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A functional equation related to inner product spaces in non-Archimedean L-random normed spaces

Abstract

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

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

1 Introduction

One of the most interesting questions in the theory of functional analysis concerning the Ulam stability problem of functional equations is as follows: When is it true that a mapping satisfying a functional equation approximately must be close to an exact solution of the given functional equation?

The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940 and affirmatively solved by Hyers [2]. The result of Hyers was generalized by Aoki [3] for approximate additive mappings and by ThM Rassias [4] for approximate linear mappings by allowing the difference Cauchy equation f( x 1 + x 2 )f( x 1 )f( x 2 ) to be controlled by ε( x 1 p + x 2 p ). In 1994, a generalization of the ThM Rassias‘ theorem was obtained by Gǎvruta [5], who replaced ε( x 1 p + x 2 p ) by a general control function φ( x 1 , x 2 ).

Quadratic functional equations were used to characterize inner product spaces [6]. A square norm on an inner product space satisfies the parallelogram equality x 1 + x 2 2 + x 1 x 2 2 =2( x 1 2 + x 1 2 ). The functional equation

f(x+y)+f(xy)=2f(x)+2f(y)
(1.1)

is related to a symmetric bi-additive mapping [7, 8]. It is natural that this equation is called a quadratic functional equation, and every solution of the quadratic equation (1.1) is said to be a quadratic mapping.

It was shown by ThM Rassias [9] that the norm defined over a real vector space X is induced by an inner product if and only if for a fixed integer n2

i = 1 n x i 1 n j = 1 n x j 2 = i = 1 n x i 2 n 1 n i = 1 n x i 2

for all x 1 ,, x n X.

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

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

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

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

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

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

(NA1) x=0 if and only if x=0;

(NA2) rx=|r|x for all and xX;

(NA3) the strong triangle inequality (ultra-metric); namely,

x+ymax { x , y } (x,yX).

Then (X,) is called a non-Archimedean space.

Thanks to the inequality

x m x l max { x ȷ + 1 x ȷ : l ȷ m 1 } (m>l)

a sequence { x m } is Cauchy in X if and only if { x m + 1 x m } converges to zero in a non-Archimedean space. By a complete non-Archimedean space, we mean a non-Archimedean space in which every Cauchy sequence is convergent.

In 1897, Hensel [10] introduced a normed space, which does not have the Archimedean property.

During the last three decades, the theory of non-Archimedean spaces has gained the interest of physicists for their research in particular in problems coming from quantum physics, p-adic strings, and superstrings [11]. Although many results in the classical normed space theory have a non-Archimedean counterpart, but their proofs are essentially different and require an entirely new kind of intuition [1216].

The main objective of this paper is to prove the Hyers-Ulam stability of the following functional equation related to inner product spaces:

i = 1 n f ( x i 1 n j = 1 n x j ) = i = 1 n f( x i )nf ( 1 n i = 1 n x i )
(1.2)

(nN, n2) in non-Archimedean normed spaces. Interesting new results concerning functional equations related to inner product spaces have recently been obtained by Najati and ThM Rassias [18] as well as for the fuzzy stability of a functional equation related to inner product spaces by Park [19] and Gordji and Khodaei [20]. During the last decades, several stability problems for various functional equations have been investigated by many mathematicians; [2156].

2 Preliminaries

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

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

The subspace D L + Δ L + is defined as D L + ={F Δ L + : l F(+)= 1 L }, where l f(x) denotes the left limit of the function f at the point x. The space Δ L + is partially ordered by the usual point-wise ordering of functions, that is, FG if and only if F(t) L G(t) for all t in R. The maximal element for Δ L + in this order is the distribution function given by

ε 0 (t)={ 0 L , if  t 0 , 1 L , if  t > 0 .

Definition 2.1[66]

A triangular norm (t-norm) on L is a mapping T: ( L ) 2 L satisfying the following conditions:

(1) (xL)(T(x, 1 L )=x) (: boundary condition);

(2) ((x,y) ( L ) 2 )(T(x,y)=T(y,x)) (: commutativity);

(3) ((x,y,z) ( L ) 3 )(T(x,T(y,z))=T(T(x,y),z)) (: associativity);

(4) ((x, x ,y, y ) ( L ) 4 )(x L x  and y L y T(x,y) L T( x , y )) (: monotonicity).

