Open Access

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

Journal of Inequalities and Applications20122012:249

https://doi.org/10.1186/1029-242X-2012-249

Received: 20 April 2012

Accepted: 11 October 2012

Published: 29 October 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 -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.

Keywords

Hyers-Ulam stabilitycubic function-fuzzy normed space

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 : X Y 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 ( 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 [31] 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.

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 ( n ) by the recursion equation x T ( n ) = T ( x T ( n 1 ) , x ) for all n N . A t-norm T is said to be of Hadžić type (we denote it by T H ) 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 : U L . For any u U , 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 ) ) : 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, 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 : = inf L and 1 L : = sup L .

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

    ( x L ) ( 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 n N { 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 n 2 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 : L L 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 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 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 , y V and t , s ( 0 , ) ,
  1. (i)

    0 L < L P ( x , t ) ;

     
  2. (ii)

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

     
  3. (iii)

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

     
  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 ε L { 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 x V 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 ( x y , t ) = P ( y x , 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 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 -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 : X Y 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 , y X 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 , y X and t > 0 , then there exists a unique cubic mapping C : X Y 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 x X 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 x X 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 x X , t > 0 and all k N . Since n 2 , we get 1 > 1 n + + 1 n r for all r N . 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 x X , t > 0 and all r N .

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 x X , t > 0 and all r N . 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 x X .

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

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 x X and all n N . 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 x X 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 : X Y 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 , y X 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 , 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 i = 1 ( P ( x , n i + 2 t ) )

for all x X and all t > 0 .

Declarations

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).

Authors’ Affiliations

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

References

  1. Zadeh LA: Fuzzy sets. Inf. Control 1965, 8: 338–353. 10.1016/S0019-9958(65)90241-XMathSciNetView ArticleMATHGoogle Scholar
  2. George A, Veeramani P: On some result in fuzzy metric space. Fuzzy Sets Syst. 1994, 64: 395–399. 10.1016/0165-0114(94)90162-7MathSciNetView ArticleMATHGoogle Scholar
  3. George A, Veeramani P: On some result of analysis for fuzzy metric spaces. Fuzzy Sets Syst. 1997, 90: 365–368. 10.1016/S0165-0114(96)00207-2MathSciNetView ArticleMATHGoogle Scholar
  4. Gregory V, Romaguera S: Some properties of fuzzy metric spaces. Fuzzy Sets Syst. 2000, 115: 485–489. 10.1016/S0165-0114(98)00281-4View ArticleMathSciNetMATHGoogle Scholar
  5. Gregory V, Romaguera S: On completion of fuzzy metric spaces. Fuzzy Sets Syst. 2002, 130: 399–404. 10.1016/S0165-0114(02)00115-XView ArticleMathSciNetMATHGoogle Scholar
  6. Gregory V, Romaguera S: Characterizing completable fuzzy metric spaces. Fuzzy Sets Syst. 2004, 144: 411–420. 10.1016/S0165-0114(03)00161-1View ArticleMathSciNetMATHGoogle Scholar
  7. Hu C: C -structure of FTS. V: fuzzy metric spaces. J. Fuzzy Math. 1995, 3: 711–721.MathSciNetMATHGoogle Scholar
  8. Lowen R: Fuzzy Set Theory. Kluwer Academic, Dordrecht; 1996.View ArticleMATHGoogle Scholar
  9. Schweizer B, Sklar A: Statistical metric spaces. Pac. J. Math. 1960, 10: 314–334.MATHGoogle Scholar
  10. Katsaras AK: Fuzzy topological vector spaces II. Fuzzy Sets Syst. 1984, 12: 143–154. 10.1016/0165-0114(84)90034-4MathSciNetView ArticleMATHGoogle Scholar
  11. Wu C, Fang J: Fuzzy generalization of Kolmogoroff’s theorem. J. Harbin Inst. Technol. 1984, 1: 1–7. (in Chinese, English abstract)Google Scholar
  12. Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, Dordrecht; 2001.MATHGoogle Scholar
  13. Amini M, Saadati R: Topics in fuzzy metric space. J. Fuzzy Math. 2003, 4: 765–768.MathSciNetMATHGoogle Scholar
  14. Gregory V, Romaguera S: Some properties of fuzzy metric spaces. Fuzzy Sets Syst. 2000, 115: 485–489. 10.1016/S0165-0114(98)00281-4View ArticleMathSciNetMATHGoogle Scholar
  15. Saadati R, Park C: Non-Archimedean -fuzzy normed spaces and stability of functional equations. Comput. Math. Appl. 2010, 60: 2488–2496. 10.1016/j.camwa.2010.08.055MathSciNetView ArticleMATHGoogle Scholar
  16. Deschrijver G, Kerre EE: On the relationship between some extensions of fuzzy set theory. Fuzzy Sets Syst. 2003, 123: 227–235.MathSciNetView ArticleMATHGoogle Scholar
  17. Goguen J: -fuzzy sets. J. Math. Anal. Appl. 1967, 18: 145–174. 10.1016/0022-247X(67)90189-8MathSciNetView ArticleMATHGoogle Scholar
  18. Park J: Intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2004, 22: 1039–1046. 10.1016/j.chaos.2004.02.051MathSciNetView ArticleMATHGoogle Scholar
  19. Ulam SM Science Editions. In Problems in Modern Mathematics. Wiley, New York; 1964. Chapter VIGoogle Scholar
  20. Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar
  21. Aoki T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 1950, 2: 64–66. 10.2969/jmsj/00210064View ArticleMathSciNetMATHGoogle Scholar
  22. Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1View ArticleMathSciNetMATHGoogle Scholar
  23. Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge; 2002.View ArticleMATHGoogle Scholar
  24. Ebadian A, Ghobadipour N, Savadkouhi MB, Eshaghi Gordji M: Stability of a mixed type cubic and quartic functional equation in non-Archimedean -fuzzy normed spaces. Thai J. Math. 2011, 9: 243–259.MathSciNetMATHGoogle Scholar
  25. Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.View ArticleMATHGoogle Scholar
  26. Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York; 2011.View ArticleMATHGoogle Scholar
  27. Moslehian MS, Rassias TM: Stability of functional equations in non-Archimedean spaces. Appl. Anal. Discrete Math. 2007, 1: 325–334. 10.2298/AADM0702325MMathSciNetView ArticleMATHGoogle Scholar
  28. Rassias TM: Functional Equations, Inequalities and Applications. Kluwer Academic, Dordrecht; 2003.View ArticleMATHGoogle Scholar
  29. Saadati R, Cho YJ, Vahidi J: The stability of the quartic functional equation in various spaces. Comput. Math. Appl. 2010, 60: 1994–2002. 10.1016/j.camwa.2010.07.034MathSciNetView ArticleMATHGoogle Scholar
  30. Shakeri, S, Saadati, R, Park, C: Stability of the quadratic functional equation in non-Archimedean -fuzzy normed spaces. PreprintGoogle Scholar
  31. Cho, YJ, Saadati, R, Wang, S: Nonlinear -fuzzy stability of functional equations. PreprintGoogle Scholar
  32. Hadžić O, Pap E, Budincević M: Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces. Kybernetika 2002, 38: 363–381.MathSciNetMATHGoogle Scholar
  33. Atanassov KT: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20: 87–96. 10.1016/S0165-0114(86)80034-3MathSciNetView ArticleMATHGoogle Scholar

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