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

Abstract

We establish some stability results concerning the functional equation

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

where n2 is a fixed integer in the setting of non-Archimedean L-fuzzy normed spaces.

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

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:XY 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(2x+y)+f(2xy)=2f(x+y)+2f(xy)+12f(x).

Hence, the above equation is called the cubic functional equation. Recently, Cho, Saadati and Wang [5] introduced the functional equation

3f(x+3y)+f(3xy)=15 [ f ( x + y ) + f ( x y ) ] +80f(y),

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

In this paper, we investigate the Hyers-Ulam stability of the functional equation as follows:

nf(x+ny)+f(nxy)= n ( n 2 + 1 ) 2 [ f ( x + y ) + f ( x y ) ] + ( n 4 1 ) f(y),
(1.1)

where n2 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 bc (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+b1,0}, T P (a,b):=ab, 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 rN. A t-norm T is said to be Hadžić type (we denote it by TH) 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:UL. For any uU, 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)):uU}, where ζ A (u)[0,1] and η A (u)[0,1] for all uU 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 :=infL and 1 L :=supL.

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 xL (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 rN 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 r2 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:LL 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 xL, then N is called an involution negator.

  3. (3)

    The negator N s on ([0,1],) defined as N s (x)=1x 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,yV and t,s(0,),

  1. (a)

    0 L < L P(x,t);

  2. (b)

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

  3. (c)

    P(αx,t)=P(x, t | α | ) for all α0;

  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 εL{ 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 xV 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(xy,t)=P(yx,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 xV and radius rL{ 0 L , 1 L } as

B(x,r,t)= { y V : N ( r ) < L P ( x y , t ) } .

A subset AV is called open if, for all xA, there exist t>0 and rL{ 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,bK,

  1. (1)

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

  2. (2)

    |ab|=|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,yV and t,s(0,),

  1. (a)

    0 L < L P(x,t);

  2. (b)

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

  3. (c)

    P(αx,t)=P(x, t | α | ) for all α0;

  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(cx,cx,t) L Ψ ( x , x , t | c | )

and

lim t Ψ(x,y,t)= 1 L

for all x,yX, all t>0 and all cK{0}.

Definition 4.3 A mapping f:XY is said to be Ψ-approximately cubic if

(4.1)

for all x,yX and all t>0.

The following is the main result in this paper.

Theorem 4.4 Let f:XY be a Ψ-approximately cubic mapping. If nK{0} and there are α(0,) and an integer k (k2) with | n | k <α such that

Ψ ( n k x , n k y , t ) L Ψ(x,y,αt)
(4.2)

and

lim r T j = r M ( x , α j t | n | k j ) = 1 L

for all x,yX and all t>0, then there exists a unique cubic mapping C:XY 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 xX 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 xX and all t>0.

Proof Let T 0 :LL be the identity mapping I L :LL. 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 xX, all t>0 and all j1. Putting y=0 in (4.1), we obtain

P ( f ( n x ) n 3 f ( x ) , t ) L Ψ(x,0,t)

for all xX and all t>0. This proves (4.4) for j=1. Let (4.4) hold for some j1. 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 xX and all t>0. Since | n | 3 1, it follows that

for all xX and all t>0. Thus (4.4) holds for all j1. In particular, we have

P ( f ( n k x ) n 3 k f ( x ) , t ) L M(x,t)

for all xX 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 xX, all t>0 and all r0, and so

for all xX, all t>0 and all r0. Hence it follows that

for all xX, all t>0 and all r0. Since lim r T j = r M(x, α j + 1 | n | k j t)= 1 L for all xX 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:XY such that

lim r P ( n 3 k r f ( x n k r ) C ( x ) , t ) = 1 L
(4.5)

for all xX and all t>0. Next, for all r1, all xX 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 xX 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 xX 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 xX 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,yX 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,yX 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 :XY be another cubic mapping such that

P ( C ( x ) f ( x ) , t ) L M(x,t)

for all xX and all t>0. Then we have, for all x,yX and all t>0,

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

Corollary 4.5 Let TH and let f:XY be a Ψ-approximately cubic mapping. If nK{0} and there are α(0,) and an integer k (k2) with | n | k <α such that

Ψ ( n k x , n k y , t ) L Ψ(x,y,αt)

and

lim r M ( x , α r t | n | k r ) = 1 L

for all x,yX and all t>0, then there exists a unique cubic mapping C:XY such that

P ( f ( x ) C ( x ) , t ) L T j = 1 M ( x , α j + 1 | n | k j t )

for all xX 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 xX and all t>0.

Proof Since

lim r M ( x , α r | n | k r t ) = 1 L

for all xX and all t>0 and T is of Had 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 xX 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 xX 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,yX 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 xX 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 xX and all t>0. Let f:XY be a Ψ-approximately cubic mapping. A straightforward computation shows that

for all x,yX and all t>0. Therefore, all the conditions of Theorem 4.4 hold, and so there exists a unique cubic mapping C:XY such that

P μ , ν ( f ( x ) C ( x ) , t ) L ( t t + | n | k , | n | k t + | n | k )

for all xX and all t>0.

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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).

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Correspondence to Won-Gil Park.

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The authors declare that they have no competing interests.

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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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Bae, JH., Lee, SB. & Park, WG. Stability results in non-Archimedean L-fuzzy normed spaces for a cubic functional equation. J Inequal Appl 2012, 193 (2012). https://doi.org/10.1186/1029-242X-2012-193

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Keywords

  • Hyers-Ulam stability
  • cubic functional equation