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

lim n T( x n ,y)=T(x,y),

for any yL.

A t-norm T can be extended (by associativity) in a unique way to an n-array operation taking for ( x 1 ,, x n ) L n the value T( x 1 ,, x n ) defined by

T i = 1 0 x i =1, T i = 1 n x i =T ( T i = 1 n 1 x i , x n ) =T( x 1 ,, x n ).

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

T i = 1 x i = lim n T i = 1 n x i .
(2.1)

The limit on the right side of (2.1) exists since the sequence ( T i = 1 n x i ) n N is non-increasing and bounded from below.

Note that we put T=T whenever L=[0,1]. If T is a t-norm then, for all x[0,1] and nN{0}, x T ( n ) is defined by 1 if n=0 and T( x T ( n 1 ) ,x) if n1. A t-norm T is said to be of Hadžić-type (we denote by TH) if the family ( x T ( n ) ) n N is equi-continuous at x=1 (see [67]).

Definition 2.2[66]

A continuous t-norm T on L= [ 0 , 1 ] 2 is said to be continuous t-representable if there exist a continuous t-norm and a continuous t-co-norm on [0,1] such that, for all x=( x 1 , x 2 ), y=( y 1 , y 2 )L,

T(x,y)=( x 1 y 1 , x 2 y 2 ).

For example,

T(a,b)= ( a 1 b 1 , min { a 2 + b 2 , 1 } )

and

M(a,b)= ( min { a 1 , b 1 } , max { a 2 , b 2 } )

for all a=( a 1 , a 2 ), b=( b 1 , b 2 ) [ 0 , 1 ] 2 are continuous t-representable.

Define the mapping T from L 2 to L by

T (x,y)=min(x,y)={ x , if  y L x , y , if  x L y .

Recall (see [67, 68]) that, if { x n } is a given sequence in L, then ( T ) i = 1 n x i is defined recurrently by ( T ) i = 1 1 x i = x 1 and ( T ) i = 1 n x i = T ( ( T ) i = 1 n 1 x i , x n ) for all n2.

A negation on L is any decreasing mapping N:LL satisfying N( 0 L )= 1 L and N( 1 L )= 0 L . If N(N(x))=x for all xL, then N is called an involutive negation. In the following, L is endowed with a (fixed) negation N.

Definition 2.3 A latticetic random normed space is a triple (X,μ, T ), where X is a vector space and μ is a mapping from X into D L + satisfying the following conditions:

(LRN1) μ x (t)= ε 0 (t) for all t>0 if and only if x=0;

(LRN2) μ α x (t)= μ x ( t | α | ) for all x in X, α0 and t0;

(LRN3) μ x + y (t+s) L T ( μ x (t), μ y (s)) for all x,yX and t,s0.

We note that, from (LPN2), it follows that μ x (t)= μ x (t) for all xX and t0.

Example 2.4 Let L=[0,1]×[0,1] and an operation L be defined by

L = { ( a 1 , a 2 ) : ( a 1 , a 2 ) [ 0 , 1 ] × [ 0 , 1 ]  and  a 1 + a 2 1 } , ( a 1 , a 2 ) L ( b 1 , b 2 ) a 1 b 1 , a 2 b 2 , a = ( a 1 , a 2 ) , b = ( b 1 , b 2 ) L .

Then (L, L ) is a complete lattice (see [66]). In this complete lattice, we denote its units by 0 L =(0,1) and 1 L =(1,0). Let (X,) be a normed space. Let T(a,b)=(min{ a 1 , b 1 },max{ a 2 , b 2 }) for all a=( a 1 , a 2 ), b=( b 1 , b 2 )[0,1]×[0,1] and μ be a mapping defined by

μ x (t)= ( t t + x , x t + x ) ,t R + .

Then (X,μ,T) is a latticetic random normed space.

If (X,μ, T ) is a latticetic random normed space, then we have

V= { V ( ε , λ ) : ε > L 0 L , λ L { 0 L , 1 L } }

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

V(ε,λ)= { x X : F x ( ε ) > L N ( λ ) } .

