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Fixed point theorems for contractions in fuzzy normed spaces and intuitionistic fuzzy normed spaces

Abstract

In this paper, we prove that some coupled fixed point theorems and coupled coincidence point theorems for contractions in fuzzy normed spaces and intuitionistic fuzzy normed spaces can be directly deduced from fixed point theorems for contractions in fuzzy normed spaces. We also prove that these results are equivalent.

MSC:47H10, 54A40, 54E50, 54H25.

1 Introduction

The well-known Banach contraction mapping principle [1] is a powerful tool in nonlinear analysis; many mathematicians have much contributed to the improvement and generalization of this principle in many ways. Especially, some recent meaningful results have been obtained in [218].

In this paper, we first prove a simple fixed point theorem for an increasing mapping defined on fuzzy normed spaces, and by using this result, we can easily prove some coupled fixed point theorems and coupled coincidence point theorems in fuzzy normed spaces and intuitionistic fuzzy normed spaces. Also, we prove that these results are essentially equivalent. Finally, we give an example to show that our contractive conditions is a real improvement over the contractive conditions used in [13] and [17]. Our results are also an improvement over the results in [13] and [17].

For the reader’s convenience, we restate some definitions and results that will be used in this paper.

Definition 1.1 ([13])

A binary operation :[0,1]×[0,1][0,1] is a continuous t-norm if satisfies the following conditions:

  1. (i)

    is commutative and associative;

  2. (ii)

    is continuous;

  3. (iii)

    a1=a, a[0,1];

  4. (iv)

    abcd, whenever ac and bd for all a,b,c,d[0,1].

Definition 1.2 (cf. [14, 18])

A t-norm is said to be of H-type if the sequence of functions { n a } n = 1 is equicontinuous at a=1.

The t-norm m defined by a m b=min{a,b} is an example of an H-type t-norm .

Definition 1.3 ([13])

A binary operation :[0,1]×[0,1][0,1] is a continuous t-conorm if satisfies the following conditions:

  1. (i)

    is commutative and associative;

  2. (ii)

    is continuous;

  3. (iii)

    a0=a, a[0,1];

  4. (iv)

    abcd, whenever ac and bd for all a,b,c,d[0,1].

Definition 1.4 ([23])

A fuzzy normed space (briefly, FNS) is a triple (X,μ,), where X is a vector space, is a continuous t-norm and μ:X×(0,)[0,1] is a fuzzy set such that, for all x,yX and t,s>0,

  1. (i)

    μ(x,t)>0;

  2. (ii)

    μ(x,t)=1 if and only if x=θ;

  3. (iii)

    μ(cx,t)=μ(x, t | c | ) for all c0;

  4. (iv)

    μ(x,s)μ(y,t)μ(x+y,s+t);

  5. (v)

    μ(x,) is a continuous function of R + and

    lim t μ(x,t)=1, lim t 0 μ(x,t)=0.

By the results in George and Veeramani [19], we can know that every fuzzy norm (μ,) on X generates a Hausdorff first countable topology τ μ on X which has as a base the family of open sets of the form

{ B ( x , r , t ) : x X , r ( 0 , 1 ) , t > 0 } ,

where B(x,r,t)={yX:μ(x,y,t)>1r} for all xX, r(0,1) and t>0.

Definition 1.5 (cf. [13, 21])

The 5-tuple (X,μ,ν,,) is said to be an intuitionistic fuzzy normed space (for short, IFNS) if X is a linear space, is a continuous t-norm, is a continuous t-conorm and μ, ν are fuzzy sets on X×(0,) satisfying the following conditions:

  1. (i)

    μ(x,t)+ν(x,t)1, (x,t)X×(0,);

  2. (ii)

    μ(x,t)>0;

  3. (iii)

    μ(x,t)=1 if and only if x=θ;

  4. (iv)

    μ(cx,t)=μ(x, t | c | ) for all c0;

  5. (v)

    μ(x,s)μ(y,t)μ(x+y,s+t);

  6. (vi)

    μ(x,) is a continuous function of R + and

    lim t μ(x,t)=1, lim t 0 μ(x,t)=0;
  7. (vii)

    ν(x,t)<1;

  8. (viii)

    ν(x,t)=0 if and only if x=θ;

  9. (ix)

    ν(cx,t)=ν(x, t | c | ) for all c0;

  10. (x)

    ν(x,s)ν(y,t)ν(x+y,s+t);

  11. (xi)

    ν(x,) is a continuous function of R + and

    lim t ν(x,t)=0, lim t 0 ν(x,t)=1.

