# 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 $K$ be a field. A valuation mapping on $K$ is a function $|⋅|:K→R$ such that for any $r,s∈K$ the following conditions are satisfied: (i) $|r|≥0$ and equality holds if and only if $r=0$; (ii) $|rs|=|r|⋅|s|$; (iii) $|r+s|≤|r|+|s|$.

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

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

Throughout this paper, we assume that $K$ is a valued field, $X$ and $Y$ are vector spaces over $K$, $a,b∈K$ are fixed with $λ:= a 2 + b 2 ≠0,1$ ($λ 1 :=2a≠0,1$ if $a=b$) and n is a positive integer. Moreover, $N$ stands for the set of all positive integers and $R$ (respectively, $Q$) denotes the set of all reals (respectively, rationals).

A mapping $f: X n →Y$ 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 $i=1,…,n$ and all $x 1 ,…, x i − 1 , x i , x i ′ , x i + 1 ,…, x n ∈X$. Letting $x i = x i ′ =0$ in (1.1), we get $f( x 1 ,…, x i − 1 ,0, x i + 1 ,…, x n )=0$. Putting $x i ′ =0$ in (1.1), we have

$f( x 1 ,…, x i − 1 ,a x i , x i + 1 ,…, x n )+f( x 1 ,…, x i − 1 ,b x i , x i + 1 ,…, x n )=λf( x 1 ,…, x n ).$
(1.2)

Replacing $x i$ by $a x i$ and $x i ′$ by $b x i$ in (1.1), respectively, we obtain (1.3)

From (1.2) and (1.3), one gets

$f( x 1 ,…, x i − 1 ,λ x i , x i + 1 ,…, x n )= λ 2 f( x 1 ,…, x n )$
(1.4)

for all $i=1,…,n$ and all $x 1 ,…, x n ∈X$. If $a=b$ in (1.1), then we have (1.5)

Letting $x i ′ = x i$ in (1.5), we obtain

$f( x 1 ,…, x i − 1 , λ 1 x i , x i + 1 ,…, x n )= λ 1 2 f( x 1 ,…, x n )$
(1.6)

for all $i=1,…,n$ and all $x 1 ,…, x n ∈X$.

The study of stability problems for functional equations is related to a question of Ulam  concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers . The result of Hyers was generalized by Aoki  for approximate additive mappings and by Rassias  for approximate linear mappings by allowing the Cauchy difference operator $CDf(x,y)=f(x+y)−[f(x)+f(y)]$ to be controlled by $ϵ( ∥ x ∥ p + ∥ y ∥ p )$. In 1994, a further generalization was obtained by Găvruţa , who replaced $ϵ( ∥ x ∥ p + ∥ y ∥ p )$ by a general control function $φ(x,y)$. We refer the reader to see, for instance, [1, 47, 1416, 18, 20, 22, 23, 25, 26, 28, 3137] 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 $X$ be a linear space over a field $K$ with a non-Archimedean valuation $|⋅|$. A function $∥⋅∥:X→[0,∞)$ is said to be a non-Archimedean norm if it satisfies the following conditions:

1. (i)

$∥x∥=0$ if and only if $x=0$;

2. (ii)

$∥rx∥=|r|∥x∥$, $r∈K$, $x∈X$;

3. (iii)

the strong triangle inequality

$∥x+y∥≤max{∥x∥,∥y∥},x,y∈X.$

Then $(X,∥⋅∥)$ 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 $Q$. The completion of $Q$ with respect to the metric $d(a,b)= | a − b | p$ is denoted by $Q p$ 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 ).

A triangular norm (shorter t-norm, ) is a binary operation $T:[0,1]×[0,1]→[0,1]$ which satisfies the following conditions: (a) T is commutative and associative; (b) $T(a,1)=a$ for all $a∈[0,1]$; (c) $T(a,b)≤T(c,d)$ whenever $a≤c$ and $b≤d$ for all $a,b,c,d∈[0,1]$. 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 $[0,1]×[0,1]$.

