

for all $x_1,\dots ,x_i,x_i^{\prime },x_{i+1},\dots ,x_n\in \mathcal{X}$ and $t>0$. If $f:{\mathcal{X}}^n\to \mathcal{Y}$ is a mapping satisfying (3.3) and

(3.11)

for all $x_1,\dots ,x_i,x_i^{\prime },x_{i+1},\dots ,x_n\in \mathcal{X}$, $i\in \{1,2,\dots ,n\}$ and $t\in [0,\mathrm{\infty })$, then for every $i\in \{1,2,\dots ,n\}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q_i:{\mathcal{X}}^n\to \mathcal{Y}$ satisfying the functional equation (1.5) and such that

(3.12)

for all $x_1,\dots ,x_n\in \mathcal{X}$ and $t>0$.

Proof Fix $x_1,\dots ,x_n\in \mathcal{X}$, $j\in \mathbb{N}\cup \{0\}$, $i\in \{1,2,\dots ,n\}$ and $t>0$. Putting $x_i^{\prime }=x_i$ in (3.11), we get

(3.13)

Hence,

Therefore one can get

and thus by (3.10) it follows that ${\{\frac{1}{{\lambda }_1^{2j}}f(x_1,\dots ,x_{i-1},{\lambda }_1^j{x}_i,x_{i+1},\dots ,x_n)\}}_{j\in \mathbb{N}}$ is a Cauchy sequence in a complete non-Archimedean fuzzy normed space. Hence, we can define a mapping $Q_i:{\mathcal{X}}^n\to \mathcal{Y}$ such that

$$\underset{j\to \mathrm{\infty }}{lim}N(\frac{1}{{\lambda }_1^{2j}}f(x_1,\dots ,x_{i-1},{\lambda }_1^j{x}_i,x_{i+1},\dots ,x_n)-Q_i(x_1,\dots ,x_n),t)=1.$$

Using (3.13) and induction, one can show that for any $k\in \mathbb{N}$ we have

Therefore,

Letting $k\to \mathrm{\infty }$ in this inequality, we obtain (3.12). The rest of the proof of this theorem is omitted as being similar to the corresponding that of Theorem 3.1. □

Let $(\mathcal{Y},N,T)$ be a complete non-Archimedean fuzzy normed space over a non-Archimedean field $\mathbb{K}$. In any such space, a sequence ${\{x_k\}}_{k\in \mathbb{N}}$ is Cauchy if and only if ${\{x_{k+1}-x_k\}}_{k\in \mathbb{N}}$ converges to zero. Analysis similar to that in the proof of Theorem 3.2 gives the following.

Theorem 3.3 Let $\mathbb{K}$ be a non-Archimedean field, $\mathcal{X}$ be a vector space over $\mathbb{K}$ and $(\mathcal{Y},N,T)$ be a complete non-Archimedean fuzzy normed space over $\mathbb{K}$. Assume also that, for every $i\in \{1,2,\dots ,n\}$, ${\mathrm{\Psi }}_i:{\mathcal{X}}^{n+1}\times [0,\mathrm{\infty })\to [0,1]$ is a mapping such that (3.9) holds and

$$\underset{k\to \mathrm{\infty }}{lim}{T}_{j=1}^{\mathrm{\infty }}{\mathrm{\Psi }}_i(x_1,\dots ,x_{i-1},{\lambda }_1^{k+j-1}{x}_i,{\lambda }_1^{k+j-1}{x}_i,x_{i+1},\dots ,x_n,{|{\lambda }_1|}^{2k+2j}t)=1$$

for all $x_1,\dots ,x_i,x_i^{\prime },x_{i+1},\dots ,x_n\in \mathcal{X}$ and $t>0$. If $f:{\mathcal{X}}^n\to \mathcal{Y}$ is a mapping satisfying (3.3) and (3.11), then for every $i\in \{1,2,\dots ,n\}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q_i:{\mathcal{X}}^n\to \mathcal{Y}$ satisfying (1.5) and (3.12).

