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# Non-Archimedean Hyers-Ulam-Rassias stability of m-variable functional equation

## Abstract

The main goal of this paper is to study the Hyers-Ulam-Rassias stability of the following Euler-Lagrange type additive functional equation:

$∑ j = 1 m f ( − r j x j + ∑ 1 ≤ i ≤ m , i ≠ j r i x i ) +2 ∑ i = 1 m r i f( x i )=mf ( ∑ i = 1 m r i x i ) ,$

where $r 1 ,…, r m ∈R$, $∑ i = k m r k ≠0$, and $r i , r j ≠0$ for some $1≤i, in non-Archimedean Banach spaces.

MSC:39B22, 39B52, 46S10.

## 1 Introduction and preliminaries

A valuation is a function $|⋅|$ from a field $K$ into $[0,∞)$ such that 0 is the unique element having the 0 valuation, $|rs|=|r||s|$ and the triangle inequality is replaced by $|r+s|≤max{|r|,|s|}$.

The field $K$ is called a valued field if $K$ carries a valuation. The usual absolute values of $R$ and $C$ are the examples of valuations.

Let us consider the valuation which satisfies a stronger condition than the triangle inequality. If the triangle inequality is replaced by $|r+s|≤max{|r|,|s|}$ for all $r,s∈K$, then the function $|⋅|$ is called a non-Archimedean valuation and the field is called a non-Archimedean field. Clearly, $|1|=|−1|=1$ and $|n|≤1$ for all integers $n≥1$. A trivial example of a non-Archimedean valuation is the function $|⋅|$ taking everything except for 0 into 1 and $|0|=0$.

Definition 1.1 Let X be a vector space over a field $K$ with a non-Archimedean valuation $|⋅|$. A function $∥⋅∥:X→[0,∞)$ is called a non-Archimedean norm if the following conditions hold:

1. (a)

$∥x∥=0$ if and only if $x=0$ for all $x∈X$;

2. (b)

$∥rx∥=|r|∥x∥$ for all $r∈K$ and $x∈X$;

3. (c)

the strong triangle inequality holds:

$∥x+y∥≤max { ∥ x ∥ , ∥ y ∥ }$

for all $x,y∈X$.

Then $(X,∥⋅∥)$ is called a non-Archimedean normed space.

Definition 1.2 Let ${ x n }$ be a sequence in a non-Archimedean normed space X.

1. (a)

A sequence ${ x n } n = 1 ∞$ in a non-Archimedean space is a Cauchy sequence iff the sequence ${ x n + 1 − x n } n = 1 ∞$ converges to zero;

2. (b)

The sequence ${ x n }$ is said to be convergent if, for any $ε>0$, there is a positive integer N and $x∈X$ such that $∥ x n −x∥≤ε$, for all $n≥N$. Then the point $x∈X$ is called the limit of the sequence ${ x n }$, which is denoted by $lim n → ∞ x n =x$;

3. (c)

If every Cauchy sequence in X converges, then the non-Archimedean normed space X is called a non-Archimedean Banach space.

Example 1.1 Fix a prime number p. For any nonzero rational number x, there exists a unique integer $n x ∈Z$ such that $x= a b p n x$, where a and b are integers not divisible by p. Then $| x | p := p − n x$ defines a non-Archimedean norm on $Q$. The completion of $Q$ with respect to the metric $d(x,y)= | x − y | p$ is denoted by $Q p$ which is called the p-adic number field. In fact, $Q p$ is the set of all formal series $x= ∑ k ≥ n x ∞ a k p k$ where $| a k |≤p−1$ are integers. The addition and multiplication between any two elements of $Q p$ are defined naturally. The norm $| ∑ k ≥ n x ∞ a k p k | p = p − n x$ is a non-Archimedean norm on $Q p$ and it makes $Q p$ a locally compact field.

Theorem 1.1 Let$(X,d)$be a complete generalized metric space and$J:X→X$be a strictly contractive mapping with Lipschitz constant$L<1$. Then, for all$x∈X$, either$d( J n x, J n + 1 x)=∞$for all nonnegative integers n or there exists a positive integer$n 0$such that:

1. (a)

$d( J n x, J n + 1 x)<∞$for all$n 0 ≥ n 0$;

2. (b)

the sequence${ J n x}$converges to a fixed point$y ∗$of J;

3. (c)

$y ∗$is the unique fixed point of J in the set$Y={y∈X:d( J n 0 x,y)<∞}$;

4. (d)

$d(y, y ∗ )≤ 1 1 − L d(y,Jy)$for all$y∈Y$.