Park proved in [22], among other results, that each intuitionistic fuzzy norm (μ,ν) on X generates a Hausdorff first countable topology τ ( μ , ν ) on X which has as a base the family of open sets of the form {B(x,r,t):xX,r(0,1),t>0}, where B(x,r,t)={yX:μ(x,y,t)>1r,υ(x,y,t)<r} for all xX, r(0,1) and t>0. According to this topology, Park [22] gave the following definitions.

Definition 1.6 A sequence { x n } in an intuitionistic fuzzy normed linear space (X,μ,ν,,) is said to converge to xX with respect to the intuitionistic fuzzy norm (μ,ν) if, for any ε>0, t>0, 0<ε<1, there exists an integer n 0 N such that

μ( x n x,t)>1εandν( x n x,t)<εfor all n n 0 .

Definition 1.7 A sequence { x n } in an intuitionistic fuzzy normed linear space (X,μ,ν,,) is said to be a Cauchy sequence with respect to the intuitionistic fuzzy norm (μ,ν) if, for any ε>0, t>0, 0<ε<1, there exists an integer n 0 N such that

μ( x m x n ,t)>1εandν( x m x n ,t)<εfor all m,n n 0 .

Definition 1.8 Let (X,μ,ν,,) be an IFNS. Then (X,μ,ν,,) is said to be complete if every Cauchy sequence in (X,μ,ν,,) is convergent.

Definition 1.9 Let X and Y be two intuitionistic fuzzy normed spaces. A mapping f:XY is said to be continuous at x 0 X if, for any sequence { x n } in X converging to x 0 , the sequence {f( x n )} in Y converges to f( x 0 )Y. If f:XY is continuous at each xX, then f is said to be continuous on X.

For the topology τ ( μ , ν ) , Gregori et al. [20] proved the following result.

Lemma 1.1 Let (X,μ,ν,,) be an intuitionistic fuzzy metric space. Then the topologies τ ( μ , ν ) and τ μ coincide on X.

The following lemma was proved by Haghi et al. [15].

Lemma 1.2 Let X be a nonempty set, and let g:XX be a mapping. Then there exists a subset EX such that g(E)=g(X) and g:EX is one-to-one.

Definition 1.10 ([16])

A point (x,y)X×X is called a coupled coincidence point of the mappings F:X×XX and g:XX if

F(x,y)=gx,F(y,x)=gy.

Definition 1.11 ([16])

Let (X,) be a partially ordered set, and let F:X×XX, g:XX be two mappings. Then F is said to have the mixed g-monotone property if F is monotone g-non-decreasing in the first argument and is monotone g-non-increasing in the second argument, that is, for any x,yX,

x 1 , x 2 X,g( x 1 )g( x 2 )F( x 1 ,y)F( x 2 ,y),

and

y 1 , y 2 X,g( y 1 )g( y 2 )F(x, y 1 )F(x, y 2 ).

If g:XX is an identity mapping, we say that F has the mixed monotone property.

2 Main results

Theorem 2.1 Let (X,) be a partially ordered set, and let (X,μ,) be a complete FNS such that the t-norm is of H-type. Let F:XX be a mapping such that F is non-decreasing and

μ ( F ( x ) F ( u ) , k t ) 2 μ(xu,t),
(2.1)

for which xu and t>0, where 0<k<1. Suppose either

  1. (a)

    F is continuous, or

  2. (b)

    if { x n } is a non-decreasing sequence and lim n x n =x, then x n x for all nN.

If there exists x 0 X such that

x 0 F( x 0 ),

then F has a fixed point in X.

Proof Let x 0 X such that x 0 F( x 0 ), and let x n =F( x n 1 ), n=1,2, , then we have that

x 0 x 1 x 2 x n x n + 1 .

Now, put

δ n (t):=μ( x n x n + 1 ,t).

Then, by using (2.1), we have

μ( x n x n + 1 ,kt)=μ ( F ( x n 1 ) F ( x n ) , k t ) 2 μ( x n 1 x n ,t)= 2 δ n 1 (t).

Thus, it follows that δ n (kt) 2 δ n 1 (t), and so

δ n (t) 2 δ n 1 ( t k ) 2 n δ 0 ( t k n ) .
(2.2)

On the other hand, we have

t(1k) ( 1 + k + + k m n 1 ) <t,m>n,0<k<1.