A t-norm T can be extended (by associativity) in a unique way to an m-array operation taking for $( x 1 ,…, x m )∈ [ 0 , 1 ] m$, the value $T( x 1 ,…, x m )$ 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 $m∈N$. T can also be extended to a countable operation, taking for any sequence ${ x i } i ∈ N$ in $[0,1]$, the value $T i = 1 ∞ x i$ is defined as $lim m → ∞ T i = 1 m x i$. The limit exists since the sequence ${ T i = 1 m x i } m ∈ N$ is non-increasing and bounded from below. $T i = m ∞ x i$ is defined as $T i = 1 ∞ x m + i$.

Definition 2.2 A t-norm T is said to be of Hadžić-type (H-type, we denote by $T∈H$) if a family of functions ${ T i = 1 m (t)}$ for all $m∈N$ is equicontinuous at $t=1$, that is, for all $ε∈(0,1)$ there exists $δ∈(0,1)$ such that

The t-norm $T M$ is a t-norm of Hadžić-type. Other important triangular norms we refer the reader to .

Proposition 2.3 (see )

1. (1)

If $T= T P$ or $T= T L$, then

$lim m → ∞ T i = 1 ∞ x m + i =1⟺ ∑ i = 1 ∞ (1− x i )<∞.$
2. (2)

If T is of Hadžić-type, then

$lim m → ∞ T i = m ∞ x i = lim m → ∞ T i = 1 ∞ x m + i =1$

for every sequence ${ x i } i ∈ N$ in $[0,1]$ such that $lim i → ∞ x i =1$.

Definition 2.4 (see )

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

(N1) $N(x,t)=0$ for all $t≤0$;

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

(N3) $N(cx,t)=N(x,t/|c|)$ if $c≠0$;

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

(N5) $lim t → ∞ N(x,t)=1$.

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

$N(x,t)≥T ( N ( 0 , t ) , N ( x , s ) ) =N(x,s)$

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

(N6) $N(x+y,s+t)≥T(N(x,s),N(y,t))$.

We repeatedly use the fact $N(−x,t)=N(x,t)$, $x∈X$, $t>0$, 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 $(X,N,T)$ be a non-Archimedean fuzzy normed space. Let ${ x m } m ∈ N$ be a sequence in $X$. Then ${ x m } m ∈ N$ is said to be convergent if there exists $x∈X$ such that $lim m → ∞ N( x m −x,t)=1$ for all $t>0$. In that case, x is called the limit of the sequence ${ x m } m ∈ N$ and we denote it by $lim m → ∞ x m =x$. The sequence ${ x m } m ∈ N$ in $X$ is said to be a Cauchy sequence if $lim m → ∞ N( x m + p − x m ,t)=1$ for all $t>0$ and $p=1,2,…$ . If every Cauchy sequence in $X$ is convergent, then the space is called a complete non-Archimedean fuzzy normed space.

Example 2.6 Let $(X,∥⋅∥)$ be a real (or non-Archimedean) normed space. For each $k>0$, consider

$N k (x,t)={ t t + k ∥ x ∥ , t > 0 , 0 , t ≤ 0 .$

Then $(X, N k , T M )$ is a non-Archimedean fuzzy normed space.

Example 2.7 (see )

Let $(X,∥⋅∥)$ be a real normed space. Then the triple $(X,N, T P )$, where

$N(x,t)={ e − ∥ x ∥ / t , t > 0 , 0 , t ≤ 0$

is a non-Archimedean fuzzy normed space. Moreover, if $(X,∥⋅∥)$ is complete, then $(X,N, T P )$ 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 $d:Ω×Ω→[0,∞]$ is called a generalized metric on Ω if d satisfies

1. (1)

$d(x,y)=0$ if and only if $x=y$; (2) $d(x,y)=d(y,x)$, $x,y∈Ω$; (3) $d(x,y)≤d(x,z)+d(y,z)$, $x,y,z∈Ω$.