In this paper, we prove the generalized Hyers-Ulam stability of the following functional equation:

$∑ j = 1 m f ( − r j x j + ∑ 1 ≤ i ≤ m , i ≠ j r i x i ) +2 ∑ i = 1 m r i f( x i )=mf ( ∑ i = 1 m r i x i ) ,$
(1.1)

where $r 1 ,…, r m ∈R$, $∑ k = 1 m r k ≠0$, and $r i , r j ≠0$ for some $1≤i, in non-Archimedean Banach spaces. A classical question in the theory of functional equations is the following: ‘When is it true that a function which approximately satisfies a functional equation D must be close to an exact solution of D?’.

If the problem accepts a solution, we say that the equation D is stable. The first stability problem concerning group homomorphisms was raised by Ulam  in 1940.

In the next year D. H. Hyres , gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces. In 1978, Th. M. Rassias  proved a generalization of Hyres’ theorem for additive mappings.

The result of Th. M. Rassias has influenced the development of what is now called the Hyers-Ulam-Rassias stability theory for functional equations. In 1994, a generalization of Rassias’ theorem was obtained by Gǎvruta  by replacing the bound $ϵ( ∥ x ∥ p + ∥ y ∥ p )$ by a general control function $φ(x,y)$.

The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see ).

## 2 Non-Archimedean stability of the functional equation (1.1): a fixed point approach

In this section, using a fixed point alternative approach, we prove the generalized Hyers-Ulam stability of the functional equation (1.1) in non-Archimedean normed spaces. Throughout this section, let X be a non-Archimedean normed space and Y be a non-Archimedean Banach space. Also $|2|≠1$.

Lemma 2.1 Let$X$and$Y$be linear spaces and let$r 1 ,…, r n$be real numbers with$∑ k = 1 n r k ≠0$and$r i , r j ≠0$for some$1≤i. Assume that a mapping$f:X→Y$satisfies the functional equation (1.1) for all$x 1 ,…, x n ∈X$. Then the mapping f is Cauchy additive. Moreover, $f( r k x)= r k f(x)$for all$x∈X$and all$1≤k≤n$.

Proof Since $∑ k = 1 n r k ≠0$, putting $x 1 =⋯= x n =0$ in (1.1), we get $f(0)=0$. Without loss of generality, we may assume that $r 1 , r 2 ≠0$. Letting $x 3 =⋯= x n =0$ in (1.1), we get

$f(− r 1 x 1 + r 2 x 2 )+f( r 1 x 1 − r 2 x 2 )+2 r 1 f( x 1 )+2 r 2 f( x 2 )=2f( r 1 x 1 + r 2 x 2 )$
(2.1)

for all $x 1 , x 2 ∈X$. Letting $x 2 =0$ in (2.1), we get

$2 r 1 f( x 1 )=f( r 1 x 1 )−f(− r 1 x 1 )$
(2.2)

for all $x 1 ∈X$. Similarly, by putting $x 1 =0$ in (2.1), we get

$2 r 2 f( x 2 )=f( r 2 x 2 )−f(− r 2 x 2 )$
(2.3)

for all $x 1 ∈X$. It follows from (2.1), (2.2) and (2.3) that (2.4)

for all $x 1 , x 2 ∈X$. Replacing $x 1$ and $x 2$ by $x r 1$ and $y r 2$ in (2.4), we get

$f(−x+y)+f(x−y)+f(x)+f(y)−f(−x)−f(−y)=2f(x+y)$
(2.5)

for all $x,y∈X$. Letting $y=−x$ in (2.5), we get that $f(−2x)+f(2x)=0$ for all $x∈X$. So the mapping L is odd. Therefore, it follows from (2.5) that the mapping f is additive. Moreover, let $x∈X$ and $1≤k≤n$. Setting $x k =x$ and $x l =0$ for all $1≤l≤n$, $l≠k$, in (1.1) and using the oddness of f, we get that $f( r k x)= r k f(x)$. □

Using the same method as in the proof of Lemma 2.1, we have an alternative result of Lemma 2.1 when $∑ k = 1 n r k =0$.

