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Stability results in -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 -fuzzy normed spaces that in turn generalize a Hyers-Ulam stability result in the framework of classical normed spaces.

MSC:39B22, 39B52, 39B72, 46S40, 47S40.

1 Introduction and preliminaries

The theory of fuzzy sets was introduced by Zadeh [1] 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 is made in the field of fuzzy topology [29]. One of problems in -fuzzy topology is to obtain an appropriate concept of -fuzzy metric spaces and -fuzzy normed spaces. In 1984, Katsaras [10] defined a fuzzy norm on a linear space, and at the same year Wu and Fang [11] also introduced a fuzzy normed space and gave the generalization of the Kolmogoroff normalized theorem for a fuzzy topological linear space. Some mathematicians have defined fuzzy metrics and norms on a linear space from various points of view [2, 3, 9, 1214]. In 1994, Cheng and Mordeson introduced the definition of a fuzzy norm on a linear space in such a manner that the corresponding induced fuzzy metric is of Kramosil and Michalek type [15]. In 2003, Bag and Samanta [16] modified the definition of Cheng and Mordeson [17] by removing a regular condition. In 2004, Park [18] introduced and studied the notion of intuitionistic fuzzy metric spaces. In 2006, Saadati and Park 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 a question of Ulam [19], concerning the stability of group homomorphisms, affirmatively answered for Banach spaces by Hyers [20]. Subsequently, the result of Hyers was generalized by Aoki [21] for additive mappings and by Rassias [22] for linear mappings by considering an unbounded Cauchy difference. We refer the interested readers for more information on such problems to the papers [19, 2330].

Let X and Y be real linear spaces and let f:XY be 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 [31] 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.

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 ( n ) by the recursion equation x T ( n ) =T( x T ( n 1 ) ,x) for all nN. A t-norm T is said to be of Hadžić type (we denote it by TH) if the family ( x T ( n ) ) n N is equicontinuous at x=1 (see [12]).

Other important triangular norms are as follows (see [32]):

  • The Sugeno-Weber family { T λ SW } λ [ 1 , ] is defined by T 1 SW := T D , T SW := T P and

    T λ SW (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 AA } λ [ 0 , ] is defined by T D , if λ=0, T M , if λ= and

    T λ AA (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 n-array operation by taking, for any ( x 1 ,, x n ) [ 0 , 1 ] 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 ).

A t-norm T can also be extended to a countable operation by taking, for any sequence ( x n ) n N in [0,1], the value

T i = 1 x i := lim n T i = 1 n x i .
(1.2)

The limit on the right-hand side of (1.2) exists since the sequence { T i = 1 n x i } n N is non-increasing and bounded from below.

Proposition 1.1 [32]

  1. (1)

    For T T L , the following equivalence holds:

    lim n T i = 1 x n + i =1 n = 1 (1 x n )<.
  2. (2)

    If T is of Hadžić type, then

    lim n T i = 1 x n + i =1

for all sequence { x n } n N in [0,1] such that lim n x n =1.

  1. (3)

    If T { T λ AA } λ ( 0 , ) { T λ D } λ ( 0 , ) , then

    lim n T i = 1 x n + i =1 n = 1 ( 1 x n ) α <.
  2. (4)

    If T { T λ SW } λ [ 1 , ) , then

    lim n T i = 1 x n + i =1 n = 1 (1 x n )<.

We give some definitions and related lemmas for our main result.

Definition 1.2 [17]

Let L=(L, L ) be a complete lattice and U a non-empty set called universe. An -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 1.3 [16]

Consider the set L and the operation L defined by

L = { ( x 1 , x 2 ) : ( x 1 , x 2 ) [ 0 , 1 ] 2  and  x 1 + x 2 1 } , ( x 1 , x 2 ) L ( y 1 , y 2 ) x 1 y 1 , x 2 y 2

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

Definition 1.4 [33]

An intuitionistic fuzzy set A ζ , η on a universe U is an object A ζ , η ={( 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, further, they satisfy ζ A (u)+ η A (u)1.

We presented the classical definition of t-norms which can be straightforwardly extended to any lattice L=(L, L ). Define first 0 L :=infL and 1 L :=supL.

Definition 1.5 A triangular norm (t-norm) on is a mapping T: L 2 L satisfying the following conditions:

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

  4. (iv)

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

A t-norm can also be defined recursively an (n+1)-array operation for each nN{0} by T 1 =T and

T n ( x ( 1 ) ,, x ( n + 1 ) )=T ( T n 1 ( x ( 1 ) , , x ( n ) ) , x ( n + 1 ) )

for all n2 and x ( i ) L.

The t-norm defined by

M(x,y)= { x , if  x L y , y , if  y L x ,

is a continuous t-norm.

Definition 1.6 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 1.7

  1. (1)

    A negator on 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 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 1.8 The 3-tuple (V,P,T) is said to be an -fuzzy normed space if V is a vector space, T is a continuous t-norm on and P is an -fuzzy set on V×(0,) satisfying the following conditions: for all x,yV and t,s(0,),

  1. (i)

    0 L < L P(x,t);

  2. (ii)

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

  3. (iii)

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

  4. (iv)

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

  5. (v)

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

  6. (vi)

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

In this case, P is called an -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 1.9 (see [15])

Let (V,P,T) be an -fuzzy normed space.

  1. (1)

    A sequence { x n } n 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 n 0 such that

    N(ε) < L P( x n + p x n ,t)

for all n n 0 and p>0, where N is a negator on .