Definition 2.5 Let (X,μ, T ) be a latticetic random normed space.

(1) A sequence { x n } in X is said to be convergent to a point xX if, for any t>0 and εL{ 0 L }, there exists a positive integer N such that μ x n x (t) > L N(ε) for all nN.

(2) A sequence { x n } in X is called a Cauchy sequence if, for any t>0 and εL{ 0 L }, there exists a positive integer N such that μ x n x m (t) > L N(ε) for all nmN.

(3) A latticetic random normed space (X,μ, T ) is said to be complete if every Cauchy sequence in X is convergent to a point in X.

Theorem 2.6 If(X,μ, T )is a latticetic random normed space and{ x n }is a sequence such that x n x, then lim n μ x n (t)= μ x (t).

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

Lemma 2.7 Let(X,μ, T )be a latticetic random normed space andxX. If

μ x (t)=C,t>0,

thenC= 1 L andx=0.

Proof Let μ x (t)=C for all t>0. Since Ran(μ) D L + , we have C= 1 L and, by (LRN1), we conclude that x=0. □

3 Hyers-Ulam stability in non-Archimedean latticetic random spaces

In the rest of this paper, unless otherwise explicitly stated, we will assume that G is an additive group and that X is a complete non-Archimedean latticetic random space. For convenience, we use the following abbreviation for a given mapping f:GX:

Δf( x 1 ,, x n )= i = 1 n f ( x i 1 n j = 1 n x j ) i = 1 n f( x i )+nf ( 1 n i = 1 n x i )

for all x 1 ,, x n G, where n2 is a fixed integer.

Lemma 3.1[18]

Let V 1 and V 2 be real vector spaces. If an odd mappingf: V 1 V 2 satisfies the functional equation (1.2), then f is additive.

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

Theorem 3.2 LetKbe a non-Archimedean field and(X,μ, T )a non-Archimedean complete LRN-space overK. Letφ: G n D L + be a distribution function such that

lim m φ 2 m x 1 , 2 m x 2 , , 2 m x n ( | 2 | m t ) = 1 L = lim m Φ 2 m 1 x ( | 2 | m t )
(3.1)

for allx, x 1 , x 2 ,, x n G, and

φ ˜ x (t)= lim m min { Φ 2 k x ( | 2 | k t ) : 0 k < m }
(3.2)

exists for allxG, where

Φ x (t):=min { φ 2 x , 0 , , 0 ( t ) , min { φ x , x , 0 , , 0 ( | 2 | t n ) , φ x , x , , x ( | 2 | t ) , φ ( x , x , , x ) } }
(3.3)

for allxG. Suppose that an odd mappingf:GXsatisfies the inequality

μ Δ f ( x 1 , , x n ) (t) L φ x 1 , x 2 , , x n (t)
(3.4)

for all x 1 , x 2 ,, x n Gandt>0. Then there exists an additive mappingA:GXsuch that

μ f ( x ) A ( x ) (t) L φ ˜ x ( | 2 | t )
(3.5)

for allxGandt>0, and if

lim lim m min { Φ 2 k x ( | 2 | k t ) : k < m + } = 1 L
(3.6)

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

Proof Letting x 1 =n x 1 , x i =n x 1 (i=2,,n) in (3.4) and using the oddness of f, we obtain that

(3.7)

for all x 1 , x 1 G and t>0. Interchanging x 1 with x 1 in (3.7) and using the oddness of f, we get

(3.8)

for all x 1 , x 1 G and t>0. It follows from (3.7) and (3.8) that

(3.9)

for all x 1 , x 1 G and t>0. Setting x 1 =n x 1 , x 2 =n x 1 , x i =0 (i=3,,n) in (3.4) and using the oddness of f, we get

(3.10)

for all x 1 , x 1 G and t>0. It follows from (3.9) and (3.10) that

(3.11)

for all x 1 , x 1 G and t>0. Putting x 1 =n( x 1 x 1 ), x i =0 (i=2,,n) in (3.4), we obtain