Lemma 2.2 Let$X$and$Y$be linear spaces and let$r 1 ,…, r n$be real numbers with$r i , r j ≠0$for some$1≤i. Assume that a mapping$f:X→Y$with$f(0)=0$satisfying the functional equation (1.1) for all$x 1 ,…, x n ∈X$. Then the mapping f is Cauchy additive. Moreover, $f( r k x)= r k f(x)$for all$x∈X$and all$1≤k≤n$.

Remark 2.1 Throughout this paper, $r 1 ,…, r m$ will be real numbers such that $r i , r j ≠0$ for fixed $1≤i and $φ i , j (x,y):=φ(0,…,0, x ⏟ i th ,0,…,0, y ⏟ j th ,0,…,0)$ for all $x,y∈X$ and all $1≤i.

Theorem 2.1 Let $φ: X m →[0,∞)$ be a function such that there exists an $L<1$ with

$φ ( x 1 2 , … , x m 2 ) ≤ L φ ( x 1 , … , x m ) | 2 |$
(2.6)

for all$x 1 ,…, x m ∈X$. Let$f:X→Y$be a mapping with$f(0)=0$satisfying the following inequality: (2.7)

for all$x 1 ,…, x m ∈X$. Then there is a unique Euler-Lagrange type additive mapping$EL:X→Y$such that

$∥ f ( x ) − EL ( x ) ∥ ≤ L | 2 | − | 2 | L max { max { φ i , j ( x 2 r i , − x 2 r j ) , φ i , j ( x 2 r i , 0 ) , φ i , j ( 0 , − x 2 r j ) } , 1 | 2 | max { φ i , j ( x r i , x r j ) , φ i , j ( x r i , 0 ) , φ i , j ( 0 , x r j ) } }$
(2.8)

for all$x∈X$.

Proof For each $1≤k≤m$ with $k≠i,j$, let $x k =0$ in (2.7). Then we get the following inequality: (2.9)

for all $x i , x j ∈X$. Letting $x i =0$ in (2.9), we get

$∥ f ( − r j x j ) − f ( r j x j ) + 2 r j f ( x j ) ∥ ≤ φ i , j (0, x j )$
(2.10)

for all $x j ∈X$. Similarly, letting $x j =0$ in (2.9), we get

$∥ f ( − r i x i ) − f ( r i x i ) + 2 r i f ( x i ) ∥ ≤ φ i , j ( x i ,0)$
(2.11)

for all $x i ∈X$. It follows from (2.9), (2.10) and (2.11) that for all $x i , x j ∈X$ (2.12)

Replacing $x i$ and $x j$ by $x r i$ and $y r j$ in (2.12), we get that (2.13)

for all $x,y∈X$. Putting $y=x$ in (2.13), we get

$∥ f ( x ) − f ( − x ) − f ( 2 x ) ∥ ≤ 1 | 2 | max { φ i , j ( x r i , x r j ) , φ i , j ( x r i , 0 ) , φ i , j ( 0 , x r j ) }$
(2.14)

for all $x∈X$. Replacing x and y by $x 2$ and $− x 2$ in (2.13) respectively, we get

$∥ f ( x ) + f ( − x ) ∥ ≤max { φ i , j ( x 2 r i , − x 2 r j ) , φ i , j ( x 2 r i , 0 ) , φ i , j ( 0 , − x 2 r j ) }$
(2.15)

for all $x∈X$. It follows from (2.14) and (2.15) that (2.16)

for all $x∈X$. Replacing x by $x 2$ in (2.16), we obtain (2.17)

Consider the set $S:={g:X→Y;g(0)=0}$ and the generalized metric d in S defined by

$d ( f , g ) = inf μ ∈ R + { ∥ g ( x ) − h ( x ) ∥ ≤ μ max { max { φ i , j ( x 2 r i , − x 2 r j ) , φ i , j ( x 2 r i , 0 ) , φ i , j ( 0 , − x 2 r j ) } , 1 | 2 | max { φ i , j ( x r i , x r j ) , φ i , j ( x r i , 0 ) , φ i , j ( 0 , x r j ) } } , ∀ x ∈ X } ,$

where $inf∅=+∞$. It is easy to show that $(S,d)$ is complete (see , Lemma 2.1). Now, we consider a linear mapping $J:S→S$ such that