  1. (2)

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

  2. (3)

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

Lemma 1.10 Let P be an -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 1.11 Let (V,P,T) be an -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 -fuzzy norm P.

2 Main results

In this section, we study the stability of the functional equation (1.1) in -fuzzy normed spaces.

Theorem 2.1 Let X be a linear space and (Y,P,T) a complete -fuzzy normed space. Let f:XY be a mapping with f(0)=0 and Q be an -fuzzy set on X 2 ×(0,) satisfying

P ( 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 ) , t ) L Q ( x , y , t )
(2.1)

for all x,yX and all t>0. If

T i = 1 ( Q ( n r + i 1 x , n r + i 1 y , n 3 r + 2 i + 1 t ) ) = 1 L

and

lim n Q ( n r x , n r y , n 3 r t ) = 1 L

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

P ( f ( x ) C ( x ) , t ) L T i = 1 ( Q ( n i 1 x , 0 , n 2 i + 1 t ) )
(2.2)

for all xX and t>0.

Proof Putting y=0 in (2.1), we have

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

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

P ( f ( n k + 1 x ) n 3 ( k + 1 ) f ( n k x ) n 3 k , t n 3 k ) L Q ( n k x , 0 , n 3 t ) ,

which implies that

P ( f ( n k + 1 x ) n 3 ( k + 1 ) f ( n k x ) n 3 k , t ) L Q ( n k x , 0 , n 3 ( k + 1 ) t ) ,

that is,

P ( f ( n k + 1 x ) n 3 ( k + 1 ) f ( n k x ) n 3 k , t n k + 1 ) L Q ( n k x , 0 , n 2 ( k + 1 ) t )

for all xX, t>0 and all kN. Since n2, we get 1> 1 n ++ 1 n r for all rN. By the triangle inequality, it follows that

P ( f ( n r x ) n 3 r f ( x ) , t ) L P ( f ( n r x ) n 3 r f ( x ) , k = 0 r 1 t n k + 1 ) L T k = 0 r 1 ( P ( f ( n k + 1 x ) n 3 ( k + 1 ) f ( n k x ) n 3 k , t n k + 1 ) ) L T k = 1 r ( Q ( n k 1 x , 0 , n 2 k t ) )
(2.3)

for all xX, t>0 and all rN.

In order to prove the convergence of the sequence { f ( n r x ) n 3 r }, we replace x with n m x in (2.3) to find that for all m,r>0,

P ( f ( n r + m x ) n 3 ( r + m ) f ( n m x ) n 3 m , t ) L T k = 1 r ( Q ( n k + m 1 x , 0 , n 2 k + 3 m t ) )

for all xX, t>0 and all rN. Since the right-hand side of the inequality tend to 1 L as m tends to infinity, the sequence { f ( n r x ) n 3 r } is a Cauchy sequence. Thus, we may define

C(x):= lim r f ( n r x ) n 3 r

for all xX.

Now we show that C is a cubic mapping. Replacing x, y with n r x and n r y, respectively, in (2.1), it follows that

P ( n f ( n r ( x + n y ) ) n 3 r + f ( n r ( n x y ) ) n 3 r n ( n 2 + 1 ) f ( n r ( x + y ) ) 2 n 3 r n ( n 2 + 1 ) f ( n r ( x y ) ) 2 n 3 r ( n 4 1 ) f ( n r y ) n 3 r , t ) L Q ( n r x , n r y , n 3 r t )

for all x,yX, t>0 and all rN. Taking the limit as r, we find that C satisfies (1.1) for all x,yX.

To prove (2.2), taking the limit as r in (2.3), we have (2.2).

Finally, to prove the uniqueness of the cubic mapping C subject to (2.2), let us assume that there exists another cubic mapping C which satisfies (2.2). Obviously, we have

C ( n r x ) = n 3 r C(x), C ( n r x ) = n 3 r C (x)

for all xX and all nN. Hence, it follows from (2.2) that

P ( C ( x ) C ( x ) , t ) = P ( C ( n r x ) C ( n r x ) , n 3 r t ) L T ( P ( C ( n r x ) f ( n r x ) , n 3 r 1 t ) , P ( f ( n r x ) C ( n r x ) , n 3 r 1 t ) ) L T ( T i = 1 ( Q ( n r + i 1 x , 0 , n 3 r + 2 i t ) ) , T i = 1 ( Q ( n r + i 1 x , 0 , n 3 r + 2 i t ) ) ) = T ( 1 L , 1 L ) = 1 L

for all xX and all t>0. This proves the uniqueness of C and completes the proof. □

Corollary 2.2 Let (X, P ,T) be an -fuzzy normed space and (Y,P,T) be a complete -fuzzy normed space. If f:XY is a mapping such that

P ( 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 ) , t ) L P ( x + y , t )

for all x,yX and all t>0. If

T i = 1 ( P ( x + y , n 2 r + i + 2 t ) ) = 1 L

and

lim n P ( x + y , n 2 r t ) = 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 i = 1 ( P ( x , n i + 2 t ) )

for all xX and all t>0.

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Acknowledgements

This research was supported by the 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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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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Lee, SB., Park, WG. & Bae, JH. Stability results in -fuzzy normed spaces for a cubic functional equation. J Inequal Appl 2012, 249 (2012). https://doi.org/10.1186/1029-242X-2012-249

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