Remark 3.4 Let $a,b\in \mathbb{N}$ and $\mathcal{X}$ be a commutative group, Theorems 3.1-3.3 also hold. For $a=1$, consider the non-Archimedean fuzzy normed space $(\mathcal{Y},N_1,T_M)$ defined as in Example 2.6, Theorem 3.3 yields Theorem 2 in [7]. If $a=b=\pm \frac{1}{\sqrt{2}}\in \mathbb{K}$, then ${\lambda }_1=2a=\pm \sqrt{2}\ne 1$ in Theorems 3.2-3.3 and $\lambda =a^2+b^2=1$ which is a singular case $\lambda =1$ of Theorem 3.1.

Analysis similar to that in the proof of Theorem 3.1 gives the following.

Theorem 3.5 Let $\mathbb{K}$ be a valued field, $\mathcal{X}$ be a vector space over $\mathbb{K}$ and $(\mathcal{Y},N,T)$ be a complete non-Archimedean fuzzy normed space over $\mathbb{K}$. Assume also that, for every $i\in \{1,2,\dots ,n\}$, ${\mathrm{\Psi }}_i:{\mathcal{X}}^{n+1}\times [0,\mathrm{\infty })\to [0,1]$ is a mapping such that

(3.14)

and

(3.15)

for all $x_1,\dots ,x_i,x_i^{\prime },x_{i+1},\dots ,x_n\in \mathcal{X}$ and $t>0$. If $f:{\mathcal{X}}^n\to \mathcal{Y}$ is a mapping satisfying (3.3) and (3.4), then for every $i\in \{1,2,\dots ,n\}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q_i:{\mathcal{X}}^n\to \mathcal{Y}$ such that

for all $x_1,\dots ,x_n\in \mathcal{X}$ and $t>0$.

Corollary 3.6 Let $\mathbb{K}$ be a non-Archimedean field with $0<|\lambda |<1$, $\mathcal{X}$ be a normed space over $\mathbb{K}$ and let $(\mathcal{Y},N,T)$ be a complete non-Archimedean fuzzy normed space over $\mathbb{K}$ under a t-norm $T\in \mathcal{H}$. Assume also that $\delta >0$ and $\alpha :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is a function such that $\alpha ({|\lambda |}^{-1})<{|\lambda |}^{-1}$ and $\alpha ({|\lambda |}^{-1}t)\le \alpha ({|\lambda |}^{-1})\alpha (t)$ for all $t\in [0,\mathrm{\infty })$.

If $f:{\mathcal{X}}^n\to \mathcal{Y}$ is a mapping satisfying (3.3) and

(3.16)

for all $x_1,\dots ,x_i,x_i^{\prime },x_{i+1},\dots ,x_n\in \mathcal{X}$, $i\in \{1,2,\dots ,n\}$ and $t\in [0,\mathrm{\infty })$, then for every $i\in \{1,2,\dots ,n\}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q_i:{\mathcal{X}}^n\to \mathcal{Y}$ such that

for all $x_1,\dots ,x_n\in \mathcal{X}$ and $t>0$.

Proof Fix $i\in \{1,2,\dots ,n\}$, $x_1,\dots ,x_i,x_i^{\prime },x_{i+1},\dots ,x_n\in \mathcal{X}$ and $t\in [0,\mathrm{\infty })$. Let ${\mathrm{\Psi }}_i:{\mathcal{X}}^{n+1}\times [0,\mathrm{\infty })\to [0,1]$ be defined by ${\mathrm{\Psi }}_i(x_1,\dots ,x_i,x_i^{\prime },x_{i+1},\dots ,x_n,t):=\frac{t}{t+\delta \alpha (\parallel x_i\parallel )\alpha (\parallel x_i^{\prime }\parallel )}$. Then we can apply Theorem 3.5 to obtain the result. □

Remark 3.7 Let $0<|\lambda |<1$ and $p\in (0,1)$. Then the mapping $\alpha :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ given by $\alpha (t):=t^p$, $t\in [0,\mathrm{\infty })$ satisfies $\alpha ({|\lambda |}^{-1})<{|\lambda |}^{-1}$ and $\alpha ({|\lambda |}^{-1}t)\le \alpha ({|\lambda |}^{-1})\alpha (t)$ for all $t\in [0,\mathrm{\infty })$.