$Jh(x):=2h ( x 2 )$

for all $x∈X$. Let $g,h∈S$ be such that $d(g,h)=ϵ$. Then

$∥ g ( x ) − h ( x ) ∥ ≤ ϵ max { max { φ i , j ( x 2 r i , − x 2 r j ) , φ i , j ( x 2 r i , 0 ) , φ i , j ( 0 , − x 2 r j ) } , 1 | 2 | max { φ i , j ( x r i , x r j ) , φ i , j ( x r i , 0 ) , φ i , j ( 0 , x r j ) } }$

for all $x∈X$, and so

$∥ J g ( x ) − J h ( x ) ∥ = ∥ 2 g ( x 2 ) − 2 h ( x 2 ) ∥ ≤ | 2 | ϵ max { max { φ i , j ( x 4 r i , − x 4 r j ) , φ i , j ( x 4 r i , 0 ) , φ i , j ( 0 , − x 4 r j ) } , 1 | 2 | max { φ i , j ( x 2 r i , x 2 r j ) , φ i , j ( x 2 r i , 0 ) , φ i , j ( 0 , x 2 r j ) } } ≤ | 2 | L ϵ | 2 | max { max { φ i , j ( x 2 r i , − x 2 r j ) , φ i , j ( x 2 r i , 0 ) , φ i , j ( 0 , − x 2 r j ) } , 1 | 2 | max { φ i , j ( x r i , x r j ) , φ i , j ( x r i , 0 ) , φ i , j ( 0 , x r j ) } }$

for all $x∈X$. Thus $d(g,h)=ϵ$ implies that $d(Jg,Jh)≤Lϵ$. This means that

$d(Jg,Jh)≤Ld(g,h)$

for all $g,h∈S$. It follows from (2.17) that

$d(f,Jf)≤ L | 2 | .$

By Theorem 1.1, there exists a mapping $EL:X→Y$ satisfying the following:

1. (1)

EL is a fixed point of J, that is,

$EL ( x 2 ) = 1 2 EL(x)$
(2.18)

for all $x∈X$. The mapping EL is a unique fixed point of J in the set

$Ω= { h ∈ S : d ( g , h ) < ∞ } .$

This implies that EL is a unique mapping satisfying (2.18) such that there exists $μ∈(0,∞)$ satisfying

$∥ f ( x ) − EL ( x ) ∥ ≤ μ max { max { φ i , j ( x 2 r i , − x 2 r j ) , φ i , j ( x 2 r i , 0 ) , φ i , j ( 0 , − x 2 r j ) } , 1 | 2 | max { φ i , j ( x r i , x r j ) , φ i , j ( x r i , 0 ) , φ i , j ( 0 , x r j ) } }$
(2.19)

for all $x∈X$.

1. (2)

$d( J n f,EL)→0$ as $n→∞$. This implies the equality

$lim n → ∞ 2 n f ( x 2 n ) =EL(x)$

for all $x∈X$.

1. (3)

$d(f,EL)≤ d ( f , J f ) 1 − L$ with $f∈Ω$, which implies the inequality

$d(f,EL)≤ L | 2 | − | 2 | L .$

This implies that the inequality (2.8) holds.

By (2.6) and (2.7), we obtain for all $x 1 ,…, x m ∈X$ and $n∈N$. So EL satisfies (1.1). Thus, the mapping $EL:X→Y$ is Euler-Lagrange type additive, as desired. □

Corollary 2.1 Let$θ≥0$and r be a real number with$0. Let$f:X→Y$be a mapping with$f(0)=0$satisfying the inequality (2.20)

for all$x 1 ,…,x∈X$. Then, the limit$EL(x)= lim n → ∞ 2 n f( x 2 n )$exists for all$x∈X$and$EL:X→Y$is a unique Euler-Lagrange additive mapping such that

$∥ f ( x ) − EL ( x ) ∥ ≤ | 2 | | 2 | r + 1 − | 2 | 2 max { max { | 2 | r θ ∥ x ∥ r ( | r i | r + | r j | r ) | 4 | r | r i r j | r , θ ∥ x ∥ r | 2 | r | r i | r , θ ∥ x ∥ r | 2 | r | r j | r } , 1 | 2 | max { θ ∥ x ∥ r ( | r i | r + | r j | r ) | r i r j | r , θ ∥ x ∥ r | r i | r , θ ∥ x ∥ r | r j | r } } ≤ θ ∥ x ∥ r ( | r i | r + | r j | r ) | r i r j | r ( | 2 | r + 1 − | 2 | 2 )$

for all$x∈X$.