By Definition 1.4, we get that

μ ( x n x m , t ) μ ( x n x m , t ( 1 k ) ( 1 + k + + k m n 1 ) ) μ ( x n x n + 1 , t ( 1 k ) ) μ ( x n + 1 x m , t ( 1 k ) ( k + + k m n 1 ) ) μ ( x n x n + 1 , t ( 1 k ) ) μ ( x n + 1 x n + 2 , t ( 1 k ) k ) μ ( x m 1 x m , t ( 1 k ) k m n 1 ) .
(2.3)

It follows from (2.2) and (2.3) that

μ ( x n x m , t ) μ ( x n x n + 1 , t ( 1 k ) ) μ ( x n + 1 x n + 2 , t ( 1 k ) k ) μ ( x m 1 x m , t ( 1 k ) k m n 1 ) [ 2 n δ 0 ( t ( 1 k ) k n ) ] [ 2 m 1 δ 0 ( t ( 1 k ) k n ) ] = 2 m 2 n δ 0 ( t ( 1 k ) k n ) .
(2.4)

By the hypothesis, the t-norm is of H-type; for all ε(0,1), there exists η>0 such that

p (s)>1ε,

for all s(1η,1] and for all p. Note that

lim n δ 0 ( t ( 1 k ) k n ) =1

for all t>0 and 0<k<1, we have that there exists n 0 such that

μ( x n x m ,t)>1ε,

for all m>n> n 0 . Thus, { x n } is a Cauchy sequence. Since X is complete, there exists xX such that

lim n x n =x.

If the assumption (a) holds, then by the continuity of F, we get that

x= lim n x n + 1 = lim n F( x n )=F(x).

If the assumption (b) holds, then we have that x n x for all nN. It follows from (2.1) that

lim n μ ( x n + 1 F ( x ) , k t ) = lim n μ ( F ( x n ) F ( x ) , k t ) lim n 2 μ( x n x,t)=1.

Thus, μ(xF(x),kt)=1, that is, x=F(x). Therefore, x is a fixed point of F. The proof is completed. □

Theorem 2.2 Let (X,) be a partially ordered set, and let (X,μ,) be a complete FNS such that the t-norm is of H-type and ab>0 for any a,b(0,1]. Let F:X×XX be a mapping such that F has the mixed monotone property and

μ ( F ( x , y ) F ( u , v ) , k t ) μ(xu,t)μ(yv,t),
(2.5)

for which xu,yv and t>0, where 0<k<1. Suppose either

  1. (a)

    F is continuous, or

  2. (b)

    X has the following property:

  3. (i)

    if { x n } is a non-decreasing sequence and lim n x n =x, then x n x for all nN,

  4. (ii)

    if { y n } is a non-increasing sequence and lim n y n =y, then y n y for all nN.

If there exist x 0 , y 0 X such that

x 0 F( x 0 , y 0 ), y 0 F( y 0 , x 0 ),

then F has a coupled fixed point x,yX, that is,

x=F(x,y),y=F(y,x).

Proof First, we define a partial order on X×X as follows: (x,y)(u,v) if and only if xu and yv. Second, we define a fuzzy set on X×X as follows: μ ˜ ((x,y),t)=μ(x,t)μ(y,t) for any (x,y)X×X and any t>0. Since (X,μ,) is a complete FNS, we can easily prove that (X×X, μ ˜ ,) is a complete FNS. Lastly, we define a mapping F ˜ :X×XX×X by

F ˜ (x,y)= ( F ( x , y ) , F ( y , x ) ) ,(x,y)X×X.

Since F has the mixed monotone property, if (x,y)(u,v), we have that

that is, F ˜ (x,y) F ˜ (u,v). Therefore, F ˜ :X×XX×X is a non-decreasing mapping. Since x 0 , y 0 X and

x 0 F( x 0 , y 0 ), y 0 F( y 0 , x 0 ),

we have that ( x 0 , y 0 )(F( x 0 , y 0 ),F( y 0 , x 0 ))= F ˜ ( x 0 , y 0 ). If (x,y)(u,v), by (2.5) we have that

μ ˜ ( F ˜ ( x , y ) F ˜ ( u , v ) , k t ) = μ ˜ ( ( F ( x , y ) F ( u , v ) , F ( v , u ) F ( y , x ) ) , k t ) = μ ( F ( x , y ) F ( u , v ) , k t ) μ ( F ( v , u ) F ( y , x ) , k t ) ( μ ( x u , t ) μ ( y v , t ) ) ( μ ( x u , t ) μ ( y v , t ) ) = 2 μ ˜ ( ( x , y ) ( u , v ) , t ) .

