Open Access

Stability results in non-Archimedean L -fuzzy normed spaces for a cubic functional equation

Journal of Inequalities and Applications20122012:193

https://doi.org/10.1186/1029-242X-2012-193

Received: 22 February 2012

Accepted: 31 May 2012

Published: 3 September 2012

Abstract

We establish some stability results concerning the functional equation

n f ( x + n y ) + f ( n x y ) = n ( n 2 + 1 ) 2 [ f ( x + y ) + f ( x y ) ] + ( n 4 1 ) f ( y ) ,

where n 2 is a fixed integer in the setting of non-Archimedean L -fuzzy normed spaces.

MSC:39B52, 46S10, 46S40, 47S10, 47S40.

Keywords

Hyers-Ulam stabilitycubic functional equation

1 Introduction

The theory of fuzzy sets was introduced by Zadeh [34] in 1965. After the pioneering work of Zadeh, there has been a great effort to obtain fuzzy analogues of classical theories. Among other fields, a progressive development has been made in the field of fuzzy topology [1, 8, 9, 1214, 18, 22, 30]. One of the problems in L -fuzzy topology is to obtain an appropriate concept of L -fuzzy metric spaces and L -fuzzy normed spaces. In 2004, Park [23] introduced and studied the notion of intuitionistic fuzzy metric spaces. In 2006, Saadati and Park [28] introduced and studied the notion of intuitionistic fuzzy normed spaces.

On the other hand, the study of stability problems for a functional equation is related to the question of Ulam [33] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [19]. Subsequently, the result of Hyers was generalized by Aoki [2] for additive mappings and by Rassias [24] for linear mappings by considering an unbounded Cauchy difference. We refer the interested readers for more information on such problems to the papers [4, 6, 11, 20, 21, 25, 29, 32, 33].

Let X and Y be real linear spaces and f : X Y a mapping. If X = Y = R , the cubic function f ( x ) = c x 3 , where c is a real constant, clearly satisfies the functional equation
f ( 2 x + y ) + f ( 2 x y ) = 2 f ( x + y ) + 2 f ( x y ) + 12 f ( x ) .
Hence, the above equation is called the cubic functional equation. Recently, Cho, Saadati and Wang [5] introduced the functional equation
3 f ( x + 3 y ) + f ( 3 x y ) = 15 [ f ( x + y ) + f ( x y ) ] + 80 f ( y ) ,

which has f ( x ) = c x 3 ( x R ) as a solution for X = Y = R .

In this paper, we investigate the Hyers-Ulam stability of the functional equation as follows:
n f ( x + n y ) + f ( n x y ) = n ( n 2 + 1 ) 2 [ f ( x + y ) + f ( x y ) ] + ( n 4 1 ) f ( y ) ,
(1.1)

where n 2 is a fixed integer.

2 Preliminaries

In this section, we recall some definitions and results for our main result in this paper.

A triangular norm (shorter t-norm) is a binary operation on the unit interval [ 0 , 1 ] , i.e., a function T : [ 0 , 1 ] × [ 0 , 1 ] [ 0 , 1 ] satisfying the following four axioms: for all a , b , c [ 0 , 1 ] ,
  1. (i)

    T ( a , b ) = T ( b , a ) (commutativity);

     
  2. (ii)

    T ( a , T ( b , c ) ) = T ( T ( a , b ) , c ) (associativity);

     
  3. (iii)

    T ( a , 1 ) = a (boundary condition);

     
  4. (iv)

    T ( a , b ) T ( a , c ) whenever b c (monotonicity).

     
Basic examples are the Łukasiewicz t-norm T L and the t-norms T P , T M and T D , where T L ( a , b ) : = max { a + b 1 , 0 } , T P ( a , b ) : = a b , T M ( a , b ) : = min { a , b } and
T D ( a , b ) : = { min { a , b } if  max { a , b } = 1 , 0 otherwise

for all a , b [ 0 , 1 ] .

For all x [ 0 , 1 ] and all t-norms T, let x T ( 0 ) : = 1 . For all x [ 0 , 1 ] and all t-norms T, define x T ( r ) by the recursion equation x T ( r ) = T ( x T ( r 1 ) , x ) for all r N . A t-norm T is said to be Hadžić type (we denote it by T H ) if the family ( x T ( r ) ) r N is equicontinuous at x = 1 (see [15]).