Proof The proof follows from Theorem 2.1 by taking $φ( x 1 ,…, x m )=θ( ∑ i = 1 m ∥ x i ∥ r )$ for all $x 1 ,…, x m ∈X$. In fact, if we choose $L= | 2 | 1 − r$, then we get the desired result. □

Theorem 2.2 Let $φ: X m →[0,∞)$ be a function such that there exists an $L<1$ with

$φ( x 1 ,…, x m )≤|2|Lφ ( x 1 2 , … , x m 2 )$
(2.21)

for all$x 1 ,…, x m ∈X$. Let$f:X→Y$be a mapping with$f(0)=0$satisfying the inequality (2.7). Then, there is a unique Euler-Lagrange additive mapping$EL:X→Y$such that

$∥ f ( x ) − EL ( x ) ∥ ≤ 1 | 2 | − | 2 | L max { max { φ i , j ( x 2 r i , − x 2 r j ) , φ i , j ( x 2 r i , 0 ) , φ i , j ( 0 , − x 2 r j ) } , 1 | 2 | max { φ i , j ( x r i , x r j ) , φ i , j ( x r i , 0 ) , φ i , j ( 0 , x r j ) } } .$
(2.22)

Proof By (2.16), we have

$∥ f ( 2 x ) 2 − f ( x ) ∥ ≤ 1 | 2 | max { max { φ i , j ( x 2 r i , − x 2 r j ) , φ i , j ( x 2 r i , 0 ) , φ i , j ( 0 , − x 2 r j ) } , 1 | 2 | max { φ i , j ( x r i , x r j ) , φ i , j ( x r i , 0 ) , φ i , j ( 0 , x r j ) } }$
(2.23)

for all $x∈X$ . Let $(S,d)$ be the generalized metric space defined in the proof of Theorem 2.1. Now, we consider a linear mapping $J:S→S$ such that

$Jh(x):= 1 2 h(2x)$

for all $x∈X$. Let $g,h∈S$ be such that $d(g,h)=ϵ$. Then

$∥ g ( x ) − h ( x ) ∥ ≤ ϵ max { max { φ i , j ( x 2 r i , − x 2 r j ) , φ i , j ( x 2 r i , 0 ) , φ i , j ( 0 , − x 2 r j ) } , 1 | 2 | max { φ i , j ( x r i , x r j ) , φ i , j ( x r i , 0 ) , φ i , j ( 0 , x r j ) } }$

for all $x∈X$, and so

$∥ J g ( x ) − J h ( x ) ∥ = ∥ g ( 2 x ) 2 − h ( 2 x ) 2 ∥ ≤ 1 | 2 | ϵ max { max { φ i , j ( x r i , − x r j ) , φ i , j ( x r i , 0 ) , φ i , j ( 0 , − x r j ) } , 1 | 2 | max { φ i , j ( 2 x r i , 2 x r j ) , φ i , j ( 2 x r i , 0 ) , φ i , j ( 0 , 2 x r j ) } } ≤ | 2 | L ϵ | 2 | max { max { φ i , j ( x 2 r i , − x 2 r j ) , φ i , j ( x 2 r i , 0 ) , φ i , j ( 0 , − x 2 r j ) } , 1 | 2 | max { φ i , j ( x r i , x r j ) , φ i , j ( x r i , 0 ) , φ i , j ( 0 , x r j ) } }$

for all $x∈X$ . Thus $d(g,h)=ϵ$ implies that $d(Jg,Jh)≤Lϵ$. This means that

$d(Jg,Jh)≤Ld(g,h)$

for all $g,h∈S$. It follows from (2.23) that

$d(f,Jf)≤ 1 | 2 | .$

By Theorem 1.1, there exists a mapping $EL:X→Y$ satisfying the following:

1. (1)

EL is a fixed point of J, that is,

$EL(2x)=2EL(x)$
(2.24)

for all $x∈X$. The mapping EL is a unique fixed point of J in the set

$Ω= { h ∈ S : d ( g , h ) < ∞ } .$

This implies that EL is a unique mapping satisfying (2.24) such that there exists $μ∈(0,∞)$ satisfying

$∥ g ( x ) − h ( x ) ∥ ≤ μ max { max { φ i , j ( x 2 r i , − x 2 r j ) , φ i , j ( x 2 r i , 0 ) , φ i , j ( 0 , − x 2 r j ) } , 1 | 2 | max { φ i , j ( x r i , x r j ) , φ i , j ( x r i , 0 ) , φ i , j ( 0 , x r j ) } }$

for all $x∈X$.

