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

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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 ||:KR such that for any r,sK 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 nN.

Throughout this paper, we assume that K is a valued field, X and Y are vector spaces over K, a,bK are fixed with λ:= a 2 + b 2 0,1 ( λ 1 :=2a0,1 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 [30] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [13]. The result of Hyers was generalized by Aoki [2] for approximate additive mappings and by Rassias [27] for approximate linear mappings by allowing the Cauchy difference operator CDf(x,y)=f(x+y)[f(x)+f(y)] to be controlled by ϵ( x p + y p ). In 1994, a further generalization was obtained by Găvruţa [9], 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, rK, xX;

  3. (iii)

    the strong triangle inequality

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

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 [21]).

A triangular norm (shorter t-norm, [29]) 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 ac and bd 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+b1,0}, 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 mN. 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 TH) if a family of functions { T i = 1 m (t)} for all mN is equicontinuous at t=1, that is, for all ε(0,1) there exists δ(0,1) such that

t>1δ T i = 1 m (t)>1ε for all  m N .

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

Proposition 2.3 (see [12])

  1. (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 [22])

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,yX and all s,tR:

(N1) N(x,t)=0 for all t0;

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

(N3) N(cx,t)=N(x,t/|c|) if c0;

(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,yX, 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), xX, 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 xX 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 [22])

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,yΩ; (3) d(x,y)d(x,z)+d(y,z), x,y,zΩ.

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

Theorem 2.8 Let (Ω,d) be a complete generalized metric space and J:ΩΩ be a strictly contractive mapping with Lipschitz constant 0<L<1, that is

d(Jx,Jy)Ld(x,y),x,yΩ.

If there exists a nonnegative integer m 0 such that d( J m 0 x, J m 0 + 1 x)< for an xΩ, 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, jN{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 kN with k1, 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<i (the same arguments apply to the case where k>i). From (3.1) and (3.4), it follows that

Hence the mapping Q i is a general multi-Euler-Lagrange quadratic mapping. Let us finally assume that Q i : 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, jN{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 kN 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,bN 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 [7]. 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 TH. 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 [19]) shows that (Ω, d i ) is a complete generalized metric space. We now define an operator J i :ΩΩ 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 ),gΩ, x 1 ,, x n X.

Let g,hΩ 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 ={gΩ: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:CC be defined by

ϕ(x)={ x 2 , for  | x | < 1 , 1 , for  | x | 1 .

Consider the function f:CC be defined by

f(x)= j = 0 ϕ ( α j x ) α 2 j

for all xC, where α>max{|a|,|b|,1}. Then f satisfies the functional inequality

(4.5)

for all x,yC, but there do not exist a general multi-Euler-Lagrange quadratic function Q:CC and a constant d>0 such that |f(x)Q(x)|d | x | 2 for all xC.

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 k1 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 |bxay|<1, α l |x|<1, α l |y|<1

for all l=0,1,,k1. 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:CC and a constant d>0 such that |f(x)Q(x)|d | x | 2 for all xC. Then there exists a constant cC 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 sN 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 ,

which contradicts (4.7).

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 [14]). For a=b=1 and n=1, Corollary 4.8 yields the main theorem in [17]. The generalized Hyers-Ulam stability problem for the case of r=p+q=2 was excluded in Corollary 4.8 (see[10]).

Note that by (4.4) one can get

Letting t 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.

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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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Keywords

  • stability of general multi-Euler-Lagrange quadratic functional equation
  • direct method
  • fixed point method
  • non-Archimedean fuzzy normed space