Other important triangular norms are as follows (see [16]):

  • The Sugeno-Weber family { T λ S W } λ [ 1 , ] is defined by T 1 S W : = T D , T S W : = T P and
    T λ S W ( x , y ) : = max { 0 , x + y 1 + λ x y 1 + λ }

if λ ( 1 , ) .

  • The Domby family { T λ D } λ [ 0 , ] is defined by T D , if λ = 0 , T M , if λ = and
    T λ D ( x , y ) : = 1 1 + [ ( 1 x x ) λ + ( 1 y y ) λ ] 1 / λ

if λ ( 0 , ) .

  • The Aczel-Alsina family { T λ A A } λ [ 0 , ] is defined by T D , if λ = 0 , T M , if λ = and
    T λ A A ( x , y ) : = e ( | log x | λ + | log y | λ ) 1 / λ

if λ ( 0 , ) .

A t-norm T can be extended (by associativity) in a unique way to an r-array operation by taking, for any ( x 1 , , x r ) [ 0 , 1 ] r , the value T ( x 1 , , x r ) defined by
T j = 1 0 x j : = 1 , T j = 1 r x j : = T ( T j = 1 r 1 x j , x r ) = T ( x 1 , , x r ) .
A t-norm T can also be extended to a countable operation by taking, for any sequence ( x r ) r N in [ 0 , 1 ] , the value
T j = 1 x j : = lim r T j = 1 r x j .
(2.1)

The limit on the right side of (2.1) exists since the sequence { T j = 1 r x j } r N is non-increasing and bounded below.

Proposition 2.1 [16]

  1. (1)
    For T T L , the following equivalence holds:
    lim r T j = 1 x r + j = 1 r = 1 ( 1 x r ) < .
     
  2. (2)
    If T is of Hadžić type, then
    lim r T j = 1 x r + j = 1
     
for all sequence { x r } r N in [ 0 , 1 ] such that lim r x r = 1 .
  1. (3)
    If T { T λ A A } λ ( 0 , ) { T λ D } λ ( 0 , ) , then
    lim r T j = 1 x r + j = 1 r = 1 ( 1 x r ) α < .
     
  2. (4)
    If T { T λ S W } λ [ 1 , ) , then
    lim r T j = 1 x r + j = 1 r = 1 ( 1 x r ) < .
     

3 L -fuzzy normed spaces

In this section, we give some definitions and related lemmas for our main result.

Definition 3.1 [10]

Let L = ( L , L ) be a complete lattice and U a non-empty set called a universe. An L -fuzzy set A on U is defined by a mapping A : U L . For any u U , A ( u ) represents the degree (in L) to which u satisfies A .

Lemma 3.2 [7]

Consider the set L and the operation L defined by

for all ( x 1 , x 2 ) , ( y 1 , y 2 ) L . Then ( L , L ) is a complete lattice.

Definition 3.3 [3]

An intuitionistic fuzzy set A ζ , η on a universe U is an object A ζ , η = { ( u , ζ A ( u ) , η A ( u ) ) : u U } , where ζ A ( u ) [ 0 , 1 ] and η A ( u ) [ 0 , 1 ] for all u U are called the membership degree and the non-membership degree, respectively, of u in A ζ , η , and furthermore, they satisfy ζ A ( u ) + η A ( u ) 1 .

In Section 2, we presented the classical definition of t-norms, which can be straight-forwardly extended to any lattice L = ( L , L ) . Define first 0 L : = inf L and 1 L : = sup L .

Definition 3.4 A triangular norm (t-norm) on L is a mapping T : L 2 L satisfying the following conditions:
  1. (i)

    T ( x , 1 L ) = x for all x L (boundary condition);

     
  2. (ii)

    T ( x , y ) = T ( y , x ) for all ( x , y ) L 2 (commutativity);

     
  3. (iii)

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

     
  4. (iv)

    x L x and y L y T ( x , y ) L T ( x , y ) for all ( x , x , y , y ) L 4 (monotonicity).

     
A t-norm can also be defined recursively as an ( r + 1 ) -array operation for each r N by T 1 = T and
T r ( x 1 , , x r + 1 ) = T ( T r 1 ( x 1 , , x r ) , x r + 1 )

for all r 2 and x j L .

The t-norm M defined by
M ( x , y ) = { x if  x L y , y if  y L x

is a continuous t-norm.

Definition 3.5 A t-norm T on L is said to be t-representable if there exist a t-norm T and a t-conorm S on [ 0 , 1 ] such that
T ( x , y ) = ( T ( x 1 , y 1 ) , S ( x 2 , y 2 ) )

for all x = ( x 1 , x 2 ) , y = ( y 1 , y 2 ) L .