1. (2)

$d( J n f,EL)→0$ as $n→∞$. This implies the equality

$lim n → ∞ f ( 2 n x ) 2 n =EL(x)$

for all $x∈X$.

1. (3)

$d(f,EL)≤ d ( f , J f ) 1 − L$ with $f∈Ω$, which implies the inequality

$d(f,EL)≤ 1 | 2 | − | 2 | L .$

This implies that the inequality (2.22) holds. The rest of the proof is similar to the proof of Theorem 2.1. □

Corollary 2.2 Let$θ≥0$and r be a real number with$r>1$. Let$f:X→Y$be a mapping with$f(0)=0$satisfying (2.20). Then, the limit$EL(x)= lim n → ∞ f ( 2 n x ) 2 n$exists for all$x∈X$and$EL:X→Y$is a unique cubic mapping such that

$∥ f ( x ) − EL ( x ) ∥ ≤ 1 | 2 | − | 2 | r max { max { | 2 | r θ ∥ x ∥ r ( | r i | r + | r j | r ) | 4 | r | r i r j | r , θ ∥ x ∥ r | 2 | r | r i | r , θ ∥ x ∥ r | 2 | r | r j | r } , 1 | 2 | max { θ ∥ x ∥ r ( | r i | r + | r j | r ) | r i r j | r , θ ∥ x ∥ r | r i | r , θ ∥ x ∥ r | r j | r } } ≤ θ ∥ x ∥ r ( | r i | r + | r j | r ) | r i r j | r ( | 2 | r + 1 − | 2 | r + 2 )$

for all$x∈X$ .

Proof The proof follows from Theorem 2.2 by taking $φ( x 1 ,…, x m )=θ( ∑ i = 1 m ∥ x i ∥ r )$ for all $x 1 ,…, x m ∈X$. In fact, if we choose $L= | 2 | r − 1$, then we get the desired result. □

## 3 Non-Archimedean stability of the functional equation (1.1): a direct method

In this section, using a direct method, we prove the generalized Hyers-Ulam stability of the cubic functional equation (1.1) in non-Archimedean normed spaces. Throughout this section, we assume that G is an additive semigroup and X is a non-Archimedean Banach space.

Theorem 3.1 Let $φ: G m →[0,+∞)$ be a function such that

$lim n → ∞ | 2 | n φ ( x 1 2 n , … , x m 2 n ) =0$
(3.1)

for all $x 1 ,…, x m ∈G$ and let for each $x∈G$ the limit

$Θ ( x ) = lim n → ∞ max { | 2 | k max { max { φ i , j ( x 2 k + 2 r i , − x 2 k + 2 r j ) , φ i , j ( x 2 k + 2 r i , 0 ) , φ i , j ( 0 , − x 2 k + 2 r j ) } , 1 | 2 | max { φ i , j ( x 2 k + 1 r i , x 2 k + 1 r j ) , φ i , j ( x 2 k + 1 r i , 0 ) , φ i , j ( 0 , x 2 k + 1 r j ) } } | 0 ≤ k < n }$
(3.2)

exist. Suppose that$f:G→X$is a mapping with$f(0)=0$satisfying the following inequality: (3.3)

for all$x 1 ,…, x m ∈X$. Then, the limit$EL(x):= lim n → ∞ 2 n f( x 2 n )$exists for all$x∈G$and defines an Euler-Lagrange type additive mapping$EL:G→X$such that

$∥ f ( x ) − EL ( x ) ∥ ≤Θ(x).$
(3.4)

Moreover, if then EL is the unique mapping satisfying (3.4).