μ f ( n ( x 1 x 1 ) ) f ( ( n 1 ) ( x 1 x 1 ) ) f ( ( x 1 x 1 ) ) (t) L φ n ( x 1 x 1 ) , 0 , , 0 (t)
(3.12)

for all x 1 , x 1 G and t>0. It follows from (3.11) and (3.12) that

μ f ( n ( x 1 x 1 ) ) f ( n x 1 ) + f ( n x 1 ) ( t ) L min { φ n ( x 1 x 1 ) , 0 , , 0 ( t ) , φ n x 1 , n x 1 , 0 , , 0 ( | 2 | n t ) , min { φ n x 1 , n x 1 , , n x 1 ( | 2 | n t ) , φ n x 1 , n x 1 , , n x 1 ( | 2 | n t ) } }
(3.13)

for all x 1 , x 1 G and t>0. Replacing x 1 and x 1 by x n and x n in (3.13), respectively, we obtain

μ f ( 2 x ) 2 f ( x ) (t) L min { φ 2 x , 0 , , 0 ( t ) , min { φ x , x , 0 , , 0 ( | 2 | n t ) , φ x , x , , x ( t ) , φ x , x , , x ( t ) } }

for all xG and t>0. Hence,

μ f ( 2 x ) 2 f ( x ) (t) L Φ x ( | 2 | t )
(3.14)

for all xG and t>0. Replacing x by 2 m 1 x in (3.14), we have

μ f ( 2 m 1 x ) 2 m 1 f ( 2 m x ) 2 m (t) L Φ 2 m 1 x ( | 2 | m t )
(3.15)

for all xG and t>0. It follows from (3.1) and (3.15) that the sequence { f ( 2 m x ) 2 m } is Cauchy. Since X is complete, we conclude that { f ( 2 m x ) 2 m } is convergent. So one can define the mapping A:GX by A(x):= lim m f ( 2 m x ) 2 m for all xG. It follows from (3.14) and (3.15) that

μ f ( x ) f ( 2 m x ) 2 m (t) L min { Φ 2 k x ( | 2 | k + 1 t ) : 0 k < m }
(3.16)

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

μ Δ A ( x 1 , x 2 , , x n ) ( t ) = lim m μ Δ f ( 2 m x 1 , 2 m x 2 , , 2 m x n ) ( | 2 | m t ) L lim m φ 2 m x 1 , 2 m x 2 , , 2 m x n ( | 2 | m t ) = 1 L

for all x 1 , x 2 ,, x n G and t>0. Thus the mapping A satisfies (1.2). By Lemma 3.1, A is additive.

If A is another additive mapping satisfying (3.5), then

μ A ( x ) A ( x ) ( t ) = lim μ A ( 2 x ) A ( 2 x ) ( | 2 | t ) L lim min { μ A ( 2 x ) f ( 2 x ) ( | 2 | t ) , μ f ( 2 x ) Q ( 2 x ) ( | 2 | t ) } L lim lim m min { φ ˜ 2 k x ( | 2 | k + 1 ) : k < m + } = 0

for all xG, thus, A= A . □

Corollary 3.3 Let ρ:[0,)[0,) be a function satisfying

(i) ρ(|2|t)ρ(|2|)ρ(t)for allt0,

(ii) ρ(|2|)<|2|.

Letε>0and let(G,μ, T )be an LRN-space in whichL= D + . Suppose that an odd mappingf:GXsatisfies the inequality

μ Δ f ( x 1 , , x n ) (t) L t t + ε i = 1 n ρ ( x i )

for all x 1 ,, x n Gandt>0. Then there exists a unique additive mappingA:GXsuch that

μ f ( x ) A ( x ) (t) L t t + 2 n | 2 | 2 ε ρ ( x )

for allxGandt>0.

Proof Defining φ: G n D + by φ x 1 , , x n (t):= t t + ε i = 1 n ρ ( x i ) , we have

lim m φ 2 m x 1 , , 2 m x n ( | 2 | m t ) L lim m φ x 1 , , x n ( ( | 2 | ρ ( | 2 | ) ) m ) = 1 L

for all x 1 ,, x n G and t>0. So, we have

φ ˜ x (t):= lim m min { Φ 2 k x ( | 2 | k ) : 0 k < m } = Φ x (t)

and

lim lim m min { Φ 2 k x ( | 2 | k ) : k < m + } = lim Φ 2 x ( | 2 | ) = 1 L

for all xG and t>0. It follows from (3.3) that

Φ x (t)=min { t t + ε ρ ( 2 x ) , t t + 1 | 2 | 2 n ε ρ ( x ) } = | 2 | t | 2 | t + 2 n ε ρ ( x ) .