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

Theorem 2.8 Let $(Ω,d)$ be a complete generalized metric space and $J:Ω→Ω$ be a strictly contractive mapping with Lipschitz constant $0, that is

$d(Jx,Jy)≤Ld(x,y),x,y∈Ω.$

If there exists a nonnegative integer $m 0$ such that $d( J m 0 x, J m 0 + 1 x)<∞$ for an $x∈Ω$, then

1. (1)

the sequence ${ J m x } m ∈ N$ converges to a fixed point $x ∗$ of J;

2. (2)

$x ∗$ is the unique fixed point of J in the set $Ω ∗$,

$Ω ∗ := { y ∈ Ω | d ( J m 0 x , y ) < ∞ } ;$
3. (3)

if $y∈ Ω ∗$, then

$d ( y , x ∗ ) ≤ 1 1 − L d(y,Jy).$

## 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 $K$ be a valued field, $X$ be a vector space over $K$ and $(Y,N,T)$ be a complete non-Archimedean fuzzy normed space over $K$. Assume also that, for every $i∈{1,2,…,n}$, $Ψ i : X n + 1 ×[0,∞)→[0,1]$ is a mapping such that (3.1)

and (3.2)

for all $x 1 ,…, x i , x i ′ , x i + 1 ,…, x n ∈X$ and $t>0$. If $f: X n →Y$ is a mapping satisfying

$f( x 1 ,…, x i − 1 ,0, x i + 1 ,…, x n )=0,$
(3.3)

and (3.4)

for all $x 1 ,…, x i , x i ′ , x i + 1 ,…, x n ∈X$, $i∈{1,2,…,n}$ and $t∈[0,∞)$, then for every $i∈{1,2,…,n}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q i : X n →Y$ such that (3.5)

for all $x 1 ,…, x n ∈X$ and $t>0$.

Proof Fix $x 1 ,…, x n ∈X$, $j∈N∪{0}$, $i∈{1,2,…,n}$ and $t>0$. Putting $x i ′ =0$ in (3.4), we get (3.6)

Replacing $x i$ by $a x i$ and $x i ′$ 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 ${ 1 λ 2 j f ( x 1 , … , x i − 1 , λ j x i , x i + 1 , … , x n ) } j ∈ N$ is a Cauchy sequence in a complete non-Archimedean fuzzy normed space. Hence, we can define a mapping $Q i : X n →Y$ such that

$lim j → ∞ N ( 1 λ 2 j f ( x 1 , … , x i − 1 , λ j x i , x i + 1 , … , x n ) − Q i ( x 1 , … , x n ) , t ) =1.$

Next, for each $k∈N$ with $k≥1$, we have Therefore, Letting $k→∞$ in this inequality, we obtain (3.5). Now, fix also $x i ′ ∈X$, from (3.1) and (3.4) it follows that Next, fix $k∈{1,…,n}∖{i}$, $x k ′ ∈X$, and assume, without loss of generality, that $k (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 $Q i ′ : X n →Y$ is another multi-Euler-Lagrange quadratic mapping satisfying (3.5). Then, by (1.4), (3.5) and (3.2), it follows that and therefore $Q i = Q i ′$. □

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

Theorem 3.2 Let $K$ be a valued field, $X$ be a vector space over $K$ and $(Y,N,T)$ be a complete non-Archimedean fuzzy normed space over $K$. Assume also that, for every $i∈{1,2,…,n}$, $Ψ i : X n + 1 ×[0,∞)→[0,1]$ is a mapping such that (3.9)

and (3.10)

for all $x 1 ,…, x i , x i ′ , x i + 1 ,…, x n ∈X$ and $t>0$. If $f: X n →Y$ is a mapping satisfying (3.3) and (3.11)

for all $x 1 ,…, x i , x i ′ , x i + 1 ,…, x n ∈X$, $i∈{1,2,…,n}$ and $t∈[0,∞)$, then for every $i∈{1,2,…,n}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q i : X n →Y$ satisfying the functional equation (1.5) and such that (3.12)

for all $x 1 ,…, x n ∈X$ and $t>0$.