Proof By (2.17), we know

$∥ f ( x ) − 2 f ( x 2 ) ∥ ≤ max { max { φ i , j ( x 4 r i , − x 4 r j ) , φ i , j ( x 4 r i , 0 ) , φ i , j ( 0 , − x 4 r j ) } , 1 | 2 | max { φ i , j ( x 2 r i , x 2 r j ) , φ i , j ( x 2 r i , 0 ) , φ i , j ( 0 , x 2 r j ) } }$
(3.5)

for all $x∈G$. Replacing x by $x 2 n$ in (3.5), we obtain (3.6)

It follows from (3.1) and (3.6) that the sequence ${ 2 n f ( x 2 n ) } n ≥ 1$ is a Cauchy sequence. Since X is complete, so ${ 2 n f ( x 2 n ) } n ≥ 1$ is convergent. Set

$EL(x):= lim n → ∞ 2 n f ( x 2 n ) .$

Using induction on n, one can show that (3.7)

for all $n∈N$ and all $x∈G$. By taking n to approach infinity in (3.7) and using (3.2), one obtains (3.4). By (3.1) and (3.3), we get for all $x 1 ,…, x m ∈X$. Therefore the function $EL:G→X$ satisfies (1.1).

To prove the uniqueness property of EL, let $A:G→X$ be another function satisfying (3.4). Then for all $x∈G$. Therefore $A=EL$, and the proof is complete. □

Corollary 3.1 Let $ξ:[0,∞)→[0,∞)$ be a function satisfying

$ξ ( t | 2 | ) ≤ξ ( 1 | 2 | ) ξ(t)(t≥0)ξ ( 1 | 2 | ) < | 2 | − 1 .$
(3.8)

Let$κ>0$and$f:G→X$be a mapping with$f(0)=0$satisfying the following inequality: (3.9)

for all$x 1 ,…, x m ∈G$. Then there exists a unique Euler-Lagrange type additive mapping$EL:G→X$such that

$∥ f ( x ) − EL ( x ) ∥ ≤ κ | 4 | { ξ ( | x r i | ) + ξ ( | x r j | ) } .$
(3.10)

Proof Defining $ζ: G m →[0,∞)$ by $φ( x 1 ,…, x m ):=κ( ∑ k = 1 m ξ(| x k |))$, then we have

$lim n → ∞ | 2 | n φ ( x 1 2 n , … , x m 2 n ) ≤ lim n → ∞ ( | 2 | ξ ( 1 | 2 | ) ) n φ( x 1 ,…, x m )=0$

for all $x 1 ,…, x m ∈G$. On the other hand,

$Θ ( x ) = lim n → ∞ max { | 2 | k max { max { φ i , j ( x 2 k + 2 r i , − x 2 k + 2 r j ) , φ i , j ( x 2 k + 2 r i , 0 ) , φ i , j ( 0 , − x 2 k + 2 r j ) } , 1 | 2 | max { φ i , j ( x 2 k + 1 r i , x 2 n + 1 r j ) , φ i , j ( x 2 k + 1 r i , 0 ) , φ i , j ( 0 , x 2 k + 1 r j ) } } | 0 ≤ k < n } = max { max { φ i , j ( x 4 r i , − x 4 r j ) , φ i , j ( x 4 r i , 0 ) , φ i , j ( 0 , − x 4 r j ) } , 1 | 2 | max { φ i , j ( x 2 r i , x 2 r j ) , φ i , j ( x 2 r i , 0 ) , φ i , j ( 0 , x 2 r j ) } } = κ | 4 | { ξ ( | x r i | ) + ξ ( | x r j | ) }$

for all $x∈G$, exists. Also Applying Theorem 3.1, we get the desired result. □

Theorem 3.2 Let $φ: G m →[0,+∞)$ be a function such that

$lim n → ∞ φ ( 2 n x 1 , … , 2 n x m ) | 2 | n =0$
(3.11)

for all $x 1 ,…, x m ∈G$ and let for each $x∈G$ the limit

$Θ ( x ) = lim n → ∞ max { 1 | 2 | k max { max { φ i , j ( 2 k − 1 x r i , − 2 k − 1 x r j ) , φ i , j ( 2 k − 1 x r i , 0 ) , φ i , j ( 0 , − 2 k − 1 x r j ) } , 1 | 2 | max { φ i , j ( 2 k x r i , 2 k x r j ) , φ i , j ( 2 k x r i , 0 ) , φ i , j ( 0 , 2 k x r j ) } } | 0 ≤ k < n }$
(3.12)

exist. Suppose that$f:G→X$is a mapping with$f(0)=0$satisfying (3.3). Then, the limit$EL(x):= lim n → ∞ f ( 2 n x ) 2 n$exists for all$x∈G$and defines an Euler-Lagrange type additive mapping$EL:G→X$, such that

$∥ f ( x ) − EL ( x ) ∥ ≤ 1 | 2 | Θ(x).$
(3.13)

Moreover, if then EL is the unique Euler-Lagrange type additive mapping satisfying (3.13).