Thus, all the assumptions of Theorem 2.1 hold for F ˜ and (X×X, μ ˜ ,). By Theorem 2.1 we get that F ˜ has a fixed point (x,y)X×X, that is, (x,y)=(F(x,y),F(y,x)). This implies that x=F(x,y), y=F(y,x), that is, (x,y) is a coupled fixed point of F. The proof is completed. □

By using Theorem 2.2, we can prove the following coupled fixed point theorem in intuitionistic fuzzy normed spaces.

Theorem 2.3 Let (X,) be a partially ordered set, and let (X,μ,ν,,) be a complete IFNS such that the t-norm is of H-type and ab>0 for any a,b(0,1]. Let F:X×XX be a mapping such that F has the mixed monotone property and

μ ( F ( x , y ) F ( u , v ) , k t ) μ(xu,t)μ(yv,t),
(2.6)

for which xu,yv and t>0, where 0<k<1. Suppose either

  1. (a)

    F is continuous, or

  2. (b)

    X has the following property:

  3. (i)

    if { x n } is a non-decreasing sequence and lim n x n =x, then x n x for all nN,

  4. (ii)

    if { y n } is a non-increasing sequence and lim n y n =y, then y n y for all nN.

If there exist x 0 , y 0 X such that

x 0 F( x 0 , y 0 ), y 0 F( y 0 , x 0 ),

then F has a coupled fixed point x,yX, that is,

x=F(x,y),y=F(y,x).

Proof Assume that { x n } is a sequence in (X,μ,ν,,). Let t>0, 0<ε<1. If μ( x m x n ,t)>1ε, then by Definition 1.5(i) we can deduce that ν( x m x n ,t)<ε. Thus, a sequence { x n } in (X,μ,ν,,) is Cauchy if and only if { x n } is Cauchy in (X,μ,). By Lemma 1.1, we know that the topology of (X,μ,ν,,) is the same as the topology of (X,μ,). This implies that (X,μ,ν,,) is a complete IFNS if and only if (X,μ,) is a complete FNS. Therefore, by using Theorem 2.2 to (X,μ,) and F, we get that F has a coupled fixed point x,yX. The proof is completed. □

Theorem 2.4 Let (X,) be a partially ordered set, and let (X,μ,ν,,) be a complete IFNS such that the t-norm is of H-type and ab>0 for any a,b(0,1]. Let F:X×XX, g:XX be two mappings such that F has the mixed g- monotone property and

μ ( F ( x , y ) F ( u , v ) , k t ) μ(gxgu,t)μ(gygv,t),
(2.7)

for which g(x)g(u) , g(y)g(v) and t>0, where 0<k<1, F(X×X)g(X) and g is continuous. Suppose either

  1. (a)

    F is continuous, or

  2. (b)

    X has the following property:

  3. (i)

    if { x n } is a non-decreasing sequence and lim n x n =x, then g x n gx for all nN,

  4. (ii)

    if { y n } is a non-increasing sequence and lim n y n =y, then g y n gy for all nN.

If there exist x 0 , y 0 X such that

g( x 0 )F( x 0 , y 0 ),g( y 0 )F( y 0 , x 0 ),

then there exist x,yX such that

g(x)=F(x,y),g(y)=F(y,x),

that is, F and g have a coupled coincidence point in X.

Proof The conclusion of Theorem 2.4 can be proved by using Lemma 1.2 and Theorem 2.3. Since the proof is similar to the proof of Theorem 3.2 in [17], we delete the details of the proof. The proof is completed. □

Remark 2.1 It follows from the proof of the above theorems that the following implications hold: Theorem 2.1 Theorem 2.2 Theorem 2.3 Theorem 2.4. Conversely, it is clear that the following implications hold: Theorem 2.4 Theorem 2.3 Theorem 2.2. Thus, we have the following conclusion.

Theorem 2.5 Theorem  2.2-Theorem  2.4 are equivalent.

Remark 2.2 In [13] and [17], the condition abab for all a,b[0,1] is used. But this condition cannot hold in intuitionistic fuzzy normed spaces. In fact, if this condition holds, by using (iii) and (iv) in the definition of IFNS, we can get 1=1010=0, which yields a contradiction. Furthermore, the proofs of the results in [13] and [17] have the same errors as noted in [18]. Therefore, our results improve and correct results in [13] and [17].