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

Throughout this section, we prove the stability of Eq. (1.1) in complete non-Archimedean fuzzy normed spaces using the fixed point method.

Theorem 4.1 Let $\mathbb{K}$ be a valued field, $\mathcal{X}$ be a vector space over $\mathbb{K}$ and $(\mathcal{Y},N,T)$ be a complete non-Archimedean fuzzy normed space over $\mathbb{K}$. Assume also that, for every $i\in \{1,2,\dots ,n\}$, ${\mathrm{\Psi }}_i:{\mathcal{X}}^{n+1}\times [0,\mathrm{\infty })\to [0,1]$ is a mapping such that (3.1) holds and

(4.1)

for an $L_i\in (0,1)$. If $f:{\mathcal{X}}^n\to \mathcal{Y}$ is a mapping satisfying (3.3) and (3.4), then for every $i\in \{1,2,\dots ,n\}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q_i:{\mathcal{X}}^n\to \mathcal{Y}$ such that

(4.2)

for all $x_1,\dots ,x_n\in \mathcal{X}$ and $t>0$.

Proof Fix an $i\in \{1,2,\dots ,n\}$. Consider the set $\mathrm{\Omega }:=\{g:{\mathcal{X}}^n\to \mathcal{Y}\}$ and introduce the generalized metric on Ω:

$$\begin{array}{rcl}{d}_i(g,h)& =& inf\{C\in [0,\mathrm{\infty }]:N(g(x_1,\dots ,x_n)-h(x_1,\dots ,x_n),Ct)\\ \ge T({\mathrm{\Psi }}_i(x_1,\dots ,x_i,0,x_{i+1},\dots ,x_n,|\lambda |t),\\ {\mathrm{\Psi }}_i(x_1,\dots ,x_{i-1},a{x}_i,b{x}_i,x_{i+1},\dots ,x_n,{|\lambda |}^{2}t)),\\ x_1,\dots ,x_n\in \mathcal{X},t>0\},g,h\in \mathrm{\Omega }.\end{array}$$

A standard verification (see for instance [19]) shows that $(\mathrm{\Omega },d_i)$ is a complete generalized metric space. We now define an operator $J_i:\mathrm{\Omega }\to \mathrm{\Omega }$ by

$$J_{i}g(x_1,\dots ,x_n)=\frac{1}{{\lambda }^2}g(x_1,\dots ,x_{i-1},\lambda x_i,x_{i+1},\dots ,x_n),g\in \mathrm{\Omega },x_1,\dots ,x_n\in \mathcal{X}.$$

Let $g,h\in \mathrm{\Omega }$ and $C_{g,h}\in [0,\mathrm{\infty }]$ with $d_i(g,h)\le C_{g,h}$. Then

$$\begin{array}{rcl}N(g(x_1,\dots ,x_n)-h(x_1,\dots ,x_n),C_{g,h}t)& \ge & T({\mathrm{\Psi }}_i(x_1,\dots ,x_i,0,x_{i+1},\dots ,x_n,|\lambda |t),\\ {\mathrm{\Psi }}_i(x_1,\dots ,x_{i-1},a{x}_i,b{x}_i,x_{i+1},\dots ,x_n,{|\lambda |}^{2}t)),\end{array}$$

which together with (4.1) gives

$$\begin{array}{rcl}N(J_{i}g(x_1,\dots ,x_n)-J_{i}h(x_1,\dots ,x_n),t)& \ge & T({\mathrm{\Psi }}_i(x_1,\dots ,x_i,0,x_{i+1},\dots ,x_n,\frac{|\lambda |t}{L_i{C}_{g,h}}),\\ {\mathrm{\Psi }}_i(x_1,\dots ,x_{i-1},a{x}_i,b{x}_i,x_{i+1},\dots ,x_n,\frac{{|\lambda |}^{2}t}{L_i{C}_{g,h}})),\end{array}$$