Proof It follows from (2.16) that (3.14)

for all $x∈G$. Replacing x by $2 n x$ in (3.14), we obtain (3.15)

It follows from (3.11) and (3.15) that the sequence ${ f ( 2 n x ) 2 n } n ≥ 1$ is convergent. Set

$EL(x):= lim n → ∞ f ( 2 n x ) 2 n .$

On the other hand, it follows from (3.15) that for all $x∈G$ and all nonnegative integers p, q with $q>p≥0$. Letting $p=0$ and passing the limit $q→∞$ in the last inequality and using (3.12), we obtain (3.13). The rest of the proof is similar to the proof of Theorem 3.1. □

Corollary 3.2 Let $ξ:[0,∞)→[0,∞)$ be a function satisfying

$ξ ( | 2 t | ) ≤ξ ( | 2 | ) ξ(t)(t≥0)ξ ( | 2 | ) <|2|.$
(3.16)

Let$κ>0$and$f:G→X$be a mapping with$f(0)=0$satisfying the following inequality (3.9). Then there exists a unique Euler-Lagrange type additive mapping$EL:G→X$such that

$∥ f ( x ) − EL ( x ) ∥ ≤ κ | 2 | max { ξ ( | x 2 r i | ) + ξ ( | x 2 r j | ) , 1 | 2 | ξ ( | x r i | ) + ξ ( | x r j | ) } = κ | 2 | [ ξ ( | x 2 r i | ) + ξ ( | x 2 r j | ) ] .$
(3.17)

Proof Defining $ζ: G m →[0,∞)$ by $φ( x 1 ,…, x m ):=κ( ∑ k = 1 m ξ(| x k |))$, then, we have

$lim n → ∞ φ ( 2 n x 1 , … , 2 n x m ) | 2 | n ≤ lim n → ∞ ( ξ ( | 2 | ) | 2 | ) n φ( x 1 ,…, x m )=0$

for all $x 1 ,…, x m ∈G$. On the other hand,

$Θ ( x ) = lim n → ∞ max { 1 | 2 | k max { max { φ i , j ( 2 k − 1 x r i , − 2 k − 1 x r j ) , φ i , j ( 2 k − 1 x r i , 0 ) , φ i , j ( 0 , − 2 k − 1 x r j ) } , 1 | 2 | max { φ i , j ( 2 k x r i , 2 k x r j ) , φ i , j ( 2 k x r i , 0 ) , φ i , j ( 0 , 2 k x r j ) } } | 0 ≤ k < n } = max { max { φ i , j ( x 2 r i , − x 2 r j ) , φ i , j ( x 2 r i , 0 ) , φ i , j ( 0 , − x 2 r j ) } , 1 | 2 | max { φ i , j ( x r i , x r j ) , φ i , j ( x r i , 0 ) , φ i , j ( 0 , x r j ) } }$

for all $x∈G$, exists. Also Applying Theorem 3.14, we get the desired result. □

Remark 3.1 We remark that if $ξ(|2|)=0$, then $ξ=0$ identically, and so f is itself additive. Thus, for the nontrivial ξ, we observe that $ξ(|2|)≠0$ and

$1≤ξ ( | 1 | ) ≤ξ ( | 2 | ) ξ ( 1 | 2 | ) ≤|2|ξ ( 1 | 2 | )$

implies that $1 | 2 | ≤ξ( 1 | 2 | )$.

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## Acknowledgement

Dong Yun Shin was supported by the 2011 sabbatical year research grant of the University of Seoul.

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### Authors’ contributions

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

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Azadi Kenary, H., Rezaei, H., Sharifzadeh, M. et al. Non-Archimedean Hyers-Ulam-Rassias stability of m-variable functional equation. Adv Differ Equ 2012, 111 (2012). https://doi.org/10.1186/1687-1847-2012-111 