Proof Fix $x 1 ,…, x n ∈X$, $j∈N∪{0}$, $i∈{1,2,…,n}$ and $t>0$. Putting $x i ′ = x i$ in (3.11), we get (3.13)

Hence, Therefore one can get and thus by (3.10) it follows that ${ 1 λ 1 2 j f ( x 1 , … , x i − 1 , λ 1 j x i , x i + 1 , … , x n ) } j ∈ N$ is a Cauchy sequence in a complete non-Archimedean fuzzy normed space. Hence, we can define a mapping $Q i : X n →Y$ such that

$lim j → ∞ N ( 1 λ 1 2 j f ( x 1 , … , x i − 1 , λ 1 j x i , x i + 1 , … , x n ) − Q i ( x 1 , … , x n ) , t ) =1.$

Using (3.13) and induction, one can show that for any $k∈N$ we have Therefore, Letting $k→∞$ 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 $(Y,N,T)$ be a complete non-Archimedean fuzzy normed space over a non-Archimedean field $K$. In any such space, a sequence ${ x k } k ∈ N$ is Cauchy if and only if ${ x k + 1 − x k } k ∈ N$ converges to zero. Analysis similar to that in the proof of Theorem 3.2 gives the following.

Theorem 3.3 Let $K$ be a non-Archimedean field, $X$ be a vector space over $K$ and $(Y,N,T)$ be a complete non-Archimedean fuzzy normed space over $K$. Assume also that, for every $i∈{1,2,…,n}$, $Ψ i : X n + 1 ×[0,∞)→[0,1]$ is a mapping such that (3.9) holds and

$lim k → ∞ T j = 1 ∞ Ψ i ( x 1 , … , x i − 1 , λ 1 k + j − 1 x i , λ 1 k + j − 1 x i , x i + 1 , … , x n , | λ 1 | 2 k + 2 j t ) =1$

for all $x 1 ,…, x i , x i ′ , x i + 1 ,…, x n ∈X$ and $t>0$. If $f: X n →Y$ is a mapping satisfying (3.3) and (3.11), then for every $i∈{1,2,…,n}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q i : X n →Y$ satisfying (1.5) and (3.12).

Remark 3.4 Let $a,b∈N$ and $X$ be a commutative group, Theorems 3.1-3.3 also hold. For $a=1$, consider the non-Archimedean fuzzy normed space $(Y, N 1 , T M )$ defined as in Example 2.6, Theorem 3.3 yields Theorem 2 in . If $a=b=± 1 2 ∈K$, then $λ 1 =2a=± 2 ≠1$ in Theorems 3.2-3.3 and $λ= a 2 + b 2 =1$ which is a singular case $λ=1$ of Theorem 3.1.

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

Theorem 3.5 Let $K$ be a valued field, $X$ be a vector space over $K$ and $(Y,N,T)$ be a complete non-Archimedean fuzzy normed space over $K$. Assume also that, for every $i∈{1,2,…,n}$, $Ψ i : X n + 1 ×[0,∞)→[0,1]$ is a mapping such that (3.14)

and (3.15)

for all $x 1 ,…, x i , x i ′ , x i + 1 ,…, x n ∈X$ and $t>0$. If $f: X n →Y$ is a mapping satisfying (3.3) and (3.4), then for every $i∈{1,2,…,n}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q i : X n →Y$ such that for all $x 1 ,…, x n ∈X$ and $t>0$.

