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

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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 1i<jm, 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,sK, 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 n1. 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 xX;

  2. (b)

    rx=|r|x for all rK and xX;

  3. (c)

    the strong triangle inequality holds:

    x+ymax { x , y }

for all x,yX.

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 xX such that x n xε, for all nN. Then the point xX 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 |p1 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 andJ:XXbe a strictly contractive mapping with Lipschitz constantL<1. Then, for allxX, eitherd( 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 setY={yX:d( J n 0 x,y)<};

  4. (d)

    d(y, y ) 1 1 L d(y,Jy)for allyY.

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 1i<jm, 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 [34] in 1940.

In the next year D. H. Hyres [17], 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 [24] 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 [15] 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 [133]).

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 LetXandYbe linear spaces and let r 1 ,, r n be real numbers with k = 1 n r k 0and r i , r j 0for some1i<jn. Assume that a mappingf:XYsatisfies 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 allxXand all1kn.

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(xy)+f(x)+f(y)f(x)f(y)=2f(x+y)
(2.5)

for all x,yX. Letting y=x in (2.5), we get that f(2x)+f(2x)=0 for all xX. So the mapping L is odd. Therefore, it follows from (2.5) that the mapping f is additive. Moreover, let xX and 1kn. Setting x k =x and x l =0 for all 1ln, lk, 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 LetXandYbe linear spaces and let r 1 ,, r n be real numbers with r i , r j 0for some1i<jn. Assume that a mappingf:XYwithf(0)=0satisfying 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 allxXand all1kn.

Remark 2.1 Throughout this paper, r 1 ,, r m will be real numbers such that r i , r j 0 for fixed 1i<jm and φ i , j (x,y):=φ(0,,0, x i th ,0,,0, y j th ,0,,0) for all x,yX and all 1i<jm.

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. Letf:XYbe a mapping withf(0)=0satisfying the following inequality:

(2.7)

for all x 1 ,, x m X. Then there is a unique Euler-Lagrange type additive mappingEL:XYsuch 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 allxX.

Proof For each 1km with ki,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,yX. 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 xX. 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 xX. It follows from (2.14) and (2.15) that

(2.16)

for all xX. Replacing x by x 2 in (2.16), we obtain

(2.17)

Consider the set S:={g:XY;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 [18], Lemma 2.1). Now, we consider a linear mapping J:SS such that

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

for all xX. Let g,hS 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 xX, 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 xX. Thus d(g,h)=ϵ implies that d(Jg,Jh)Lϵ. This means that

d(Jg,Jh)Ld(g,h)

for all g,hS. It follows from (2.17) that

d(f,Jf) L | 2 | .

By Theorem 1.1, there exists a mapping EL:XY 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 xX. 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 xX.

  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 xX.

  1. (3)

    d(f,EL) d ( f , J f ) 1 L with fΩ, 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 nN. So EL satisfies (1.1). Thus, the mapping EL:XY is Euler-Lagrange type additive, as desired. □

Corollary 2.1 Letθ0and r be a real number with0<r<1. Letf:XYbe a mapping withf(0)=0satisfying the inequality

(2.20)

for all x 1 ,,xX. Then, the limitEL(x)= lim n 2 n f( x 2 n )exists for allxXandEL:XYis 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 allxX.

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. Letf:XYbe a mapping withf(0)=0satisfying the inequality (2.7). Then, there is a unique Euler-Lagrange additive mappingEL:XYsuch 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 xX . Let (S,d) be the generalized metric space defined in the proof of Theorem 2.1. Now, we consider a linear mapping J:SS such that

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

for all xX. Let g,hS 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 xX, 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 xX . Thus d(g,h)=ϵ implies that d(Jg,Jh)Lϵ. This means that

d(Jg,Jh)Ld(g,h)

for all g,hS. It follows from (2.23) that

d(f,Jf) 1 | 2 | .

By Theorem 1.1, there exists a mapping EL:XY satisfying the following:

  1. (1)

    EL is a fixed point of J, that is,

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

for all xX. 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 xX.

  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 xX.

  1. (3)

    d(f,EL) d ( f , J f ) 1 L with fΩ, 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 Letθ0and r be a real number withr>1. Letf:XYbe a mapping withf(0)=0satisfying (2.20). Then, the limitEL(x)= lim n f ( 2 n x ) 2 n exists for allxXandEL:XYis 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 allxX .

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 xG 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 thatf:GXis a mapping withf(0)=0satisfying the following inequality:

(3.3)

for all x 1 ,, x m X. Then, the limitEL(x):= lim n 2 n f( x 2 n )exists for allxGand defines an Euler-Lagrange type additive mappingEL:GXsuch 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 xG. 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 nN and all xG. 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:GX satisfies (1.1).

To prove the uniqueness property of EL, let A:GX be another function satisfying (3.4). Then

for all xG. 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)(t0)ξ ( 1 | 2 | ) < | 2 | 1 .
(3.8)

Letκ>0andf:GXbe a mapping withf(0)=0satisfying the following inequality:

(3.9)

for all x 1 ,, x m G. Then there exists a unique Euler-Lagrange type additive mappingEL:GXsuch 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 xG, 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 xG 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 thatf:GXis a mapping withf(0)=0satisfying (3.3). Then, the limitEL(x):= lim n f ( 2 n x ) 2 n exists for allxGand defines an Euler-Lagrange type additive mappingEL:GX, 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 xG. 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 xG and all nonnegative integers p, q with q>p0. 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)(t0)ξ ( | 2 | ) <|2|.
(3.16)

Letκ>0andf:GXbe a mapping withf(0)=0satisfying the following inequality (3.9). Then there exists a unique Euler-Lagrange type additive mappingEL:GXsuch 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 xG, 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

1ξ ( | 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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Correspondence to DY Shin.

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The authors declare that they have no competing interests.

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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Keywords

  • stability
  • non-Archimedean normed space
  • fixed point method