and consequently, $d_i(J_{i}g,J_{i}h)\le L_i{C}_{g,h}$, which means that the operator $J_i$ is strictly contractive. Moreover, from (3.8) it follows that

$$\begin{array}{rcl}N(J_{i}f(x_1,\dots ,x_n)-f(x_1,\dots ,x_n),t)& \ge & T({\mathrm{\Psi }}_i(x_1,\dots ,x_i,0,x_{i+1},\dots ,x_n,|\lambda |t),\\ {\mathrm{\Psi }}_i(x_1,\dots ,x_{i-1},a{x}_i,b{x}_i,x_{i+1},\dots ,x_n,{|\lambda |}^{2}t))\end{array}$$

and thus $d_i(J_{i}f,f)\le 1<\mathrm{\infty }$. Therefore, by Theorem 2.8, $J_i$ has a unique fixed point $Q_i:{\mathcal{X}}^n\to \mathcal{Y}$ in the set ${\mathrm{\Omega }}^{\ast }=\{g\in \mathrm{\Omega }:d(f,g)<\mathrm{\infty }\}$ such that

$$\frac{1}{{\lambda }^2}{Q}_i(x_1,\dots ,x_{i-1},\lambda x_i,x_{i+1},\dots ,x_n)=Q_i(x_1,\dots ,x_n)$$

(4.3)

and

$$Q_i(x_1,\dots ,x_n)=\underset{j\to \mathrm{\infty }}{lim}\frac{1}{{\lambda }^{2j}}f(x_1,\dots ,x_{i-1},{\lambda }^j{x}_i,x_{i+1},\dots ,x_n).$$

Furthermore, from the fact that $f\in {\mathrm{\Omega }}^{\ast }$, Theorem 2.8, and $d_i(J_{i}f,f)\le 1$, we get

$$d_i(f,Q_i)\le \frac{1}{1-L_i}{d}_i(J_{i}f,f)\le \frac{1}{1-L_i}$$

and (4.2) follows. Similar to the proof of Theorem 3.1, one can prove that the mapping $Q_i$ is also general multi-Euler-Lagrange quadratic.

Let us finally assume that $Q_i^{\prime }:{\mathcal{X}}^n\to \mathcal{Y}$ is a general multi-Euler-Lagrange quadratic mapping satisfying condition (4.2). Then $Q_i^{\prime }$ fulfills (4.3), and therefore, it is a fixed point of the operator $J_i$. Moreover, by (4.2), we have $d_i(f,Q_i^{\prime })\le \frac{1}{1-L_i}<\mathrm{\infty }$, and consequently $Q_i^{\prime }\in {\mathrm{\Omega }}^{\ast }$. Theorem 2.8 shows that $Q_i^{\prime }=Q_i$. □

Similar to Theorem 4.1, one can prove the following result.

Theorem 4.2 Let $\mathbb{K}$ be a valued field, $\mathcal{X}$ be a vector space over $\mathbb{K}$ and $(\mathcal{Y},N,T)$ be a complete non-Archimedean fuzzy normed space over $\mathbb{K}$. Assume also that, for every $i\in \{1,2,\dots ,n\}$, ${\mathrm{\Psi }}_i:{\mathcal{X}}^{n+1}\times [0,\mathrm{\infty })\to [0,1]$ is a mapping such that (3.14) holds and

for an $L_i\in (0,1)$. If $f:{\mathcal{X}}^n\to \mathcal{Y}$ is a mapping satisfying (3.3) and (3.4), then for every $i\in \{1,2,\dots ,n\}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q_i:{\mathcal{X}}^n\to \mathcal{Y}$ such that

$$\begin{array}{rcl}N(f(x_1,\dots ,x_n)-Q_i(x_1,\dots ,x_n),t)& \ge & T({\mathrm{\Psi }}_i(x_1,\dots ,x_i,0,x_{i+1},\dots ,x_n,|\lambda |(L_i^{-1}-1)t),\\ {\mathrm{\Psi }}_i(x_1,\dots ,x_{i-1},a{x}_i,b{x}_i,x_{i+1},\dots ,x_n,{|\lambda |}^2(L_i^{-1}-1)t))\end{array}$$

for all $x_1,\dots ,x_n\in \mathcal{X}$ and $t>0$.