Corollary 3.6 Let $K$ be a non-Archimedean field with $0<|λ|<1$, $X$ be a normed space over $K$ and let $(Y,N,T)$ be a complete non-Archimedean fuzzy normed space over $K$ under a t-norm $T∈H$. Assume also that $δ>0$ and $α:[0,∞)→[0,∞)$ is a function such that $α( | λ | − 1 )< | λ | − 1$ and $α( | λ | − 1 t)≤α( | λ | − 1 )α(t)$ for all $t∈[0,∞)$.

If $f: X n →Y$ is a mapping satisfying (3.3) and (3.16)

for all $x 1 ,…, x i , x i ′ , x i + 1 ,…, x n ∈X$, $i∈{1,2,…,n}$ and $t∈[0,∞)$, then for every $i∈{1,2,…,n}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q i : X n →Y$ such that for all $x 1 ,…, x n ∈X$ and $t>0$.

Proof Fix $i∈{1,2,…,n}$, $x 1 ,…, x i , x i ′ , x i + 1 ,…, x n ∈X$ and $t∈[0,∞)$. Let $Ψ i : X n + 1 ×[0,∞)→[0,1]$ be defined by $Ψ i ( x 1 ,…, x i , x i ′ , x i + 1 ,…, x n ,t):= t t + δ α ( ∥ x i ∥ ) α ( ∥ x i ′ ∥ )$. Then we can apply Theorem 3.5 to obtain the result. □

Remark 3.7 Let $0<|λ|<1$ and $p∈(0,1)$. Then the mapping $α:[0,∞)→[0,∞)$ given by $α(t):= t p$, $t∈[0,∞)$ satisfies $α( | λ | − 1 )< | λ | − 1$ and $α( | λ | − 1 t)≤α( | λ | − 1 )α(t)$ for all $t∈[0,∞)$.

## 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 $K$ be a valued field, $X$ be a vector space over $K$ and $(Y,N,T)$ be a complete non-Archimedean fuzzy normed space over $K$. Assume also that, for every $i∈{1,2,…,n}$, $Ψ i : X n + 1 ×[0,∞)→[0,1]$ is a mapping such that (3.1) holds and (4.1)

for an $L i ∈(0,1)$. If $f: X n →Y$ is a mapping satisfying (3.3) and (3.4), then for every $i∈{1,2,…,n}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q i : X n →Y$ such that (4.2)

for all $x 1 ,…, x n ∈X$ and $t>0$.

Proof Fix an $i∈{1,2,…,n}$. Consider the set $Ω:={g: X n →Y}$ and introduce the generalized metric on Ω:

$d i ( g , h ) = inf { C ∈ [ 0 , ∞ ] : N ( g ( x 1 , … , x n ) − h ( x 1 , … , x n ) , C t ) ≥ T ( Ψ i ( x 1 , … , x i , 0 , x i + 1 , … , x n , | λ | t ) , Ψ i ( x 1 , … , x i − 1 , a x i , b x i , x i + 1 , … , x n , | λ | 2 t ) ) , x 1 , … , x n ∈ X , t > 0 } , g , h ∈ Ω .$

A standard verification (see for instance ) shows that $(Ω, d i )$ is a complete generalized metric space. We now define an operator $J i :Ω→Ω$ by

$J i g( x 1 ,…, x n )= 1 λ 2 g( x 1 ,…, x i − 1 ,λ x i , x i + 1 ,…, x n ),g∈Ω, x 1 ,…, x n ∈X.$

Let $g,h∈Ω$ and $C g , h ∈[0,∞]$ with $d i (g,h)≤ C g , h$. Then

$N ( g ( x 1 , … , x n ) − h ( x 1 , … , x n ) , C g , h t ) ≥ T ( Ψ i ( x 1 , … , x i , 0 , x i + 1 , … , x n , | λ | t ) , Ψ i ( x 1 , … , x i − 1 , a x i , b x i , x i + 1 , … , x n , | λ | 2 t ) ) ,$

which together with (4.1) gives

$N ( J i g ( x 1 , … , x n ) − J i h ( x 1 , … , x n ) , t ) ≥ T ( Ψ i ( x 1 , … , x i , 0 , x i + 1 , … , x n , | λ | t L i C g , h ) , Ψ i ( x 1 , … , x i − 1 , a x i , b x i , x i + 1 , … , x n , | λ | 2 t L i C g , h ) ) ,$

and consequently, $d i ( J i g, J i h)≤ L i C g , h$, which means that the operator $J i$ is strictly contractive. Moreover, from (3.8) it follows that