Definition 3.6
  1. (1)

    A negator on L is any decreasing mapping N : L L satisfying N ( 0 L ) = 1 L and N ( 1 L ) = 0 L .

     
  2. (2)

    If a negator N on L satisfies N ( N ( x ) ) = x for all x L , then N is called an involution negator.

     
  3. (3)

    The negator N s on ( [ 0 , 1 ] , ) defined as N s ( x ) = 1 x for all x [ 0 , 1 ] is called the standard negator on ( [ 0 , 1 ] , ) .

     
Definition 3.7 The 3-tuple ( V , P , T ) is said to be an L -fuzzy normed space if V is a vector space, T is a continuous t-norm on L and P is an L -fuzzy set on V × ( 0 , ) satisfying the following conditions: for all x , y V and t , s ( 0 , ) ,
  1. (a)

    0 L < L P ( x , t ) ;

     
  2. (b)

    P ( x , t ) = 1 L x = 0 ;

     
  3. (c)

    P ( α x , t ) = P ( x , t | α | ) for all α 0 ;

     
  4. (d)

    T ( P ( x , t ) , P ( y , s ) ) L P ( x + y , t + s ) ;

     
  5. (e)

    P ( x , ) : ( 0 , ) L is continuous;

     
  6. (f)

    lim t 0 P ( x , t ) = 0 L and lim t P ( x , t ) = 1 L .

     

In this case, P is called an L -fuzzy norm. If P = P μ , ν is an intuitionistic fuzzy set and the t-norm T is t-representable, then the 3-tuple ( V , P μ , ν , T ) is said to be an intuitionistic fuzzy normed space.

Definition 3.8 (see [27])

Let ( V , P , T ) be an L -fuzzy normed space.
  1. (1)
    A sequence { x r } r N in ( V , P , T ) is called a Cauchy sequence if, for any ε L { 0 L } and for any t > 0 , there exists a positive integer r 0 such that
    N ( ε ) < L P ( x r + p x r , t )
     
for all r r 0 and p > 0 , where N is a negator on L .
  1. (2)

    A sequence { x r } r N in ( V , P , T ) is said to be convergent to a point x V in the L -fuzzy normed space ( V , P , T ) (denoted by\hspace*{-6pt} \hspace*{-6pt}) if P ( x r x , t ) 1 L wherever r for all t > 0 .

     
  2. (3)

    If every Cauchy sequence in ( V , P , T ) is convergent in V, then the L -fuzzy normed space ( V , P , T ) is said to be complete and the L -fuzzy normed space is called an L -fuzzy Banach space.

     

Lemma 3.9 [26]

Let P be an L -fuzzy norm on V. Then we have the following:
  1. (1)

    P ( x , t ) is non-decreasing with respect to t ( 0 , ) for all x in V.

     
  2. (2)

    P ( x y , t ) = P ( y x , t ) for all x, y in V and all t ( 0 , ) .

     
Definition 3.10 Let ( V , P , T ) be an L -fuzzy normed space. For any t ( 0 , ) , we define the open ball B ( x , r , t ) with center x V and radius r L { 0 L , 1 L } as
B ( x , r , t ) = { y V : N ( r ) < L P ( x y , t ) } .

A subset A V is called open if, for all x A , there exist t > 0 and r L { 0 L , 1 L } such that B ( x , r , t ) A .

Let τ P denote the family of all open subsets of V. Then τ P is called the topology induced by the L -fuzzy norm P .

4 L -fuzzy Hyers-Ulam stability for cubic functional equations in non-Archimedean L -fuzzy normed spaces

In 1897, Hensel [17] introduced the field with valuation in which does not have the Archimedean property.

Definition 4.1 Let K be a field. A non-Archimedean absolute value on K is a function | | : K [ 0 , ) such that, for any a , b K ,
  1. (1)

    | a | 0 and equality holds if and only if a = 0 ,

     
  2. (2)

    | a b | = | a | | b | ,

     
  3. (3)

    | a + b | max { | a | , | b | } (the strict triangle inequality).

     

Note that | n | 1 for each integer n. We always assume, in addition, that | | is non-trivial, i.e., there is a 0 K such that | a 0 | { 0 , 1 } .