Remark 4.3 Similar to the proof of Corollary 3.6, one can deduce from Theorem 4.2 an analog of Corollary 3.6.

As applications of Theorems 4.1 and 4.2 , we get the following corollaries.

Corollary 4.4 Let $\mathcal{X}$ be a real normed space, $\mathcal{Y}$ be a real Banach space and $(\mathcal{Y},N,T_P)$ be the complete non-Archimedean fuzzy normed space defined as in the second example in the preliminaries. Let $\delta ,r,p,q\in (0,\mathrm{\infty })$ such that $r,s:=p+q\in (2,\mathrm{\infty })$, or $r,s\in (0,2)$. If $f:{\mathcal{X}}^n\to \mathcal{Y}$ is a mapping satisfying (3.3) and (3.4), where

$${\mathrm{\Psi }}_i(x_1,\dots ,x_i,x_i^{\prime },x_{i+1},\dots ,x_n,t):=\frac{t}{t+\delta [{\parallel x_1\parallel }^r\cdots {\parallel x_{i-1}\parallel }^r({\parallel x_i\parallel }^p{\parallel x_i^{\prime }\parallel }^q){\parallel x_{i+1}\parallel }^r\cdots {\parallel x_n\parallel }^r]},$$

then for every $i\in \{1,2,\dots ,n\}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q_i:{\mathcal{X}}^n\to \mathcal{Y}$ such that

(4.4)

for all $x_1,\dots ,x_n\in \mathcal{X}$ and $t>0$.

Proof Fix $i\in \{1,2,\dots ,n\}$, $x_1,\dots ,x_i,x_i^{\prime },x_{i+1},\dots ,x_n\in \mathcal{X}$, $t\in [0,\mathrm{\infty })$ and assume that $\lambda >1$, $r,s\in (2,\mathrm{\infty })$ (the same arguments apply to the case where $\lambda <1$, $r,s\in (0,2)$). Then we can choose $L_i={\lambda }^{2-s}<1$ and apply Theorem 4.2 to obtain the result. For $\lambda >1$, $r,s\in (0,2)$, or $\lambda <1$, $r,s\in (2,\mathrm{\infty })$, the corollary follows from Theorem 4.1. □

Corollary 4.5 Let $\mathcal{X}$ be a real normed space and $\mathcal{Y}$ be a real Banach space (or $\mathcal{X}$ be a non-Archimedean normed space and $\mathcal{Y}$ be a complete non-Archimedean normed space over a non-Archimedean field $\mathbb{K}$, respectively). Let $\delta >0$ and $r\in (0,2)\cup (2,\mathrm{\infty })$. If $f:{\mathcal{X}}^n\to \mathcal{Y}$ is a mapping satisfying (3.3) and

then for every $i\in \{1,2,\dots ,n\}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q_i:{\mathcal{X}}^n\to \mathcal{Y}$ such that

$$\parallel f(x_1,\dots ,x_n)-Q_i(x_1,\dots ,x_n)\parallel \le \frac{max\{|\lambda |,{|a|}^r+{|b|}^r\}\delta ({\parallel x_1\parallel }^r\cdots {\parallel x_n\parallel }^r)}{|{|\lambda |}^r-{|\lambda |}^2|}$$

for all $x_1,\dots ,x_n\in \mathcal{X}$.

Proof Consider the non-Archimedean fuzzy normed space $(\mathcal{Y},N_1,T_M)$ defined as in the first example in the preliminaries, ${\mathrm{\Psi }}_i$ be defined by

$${\mathrm{\Psi }}_i(x_1,\dots ,x_i,x_i^{\prime },x_{i+1},\dots ,x_n,t):=\frac{t}{t+\delta [{\parallel x_1\parallel }^r\cdots {\parallel x_{i-1}\parallel }^r({\parallel x_i\parallel }^r+{\parallel x_i^{\prime }\parallel }^r){\parallel x_{i+1}\parallel }^r\cdots {\parallel x_n\parallel }^r]},$$

and apply Theorems 4.1 and 4.2.