$N ( J i f ( x 1 , … , x n ) − f ( x 1 , … , x n ) , t ) ≥ T ( Ψ i ( x 1 , … , x i , 0 , x i + 1 , … , x n , | λ | t ) , Ψ i ( x 1 , … , x i − 1 , a x i , b x i , x i + 1 , … , x n , | λ | 2 t ) )$

and thus $d i ( J i f,f)≤1<∞$. Therefore, by Theorem 2.8, $J i$ has a unique fixed point $Q i : X n →Y$ in the set $Ω ∗ ={g∈Ω:d(f,g)<∞}$ such that

$1 λ 2 Q i ( x 1 ,…, x i − 1 ,λ x i , x i + 1 ,…, x n )= Q i ( x 1 ,…, x n )$
(4.3)

and

$Q i ( x 1 ,…, x n )= lim j → ∞ 1 λ 2 j f ( x 1 , … , x i − 1 , λ j x i , x i + 1 , … , x n ) .$

Furthermore, from the fact that $f∈ Ω ∗$, Theorem 2.8, and $d i ( J i f,f)≤1$, we get

$d i (f, Q i )≤ 1 1 − L i d i ( J i f,f)≤ 1 1 − L i$

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 $Q i ′ : X n →Y$ is a general multi-Euler-Lagrange quadratic mapping satisfying condition (4.2). Then $Q i ′$ fulfills (4.3), and therefore, it is a fixed point of the operator $J i$. Moreover, by (4.2), we have $d i (f, Q i ′ )≤ 1 1 − L i <∞$, and consequently $Q i ′ ∈ Ω ∗$. Theorem 2.8 shows that $Q i ′ = Q i$. □

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

Theorem 4.2 Let $K$ be a valued field, $X$ be a vector space over $K$ and $(Y,N,T)$ be a complete non-Archimedean fuzzy normed space over $K$. Assume also that, for every $i∈{1,2,…,n}$, $Ψ i : X n + 1 ×[0,∞)→[0,1]$ is a mapping such that (3.14) holds and for an $L i ∈(0,1)$. If $f: X n →Y$ is a mapping satisfying (3.3) and (3.4), then for every $i∈{1,2,…,n}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q i : X n →Y$ such that

$N ( f ( x 1 , … , x n ) − Q i ( x 1 , … , x n ) , t ) ≥ T ( Ψ i ( x 1 , … , x i , 0 , x i + 1 , … , x n , | λ | ( L i − 1 − 1 ) t ) , Ψ i ( x 1 , … , x i − 1 , a x i , b x i , x i + 1 , … , x n , | λ | 2 ( L i − 1 − 1 ) t ) )$

for all $x 1 ,…, x n ∈X$ 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 $X$ be a real normed space, $Y$ be a real Banach space and $(Y,N, T P )$ be the complete non-Archimedean fuzzy normed space defined as in the second example in the preliminaries. Let $δ,r,p,q∈(0,∞)$ such that $r,s:=p+q∈(2,∞)$, or $r,s∈(0,2)$. If $f: X n →Y$ is a mapping satisfying (3.3) and (3.4), where

$Ψ i ( x 1 , … , x i , x i ′ , x i + 1 , … , x n , t ) := t t + δ [ ∥ x 1 ∥ r ⋯ ∥ x i − 1 ∥ r ( ∥ x i ∥ p ∥ x i ′ ∥ q ) ∥ x i + 1 ∥ r ⋯ ∥ x n ∥ r ] ,$

then for every $i∈{1,2,…,n}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q i : X n →Y$ such that (4.4)

for all $x 1 ,…, x n ∈X$ and $t>0$.