Applying Theorem 3.2, we conclude that

μ f ( x ) A ( x ) (t) L φ ˜ x ( | 2 | t ) = Φ x ( | 2 | t ) = t t + 2 n | 2 | 2 ε ρ ( x )

for all xG and t>0. □

Lemma 3.4[18]

Let V 1 and V 2 be real vector spaces. If an even mappingf: V 1 V 2 satisfies the functional equation (1.2), then f is quadratic.

In the following theorem, we prove the Hyers-Ulam stability of the functional equation (1.2) in non-Archimedean LRN-spaces for an even mapping case.

Theorem 3.5 Let φ: G n D L + be a function such that

lim m φ 2 m x 1 , 2 m x 2 , , 2 m x n ( | 2 | 2 m t ) = 1 L = lim m φ ˜ 2 m 1 x ( | 2 | 2 m t )
(3.17)

for allx, x 1 , x 2 ,, x n G, t>0and

φ ˜ x (t)= lim m min { φ ˜ 2 k x ( | 2 | 2 k t ) : 0 k < m }
(3.18)

exists for all xG and t>0 where

φ ˜ x ( t ) : = min { φ n x , n x , 0 , , 0 ( | 2 n 2 | t ) , φ n x , 0 , , 0 ( | n 1 | t ) , φ x , ( n 1 ) x , 0 , , 0 ( | n 1 | t ) , Ψ x ( | n 1 | t ) }
(3.19)

and

Ψ x (t):=min { n φ n x , 0 , , 0 ( | 2 | n t ) , φ n x , 0 , , 0 ( | 2 | t ) , φ 0 , n x , , n x ( | 2 | t ) }
(3.20)

for allxGandt>0. Suppose that an even mappingf:GXwithf(0)=0satisfies the inequality (3.4) for all x 1 , x 2 ,, x n Gandt>0. Then there exists a quadratic mappingQ:GXsuch that

μ f ( x ) Q ( x ) (t) L φ ˜ x ( | 2 | 2 t )
(3.21)

for allxG, t>0and if

lim lim m min { φ ˜ 2 k x ( | 2 | 2 k t ) : k < m + } = 1 L
(3.22)

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

Proof Replacing x 1 by n x 1 , and x i by n x 2 (i=2,,n) in (3.4) and using the evenness of f, we obtain

(3.23)

for all x 1 , x 2 G and t>0. Interchanging x 1 with x 2 in (3.23) and using the evenness of f, we obtain

(3.24)

for all x 1 , x 2 G and t>0. It follows from (3.23) and (3.24) that

(3.25)

for all x 1 , x 2 G and t>0. Setting x 1 =n x 1 , x 2 =n x 2 , x i =0 (i=3,,n) in (3.4) and using the evenness of f, we obtain

(3.26)

for all x 1 , x 2 G and t>0. So, it follows from (3.25) and (3.26) that

(3.27)

for all x 1 , x 2 G and t>0. Setting x 1 =x, x 2 =0 in (3.27), we obtain

(3.28)

for all xG and t>0. Putting x 1 =nx, x i =0 (i=2,,n) in (3.4), one obtains

μ f ( n x ) f ( ( n 1 ) x ) ( 2 n 1 ) f ( x ) (t) L φ n x , 0 , , 0 (t)
(3.29)

for all xG and t>0. It follows from (3.28) and (3.29) that

μ f ( n x ) n 2 f ( x ) ( t ) L min { φ n x , 0 , , 0 ( t ) , φ n x , 0 , , 0 ( | 2 | n t ) , φ n x , 0 , , 0 ( | 2 | t ) , φ 0 , n x , , n x ( | 2 | t ) }
(3.30)

for all xG and t>0. Letting x 2 =(n1) x 1 and replacing x 1 by x n in (3.26), we get