The following example shows that the Hyers-Ulam stability problem for the case of $r=2$ was excluded in Corollary 4.5. □

Example 4.6 Let $\varphi :\mathbb{C}\to \mathbb{C}$ be defined by

$$\varphi (x)=\{\begin{array}{cc}{x}^2,\hfill & \text{for }|x|<1,\hfill \\ 1,\hfill & \text{for }|x|\ge 1.\hfill \end{array}$$

Consider the function $f:\mathbb{C}\to \mathbb{C}$ be defined by

$$f(x)=\sum _{j=0}^{\mathrm{\infty }}\frac{\varphi ({\alpha }^{j}x)}{{\alpha }^{2j}}$$

for all $x\in \mathbb{C}$, where $\alpha >max\{|a|,|b|,1\}$. Then f satisfies the functional inequality

(4.5)

for all $x,y\in \mathbb{C}$, but there do not exist a general multi-Euler-Lagrange quadratic function $Q:\mathbb{C}\to \mathbb{C}$ and a constant $d>0$ such that $|f(x)-Q(x)|\le d{|x|}^2$ for all $x\in \mathbb{C}$.

It is clear that f is bounded by $\frac{{\alpha }^2}{{\alpha }^2-1}$ on $\mathbb{C}$. If ${|x|}^2+{|y|}^2=0$ or ${|x|}^2+{|y|}^2\ge \frac{1}{{\alpha }^2}$, then

$$|f(ax+by)+f(bx-ay)-(a^2+b^2)[f(x)+f(y)]|\le \frac{2{\alpha }^4({|a|}^2+{|b|}^2+1)}{{\alpha }^2-1}({|x|}^2+{|y|}^2).$$

Now suppose that $0<{|x|}^2+{|y|}^2<\frac{1}{{\alpha }^2}$. Then there exists an integer $k\ge 1$ such that

$$\frac{1}{{\alpha }^{2(k+2)}}\le {|x|}^2+{|y|}^2<\frac{1}{{\alpha }^{2(k+1)}}.$$

(4.6)

Hence

$${\alpha }^l|ax+by|<1,{\alpha }^l|bx-ay|<1,{\alpha }^l|x|<1,{\alpha }^l|y|<1$$

for all $l=0,1,\dots ,k-1$. From the definition of f and the inequality (4.6), we obtain that f satisfies (4.5). Now, we claim that the functional equation (1.1) is not stable for $r=2$ in Corollary 4.5. Suppose, on the contrary, that there exist a general multi-Euler-Lagrange quadratic function $Q:\mathbb{C}\to \mathbb{C}$ and a constant $d>0$ such that $|f(x)-Q(x)|\le d{|x|}^2$ for all $x\in \mathbb{C}$. Then there exists a constant $c\in \mathbb{C}$ such that $Q(x)=c{x}^2$ for all rational numbers x. So, we obtain that

$$|f(x)|\le (d+|c|){|x|}^2$$

(4.7)

for all rational numbers x. Let $s\in \mathbb{N}$ with $s+1>d+|c|$. If x is a rational number in $(0,{\alpha }^{-s})$, then ${\alpha }^{j}x\in (0,1)$ for all $j=0,1,\dots ,s$, and for this x we get

$$f(x)=\sum _{j=0}^{\mathrm{\infty }}\frac{\varphi ({\alpha }^{j}x)}{{\alpha }^{2j}}\ge \sum _{j=0}^s\frac{\varphi ({\alpha }^{j}x)}{{\alpha }^{2j}}=(s+1)x^2>(d+|c|)x^2,$$

which contradicts (4.7).