Proof Fix $i∈{1,2,…,n}$, $x 1 ,…, x i , x i ′ , x i + 1 ,…, x n ∈X$, $t∈[0,∞)$ and assume that $λ>1$, $r,s∈(2,∞)$ (the same arguments apply to the case where $λ<1$, $r,s∈(0,2)$). Then we can choose $L i = λ 2 − s <1$ and apply Theorem 4.2 to obtain the result. For $λ>1$, $r,s∈(0,2)$, or $λ<1$, $r,s∈(2,∞)$, the corollary follows from Theorem 4.1. □

Corollary 4.5 Let $X$ be a real normed space and $Y$ be a real Banach space (or $X$ be a non-Archimedean normed space and $Y$ be a complete non-Archimedean normed space over a non-Archimedean field $K$, respectively). Let $δ>0$ and $r∈(0,2)∪(2,∞)$. If $f: X n →Y$ is a mapping satisfying (3.3) and then for every $i∈{1,2,…,n}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q i : X n →Y$ such that

$∥ f ( x 1 , … , x n ) − Q i ( x 1 , … , x n ) ∥ ≤ max { | λ | , | a | r + | b | r } δ ( ∥ x 1 ∥ r ⋯ ∥ x n ∥ r ) | | λ | r − | λ | 2 |$

for all $x 1 ,…, x n ∈X$.

Proof Consider the non-Archimedean fuzzy normed space $(Y, N 1 , T M )$ defined as in the first example in the preliminaries, $Ψ i$ be defined by

$Ψ i ( x 1 , … , x i , x i ′ , x i + 1 , … , x n , t ) := t t + δ [ ∥ x 1 ∥ r ⋯ ∥ x i − 1 ∥ r ( ∥ x i ∥ r + ∥ x i ′ ∥ r ) ∥ x i + 1 ∥ r ⋯ ∥ x n ∥ r ] ,$

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 $ϕ:C→C$ be defined by

Consider the function $f:C→C$ be defined by

$f(x)= ∑ j = 0 ∞ ϕ ( α j x ) α 2 j$

for all $x∈C$, where $α>max{|a|,|b|,1}$. Then f satisfies the functional inequality (4.5)

for all $x,y∈C$, but there do not exist a general multi-Euler-Lagrange quadratic function $Q:C→C$ and a constant $d>0$ such that $|f(x)−Q(x)|≤d | x | 2$ for all $x∈C$.

It is clear that f is bounded by $α 2 α 2 − 1$ on $C$. If $| x | 2 + | y | 2 =0$ or $| x | 2 + | y | 2 ≥ 1 α 2$, then

$| f ( a x + b y ) + f ( b x − a y ) − ( a 2 + b 2 ) [ f ( x ) + f ( y ) ] | ≤ 2 α 4 ( | a | 2 + | b | 2 + 1 ) α 2 − 1 ( | x | 2 + | y | 2 ) .$

Now suppose that $0< | x | 2 + | y | 2 < 1 α 2$. Then there exists an integer $k≥1$ such that

$1 α 2 ( k + 2 ) ≤ | x | 2 + | y | 2 < 1 α 2 ( k + 1 ) .$
(4.6)

Hence

$α l |ax+by|<1, α l |bx−ay|<1, α l |x|<1, α l |y|<1$

for all $l=0,1,…,k−1$. 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 $Q:C→C$ and a constant $d>0$ such that $|f(x)−Q(x)|≤d | x | 2$ for all $x∈C$. Then there exists a constant $c∈C$ such that $Q(x)=c x 2$ for all rational numbers x. So, we obtain that