Definition 4.2 A non-Archimedean L -fuzzy normed space is a triple ( V , P , T ) , where V is a vector space, T is a continuous t-norm on L and P is an L -fuzzy set on V × ( 0 , ) satisfying the following conditions: for all x , y V and t , s ( 0 , ) ,
  1. (a)

    0 L < L P ( x , t ) ;

     
  2. (b)

    P ( x , t ) = 1 L x = 0 ;

     
  3. (c)

    P ( α x , t ) = P ( x , t | α | ) for all α 0 ;

     
  4. (d)

    T ( P ( x , t ) , P ( y , s ) ) L P ( x + y , max { t , s } ) ;

     
  5. (e)

    P ( x , ) : ( 0 , ) L is continuous;

     
  6. (f)

    lim t 0 P ( x , t ) = 0 L and lim t P ( x , t ) = 1 L .

     

From now on, let K be a non-Archimedean field, X a vector space over K and ( Y , P , T ) a non-Archimedean L -fuzzy Banach space over K . We investigate the Hyers-Ulam stability of the cubic functional equation (1.1).

Next, we define an L -fuzzy approximately cubic mapping. Let Ψ be an L -fuzzy set on X × X × ( 0 , ) such that Ψ ( x , y , ) is non-decreasing,
Ψ ( c x , c x , t ) L Ψ ( x , x , t | c | )
and
lim t Ψ ( x , y , t ) = 1 L

for all x , y X , all t > 0 and all c K { 0 } .

Definition 4.3 A mapping f : X Y is said to be Ψ-approximately cubic if
(4.1)

for all x , y X and all t > 0 .

The following is the main result in this paper.

Theorem 4.4 Let f : X Y be a Ψ-approximately cubic mapping. If n K { 0 } and there are α ( 0 , ) and an integer k ( k 2 ) with | n | k < α such that
Ψ ( n k x , n k y , t ) L Ψ ( x , y , α t )
(4.2)
and
lim r T j = r M ( x , α j t | n | k j ) = 1 L
for all x , y X and all t > 0 , then there exists a unique cubic mapping C : X Y such that
P ( f ( x ) C ( x ) , t ) L T j = 1 M ( x , α j + 1 | n | k j t )
(4.3)
for all x X and all t > 0 , where
M ( x , t ) : = T ( Ψ ( x , 0 , t ) , Ψ ( n x , 0 , t ) , , Ψ ( n k 1 x , 0 , t ) )

for all x X and all t > 0 .

Proof Let T 0 : L L be the identity mapping I L : L L . First, we show, by induction on j, that
P ( f ( n j x ) n 3 j f ( x ) , t ) L M j ( x , t ) : = T j 1 ( Ψ ( x , 0 , t ) , , Ψ ( n j 1 x , 0 , t ) )
(4.4)
for all x X , all t > 0 and all j 1 . Putting y = 0 in (4.1), we obtain
P ( f ( n x ) n 3 f ( x ) , t ) L Ψ ( x , 0 , t )
for all x X and all t > 0 . This proves (4.4) for j = 1 . Let (4.4) hold for some j 1 . Replacing y by 0 and x by 3 j x in (4.1), we get
P ( f ( n j + 1 x ) n 3 f ( n j x ) , t ) L Ψ ( n j x , 0 , t )
for all x X and all t > 0 . Since | n | 3 1 , it follows that
for all x X and all t > 0 . Thus (4.4) holds for all j 1 . In particular, we have
P ( f ( n k x ) n 3 k f ( x ) , t ) L M ( x , t )
for all x X and all t > 0 . Replacing x by x n k ( r + 1 ) in the above inequality and using the inequality (4.2), we obtain
P ( f ( x n k r ) n 3 k f ( x n k ( r + 1 ) ) , t ) L M ( x n k ( r + 1 ) , t ) L M ( x , α r + 1 t )
for all x X , all t > 0 and all r 0 , and so
for all x X , all t > 0 and all r 0 . Hence it follows that
for all x X , all t > 0 and all r 0 . Since lim r T j = r M ( x , α j + 1 | n | k j t ) = 1 L for all x X and all t > 0 , { n 3 k r f ( x n k r ) } r N is a Cauchy sequence in the non-Archimedean L -fuzzy Banach space ( Y , P , T ) . Hence we can define a mapping C : X Y such that
lim r P ( n 3 k r f ( x n k r ) C ( x ) , t ) = 1 L
(4.5)
for all x X and all t > 0 . Next, for all r 1 , all x X and all t > 0 , we have
P ( f ( x ) n 3 k r f ( x n k r ) , t ) = P ( j = 0 r 1 [ n 3 k j f ( x n k j ) n 3 k ( j + 1 ) f ( x n k ( j + 1 ) ) ] , t ) L T j = 0 r 1 P ( n 3 k j f ( x n k j ) n 3 k ( j + 1 ) f ( x n k ( j + 1 ) ) , t ) L T j = 0 r 1 M ( x , α j + 1 | n | k j t )
for all x X and all t > 0 , and so
P ( f ( x ) C ( x ) , t ) L T ( P ( f ( x ) n 3 k r f ( x n k r ) , t ) , P ( n 3 k r f ( x n k r ) C ( x ) , t ) ) L T ( T j = 0 r 1 M ( x , α j + 1 | n | k j t ) , P ( n 3 k r f ( x n k r ) C ( x ) , t ) )
for all x X and all t > 0 . Taking the limit as r in the above inequality, we obtain
P ( f ( x ) C ( x ) , t ) L T j = 0 M ( x , α j + 1 | n | k j t )