μ f ( ( n 1 ) x ) f ( ( n 2 ) x ) ( 2 n 3 ) f ( x ) (t) L φ x , ( n 1 ) x , 0 , , 0 (t)
(3.31)

for all xG and t>0. It follows from (3.28) and (3.31) that

μ f ( ( n 2 ) x ) ( n 2 ) 2 f ( x ) ( t ) L min { φ x , ( n 1 ) x , 0 , , 0 ( t ) , φ n x , 0 , , 0 ( | 2 | n t ) , φ n x , 0 , , 0 ( | 2 | t ) , φ 0 , n x , , n x ( | 2 | t ) }
(3.32)

for all xG and t>0. It follows from (3.30) and (3.32) that

μ f ( n x ) f ( ( n 2 ) x ) 4 ( n 1 ) f ( x ) (t) L min { φ n x , 0 , , 0 ( t ) , φ x , ( n 1 ) x , 0 , , 0 ( t ) , Ψ x ( t ) }
(3.33)

for all xG and t>0. Setting x 1 = x 2 =nx, x i =0 (i=3,,n) in (3.4), we obtain

μ f ( ( n 2 ) x ) + ( n 1 ) f ( 2 x ) f ( n x ) (t) L φ n x , n x , 0 , , 0 ( | 2 | t )
(3.34)

for all xG and t>0. It follows from (3.33) and (3.34) that

(3.35)

for all xG and t>0. Thus,

μ f ( x ) f ( 2 x ) 22 (t) L φ ˜ x ( | 2 | 2 t )
(3.36)

for all xG and t>0. Replacing x by 2 m 1 x in (3.36), we have

μ f ( 2 m 1 x ) 2 2 ( m 1 ) f ( 2 m x ) 2 2 m (t) L φ ˜ 2 m 1 x ( | 2 | 2 m t )
(3.37)

for all xG and t>0. It follows from (3.17) and (3.37) that the sequence { f ( 2 m x ) 2 2 m } is Cauchy. Since X is complete, we conclude that { f ( 2 m x ) 2 2 m } is convergent. So, one can define the mapping Q:GX by Q(x):= lim m f ( 2 m x ) 2 2 m for all xG. By using induction, it follows from (3.36) and (3.37) that

μ f ( x ) f ( 2 m x ) 2 2 m (t) L min { φ ˜ 2 k x ( | 2 | 2 k + 2 t ) : 0 k < m }
(3.38)

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

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

Corollary 3.6 Let η:[0,)[0,) be a function satisfying

(i) η(|l|t)η(|l|)η(t)for allt0,

(ii) η(|l|)< | l | 2 forl{2,n1,n}.

Letε>0and let(G,μ, T )be a LRN-space in whichL= D + . Suppose that an even mappingf:GXwithf(0)=0satisfies the inequality

μ Δ f ( x 1 , , x n ) (t) t t + ε i = 1 n η ( x i )

for all x 1 ,, x n Gandt>0. Then there exists a unique quadratic mappingQ:GXsuch that

μ f ( x ) Q ( x ) (t){ t t + 2 | 2 | 2 ε η ( x ) , if  n = 2 ; t t + n | 2 | 3 | n 1 | ε η ( n x ) , if  n > 2 ,

for allxGandt>0.

Proof Defining φ: G n D + by φ x 1 , , x n (t):= t t + ε i = 1 n η ( x i ) , we have

lim m φ 2 m x 1 , , 2 m x n ( | 2 | 2 m t ) lim m φ x 1 , , x n ( ( | 2 | 2 η ( | 2 | ) ) m ) = 1 L

for all x 1 ,, x n G and t>0. We have

φ ˜ x (t):= lim m min { φ ˜ 2 k x ( | 2 | 2 k t ) : 0 k < m }

and

lim lim m min { φ ˜ 2 k x ( | 2 | 2 k t ) : k < m + } = lim φ ˜ 2 x ( | 2 | 2 t ) =0

for all xG and t>0. It follows from (3.20) that

Ψ x ( t ) = min { | 2 | t | 2 | t + 2 n ε η ( n x ) , | 2 | t | 2 | t + 2 ε η ( n x ) , | 2 | t | 2 | t + 2 ( n 1 ) ε η ( n x ) } = | 2 | t | 2 | t + n ε η ( n x ) .