Corollary 4.7 Let $\mathbb{K}$ be a non-Archimedean field with $0<|\lambda |<1$, $\mathcal{X}$ be a normed space over $\mathbb{K}$ and $\mathcal{Y}$ be a complete non-Archimedean normed space over $\mathbb{K}$. Let $\delta ,p,q\in (0,\mathrm{\infty })$ such that $p+q\in (0,2)$. If $f:{\mathcal{X}}^n\to \mathcal{Y}$ is a mapping satisfying (3.3) and

then for every $i\in \{1,2,\dots ,n\}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q_i:{\mathcal{X}}^n\to \mathcal{Y}$ such that

$$\parallel f(x_1,\dots ,x_n)-Q_i(x_1,\dots ,x_n)\parallel \le \frac{\delta {|a|}^p{|b|}^q{\parallel x_i\parallel }^{p+q}}{{|\lambda |}^{p+q}-{|\lambda |}^2}$$

(4.8)

for all $x_1,\dots ,x_n\in \mathcal{X}$.

Proof Fix $i\in \{1,2,\dots ,n\}$, $x_1,\dots ,x_i,x_i^{\prime },x_{i+1},\dots ,x_n\in \mathcal{X}$ and $t\in [0,\mathrm{\infty })$. Let ${\mathrm{\Psi }}_i:{\mathcal{X}}^{n+1}\times [0,\mathrm{\infty })\to [0,1]$ be defined by ${\mathrm{\Psi }}_i(x_1,\dots ,x_i,x_i^{\prime },x_{i+1},\dots ,x_n,t):=\frac{t}{t+\delta ({\parallel x_i\parallel }^p{\parallel x_i^{\prime }\parallel }^q)}$. Consider the non-Archimedean fuzzy normed space $(\mathcal{Y},N_1,T_M)$ defined as in Example 2.6, and apply Theorem 4.2. □

Corollary 4.8 Let $\mathcal{X}$ be a real normed space and $\mathcal{Y}$ be a real Banach space. Let $\delta ,r,p,q\in (0,\mathrm{\infty })$ such that $r,p+q\in (0,2)$, or $r,p+q\in (2,\mathrm{\infty })$. If $f:{\mathcal{X}}^n\to \mathcal{Y}$ is a mapping satisfying (3.3) and

then, for every $i\in \{1,2,\dots ,n\}$, there exists a unique general multi-Euler-Lagrange quadratic mapping $Q_i:{\mathcal{X}}^n\to \mathcal{Y}$ such that

(4.9)

for all $x_1,\dots ,x_n\in \mathcal{X}$.

Proof Fix $i\in \{1,2,\dots ,n\}$, $x_1,\dots ,x_i,x_i^{\prime },x_{i+1},\dots ,x_n\in \mathcal{X}$ and $t\in [0,\mathrm{\infty })$. Let ${\mathrm{\Psi }}_i:{\mathcal{X}}^{n+1}\times [0,\mathrm{\infty })\to [0,1]$ be defined by

$${\mathrm{\Psi }}_i(x_1,\dots ,x_i,x_i^{\prime },x_{i+1},\dots ,x_n,t):=\frac{t}{t+\delta [{\parallel x_1\parallel }^r\cdots {\parallel x_{i-1}\parallel }^r({\parallel x_i\parallel }^p{\parallel x_i^{\prime }\parallel }^q){\parallel x_{i+1}\parallel }^r\cdots {\parallel x_n\parallel }^r]}.$$

Consider the non-Archimedean fuzzy normed space $(\mathcal{Y},N_1,T_M)$ defined as in Example 2.6, and apply Theorems 4.1 and 4.2. □

Remark 4.9 Theorems 4.1 and 4.2 can be regarded as a generalization of the classical stability result in the framework of normed spaces (see [14]). For $a=b=1$ and $n=1$, Corollary 4.8 yields the main theorem in [17]. The generalized Hyers-Ulam stability problem for the case of $r=p+q=2$ was excluded in Corollary 4.8 (see[10]).

Note that by (4.4) one can get

Letting $t\to \mathrm{\infty }$ in this inequality, we obtain (4.9). Thus Corollary 4.8 is a singular case of Corollary 4.4. This study indeed presents a relationship between three various disciplines: the theory of non-Archimedean fuzzy normed spaces, the theory of stability of functional equations and the fixed point theory.