for all x X and all t > 0 , which proves (4.3).

Since T is continuous, from the well-known result in an L -fuzzy (probabilistic) normed space (see [31], Chapter 12), it follows that
for all x , y X and all t > 0 . On the other hand, replacing x, y by n k r x , n k r y in (4.1) and (4.2), we get

for all x , y X and all t > 0 . Since lim r Ψ ( x , y , α r t | n | 3 k r ) = 1 L , we infer that C is a cubic mapping.

For the uniqueness of C, let C : X Y be another cubic mapping such that
P ( C ( x ) f ( x ) , t ) L M ( x , t )
for all x X and all t > 0 . Then we have, for all x , y X and all t > 0 ,

Therefore, from (4.5), we conclude that C = C . This completes the proof. □

Corollary 4.5 Let T H and let f : X Y be a Ψ-approximately cubic mapping. If n K { 0 } and there are α ( 0 , ) and an integer k ( k 2 ) with | n | k < α such that
Ψ ( n k x , n k y , t ) L Ψ ( x , y , α t )
and
lim r M ( x , α r t | n | k r ) = 1 L
for all x , y X and all t > 0 , then there exists a unique cubic mapping C : X Y such that
P ( f ( x ) C ( x ) , t ) L T j = 1 M ( x , α j + 1 | n | k j t )
for all x X and all t > 0 , where
M ( x , t ) : = T ( Ψ ( x , 0 , t ) , Ψ ( n x , 0 , t ) , , Ψ ( n k 1 x , 0 , t ) )

for all x X and all t > 0 .

Proof Since
lim r M ( x , α r | n | k r t ) = 1 L
for all x X and all t > 0 and T is of H a d z ˇ i c ´ type, it follows from Proposition 2.1 that
lim r T j = r + 1 M ( x , α j | n | k j t ) = 1 L

for all x X and all t > 0 . Therefore, if we apply Theorem 4.4, then we get the conclusion. This completes the proof. □

Example 4.6 Let ( X , ) be a non-Archimedean Banach space, ( X , P μ , ν , T M ) a non-Archimedean L -fuzzy normed space (intuitionistic fuzzy normed space) in which
P μ , ν ( x , t ) = ( t t + x , x t + x )
for all x X and all t > 0 and ( Y , P μ , ν , T M ) a complete non-Archimedean L -fuzzy normed space (intuitionistic fuzzy normed space). Define
Ψ ( x , y , t ) = ( t 1 + t , 1 1 + t )
for all x , y X and all t > 0 . It is easy to see that (4.2) holds for α = 1 . Also, since
M ( x , t ) = ( t 1 + t , 1 1 + t )
for all x X and all t > 0 , we have
lim r ( T M ) j = r M ( x , α j | n | k j t ) = lim r [ lim s ( T M ) j = r s M ( x , α j | n | k j t ) ] = lim r lim s ( t t + | n | k j , | n | k r t + | n | k r ) = ( 1 , 0 ) = 1 L
for all x X and all t > 0 . Let f : X Y be a Ψ-approximately cubic mapping. A straightforward computation shows that
for all x , y X and all t > 0 . Therefore, all the conditions of Theorem 4.4 hold, and so there exists a unique cubic mapping C : X Y such that
P μ , ν ( f ( x ) C ( x ) , t ) L ( t t + | n | k , | n | k t + | n | k )

for all x X and all t > 0 .

Declarations

Acknowledgements

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant number 2012003499).

Authors’ Affiliations

(1)
Graduate School of Education, Kyung Hee University
(2)
Department of Mathematics, Chungnam National University
(3)
Department of Mathematics Education, College of Education, Mokwon University

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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.