Hence, by using (3.19), we obtain

φ ˜ x ( t ) = min { | 2 n 2 | t | 2 n 2 | t + 2 ε η ( n x ) , | n 1 | t | n 1 | t + ε η ( n x ) , min { | 2 n 2 | t | 2 n 2 | t + n ε η ( n x ) , | n 1 | t | n 1 | t + ε ( η ( x ) + η ( ( n 1 ) x ) ) } = { t t + 2 ε η ( x ) , if  n = 2 ; | 2 | | n 1 | t | 2 | | n 1 | t + n ε η ( n x ) , if  n > 2 ,

for all xG and t>0. Applying Theorem 3.5, we conclude the required result. □

Lemma 3.7[18]

Let V 1 and V 2 be real vector spaces. A mappingf: V 1 V 2 satisfies (1.2) if and only if there exist a symmetric bi-additive mappingB: V 1 × V 1 V 2 and an additive mappingA: V 1 V 2 such thatf(x)=B(x,x)+A(x)for allx V 1 .

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

Theorem 3.8 Letφ: G n D L + be a function satisfying (3.1) for allx, x 1 , x 2 ,, x n G, and φ ˜ x (t)and φ ˜ x (t)exist for allxGandt>0, where φ ˜ x (t)and φ ˜ x (t)are defined as in Theorems  3.2 and  3.5. Suppose that a mappingf:GXwithf(0)=0satisfies the inequality (3.4) for all x 1 , x 2 ,, x n G. Then there exist an additive mappingA:GXand a quadratic mappingQ:GXsuch that

μ f ( x ) A ( x ) Q ( x ) (t) L min { φ ˜ x ( | 2 | 2 t ) , φ ˜ x ( | 2 | 2 t ) , φ ˜ x ( | 2 | t ) , 1 | 2 | φ ˜ x ( | 2 | t ) }
(3.39)

for allxGandt>0. If

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

Proof Let f e (x)= 1 2 (f(x)+f(x)) for all xG. Then

Δ f e ( x 1 , , x n ) = 1 2 ( Δ f ( x 1 , , x n ) + Δ f ( x 1 , , x n ) ) 1 | 2 | max { φ ( x 1 , , x n ) , φ ( x 1 , , x n ) }

for all x 1 , x 2 ,, x n G and t>0. By Theorem 3.5, there exists a quadratic mapping Q:GX such that

μ f e ( x ) Q ( x ) (t) L min { φ ˜ x ( | 2 | 3 t ) , φ ˜ x ( | 2 | 3 t ) }
(3.40)

for all xG and t>0. Also, let f o (x)= 1 2 (f(x)f(x)) for all xG. By Theorem 3.2, there exists an additive mapping A:GX such that

μ f o ( x ) A ( x ) (t) L min { φ ˜ x ( | 2 | 2 t ) , φ ˜ x ( | 2 | 2 t ) }
(3.41)

for all xG and t>0. Hence (3.39) follows from (3.40) and (3.41).

The rest of proof is trivial. □

Corollary 3.9 Let γ:[0,)[0,) be a function satisfying

(i) γ(|l|t)γ(|l|)γ(t)for allt0,

(ii) γ(|l|)< | l | 2 forl{2,n1,n}.

Letε>0, (G,μ, T )be an LRN-space in whichL= D + and letf:GXsatisfy

μ Δ f ( x 1 , , x n ) (t) t t + ε i = 1 n γ ( x i )

for all x 1 ,, x n G, t>0andf(0)=0. Then there exist a unique additive mappingA:GXand a unique quadratic mappingQ:GXsuch that

μ f ( x ) A ( x ) Q ( x ) (t) | 2 | 3 t | 2 | 3 t + 2 n ε γ ( x )

for allxGandt>0.

Proof The result follows from Corollaries 3.6 and 3.3. □

Authors‘ contributions

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

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Vahidi, J., Park, C. & Saadati, R. A functional equation related to inner product spaces in non-Archimedean L-random normed spaces. J Inequal Appl 2012, 168 (2012). https://doi.org/10.1186/1029-242X-2012-168

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Keywords

  • non-Archimedean spaces
  • additive and quadratic functional equation
  • Hyers-Ulam stability
  • L-random normed spaces