$| f ( x ) | ≤ ( d + | c | ) | x | 2$
(4.7)

for all rational numbers x. Let $s∈N$ with $s+1>d+|c|$. If x is a rational number in $(0, α − s )$, then $α j x∈(0,1)$ for all $j=0,1,…,s$, and for this x we get

$f(x)= ∑ j = 0 ∞ ϕ ( α j x ) α 2 j ≥ ∑ j = 0 s ϕ ( α j x ) α 2 j =(s+1) x 2 > ( d + | c | ) x 2 ,$

Corollary 4.7 Let $K$ be a non-Archimedean field with $0<|λ|<1$, $X$ be a normed space over $K$ and $Y$ be a complete non-Archimedean normed space over $K$. Let $δ,p,q∈(0,∞)$ such that $p+q∈(0,2)$. If $f: X n →Y$ is a mapping satisfying (3.3) and then for every $i∈{1,2,…,n}$ there exists a unique general multi-Euler-Lagrange quadratic mapping $Q i : X n →Y$ such that

$∥ f ( x 1 , … , x n ) − Q i ( x 1 , … , x n ) ∥ ≤ δ | a | p | b | q ∥ x i ∥ p + q | λ | p + q − | λ | 2$
(4.8)

for all $x 1 ,…, x n ∈X$.

Proof Fix $i∈{1,2,…,n}$, $x 1 ,…, x i , x i ′ , x i + 1 ,…, x n ∈X$ and $t∈[0,∞)$. Let $Ψ i : X n + 1 ×[0,∞)→[0,1]$ be defined by $Ψ i ( x 1 ,…, x i , x i ′ , x i + 1 ,…, x n ,t):= t t + δ ( ∥ x i ∥ p ∥ x i ′ ∥ q )$. Consider the non-Archimedean fuzzy normed space $(Y, N 1 , T M )$ defined as in Example 2.6, and apply Theorem 4.2. □

Corollary 4.8 Let $X$ be a real normed space and $Y$ be a real Banach space. Let $δ,r,p,q∈(0,∞)$ such that $r,p+q∈(0,2)$, or $r,p+q∈(2,∞)$. If $f: X n →Y$ is a mapping satisfying (3.3) and then, for every $i∈{1,2,…,n}$, there exists a unique general multi-Euler-Lagrange quadratic mapping $Q i : X n →Y$ such that (4.9)

for all $x 1 ,…, x n ∈X$.

Proof Fix $i∈{1,2,…,n}$, $x 1 ,…, x i , x i ′ , x i + 1 ,…, x n ∈X$ and $t∈[0,∞)$. Let $Ψ i : X n + 1 ×[0,∞)→[0,1]$ be defined by

$Ψ i ( x 1 , … , x i , x i ′ , x i + 1 , … , x n , t ) := t t + δ [ ∥ x 1 ∥ r ⋯ ∥ x i − 1 ∥ r ( ∥ x i ∥ p ∥ x i ′ ∥ q ) ∥ x i + 1 ∥ r ⋯ ∥ x n ∥ r ] .$

Consider the non-Archimedean fuzzy normed space $(Y, N 1 , T M )$ 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 ). For $a=b=1$ and $n=1$, Corollary 4.8 yields the main theorem in . The generalized Hyers-Ulam stability problem for the case of $r=p+q=2$ was excluded in Corollary 4.8 (see).

Note that by (4.4) one can get Letting $t→∞$ 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.

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## Acknowledgements

The first author was supported by the National Natural Science Foundation of China (NNSFC) (Grant No. 11171022).

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Correspondence to Tian Zhou Xu.

### Competing interests

The authors declare that they have no competing interests.

### 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.

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Xu, T.Z., Rassias, J.M. Stability of general multi-Euler-Lagrange quadratic functional equations in non-Archimedean fuzzy normed spaces. Adv Differ Equ 2012, 119 (2012). https://doi.org/10.1186/1687-1847-2012-119 