In the following, we give an example to show that our contractive conditions are a real improvement over the contractive conditions used in [13] and [17].

Example 2.1 Let X=R, μ(x,t)= t t + | x | , ν(x,t)=1 t t + | x | for every xX, and let t>0, ab=min{a,b}, ab=max{a,b} for all a,b[0,1]. Then (X,μ,ν,,) is a complete intuitionistic fuzzy normed linear space, and the t-norm and t-conorm are of H-type. If X is endowed with the usual order xyxy0, then (X,) is a partially ordered set. Let 0<k<1, and define F(x,y)= x y 4 , gx= x k for any x,yX. Then F:X×XX is a mixed g-monotone mapping, F(X,X)g(X), and g is continuous. Let x 0 =1 and y 0 =1, then

1 k =g x 0 F( x 0 , y 0 )= 1 2 , 1 2 =F( y 0 , x 0 ) 1 k =g y 0 .

For any x,y,u,vX, with gxgu,gvgy, we have

μ ( F ( x , y ) F ( u , v ) , k t ) = k t k t + | u x + y v | 4 min { k t k t + | u x | , k t k t + | y v | } = min { t t + | u x | / k , t t + | y v | / k } = min { μ ( g x g u , t ) , μ ( g y g v , t ) } = μ ( g x g u , t ) μ ( g y g v , t ) .

Thus, all the conditions of Theorem 2.4 are satisfied. By Theorem 2.4, there is ( x , y )X×X such that F( x , y )=g( x ) and F( y , x )=g( y ). But υ does not satisfy the contractive conditions in [13] and [17]. In fact, for u=x+1, y=v+1,

ν ( F ( x , y ) F ( u , v ) , k t ) = 1 k t k t + | u x + y v | 4 = 1 k t k t + 1 2 > max { 1 k t k t + | u x | , 1 k t k t + | y v | } = max { 1 t t + | u x | / k , 1 t t + | y v | / k } = max { ν ( g x g u , t ) , ν ( g y g v , t ) } = ν ( g x g u , t ) ν ( g y g v , t ) .

This shows that υ does not satisfy the contractive conditions in [13] and [17].

References

  1. Banach S: Sur les opérations dans les ensembles abstraits et leurs applications aux éuations intérales. Fundam. Math. 1922, 3: 133–181.

    Google Scholar 

  2. Ćirić LB, Agarwal R, Samet B: Mixed monotone generalized contractions in partially ordered probabilistic metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 56

    Google Scholar 

  3. Gordji ME, Savadkouhi MB: Stability of a mixed type additive, quadratic and cubic functional equation in random normed spaces. Filomat 2012, 25(3):43–54.

    Article  Google Scholar 

  4. Ćirić LB, Abbas M, Damjanovic B, Saadati R: Common fuzzy fixed point theorems in ordered metric spaces. Math. Comput. Model. 2011, 53: 1737–1741. 10.1016/j.mcm.2010.12.050

    Article  Google Scholar 

  5. Shakeri S, Ćirić LB, Saadati R: Common fixed point theorem in partially ordered L -fuzzy metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 125082

    Google Scholar 

  6. Ćirić LB: Solving Banach fixed point principle for nonlinear contractions in probabilistic metric spaces. Nonlinear Anal. 2010, 72: 2009–2018. 10.1016/j.na.2009.10.001

    MathSciNet  Article  Google Scholar 

  7. Ćirić LB, Mihet D, Saadati R: Monotone generalized contractions in partially ordered probabilistic metric spaces. Topol. Appl. 2009, 156: 2838–2844. 10.1016/j.topol.2009.08.029

    Article  Google Scholar 

  8. Ćirić LB, Lakshmikantham V: Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces. Stoch. Anal. Appl. 2009, 27(6):1246–1259. 10.1080/07362990903259967

    MathSciNet  Article  Google Scholar 

  9. Shakeri S, Jalili M, Saadati R, Vaezpour SM, Ćirić LB: Quicksort algorithms: application of fixed point theorem in probabilistic quasi-metric spaces at domain of words. J. Appl. Sci. 2009, 9: 397–400. 10.3923/jas.2009.397.400

    Article  Google Scholar 

  10. Ćirić LB: Some new results for Banach contractions and Edelstein contractive mappings on fuzzy metric spaces. Chaos Solitons Fractals 2009, 42: 146–154. 10.1016/j.chaos.2008.11.010

    MathSciNet  Article  Google Scholar 

  11. Ćirić LB, Jesic SN, Ume JS: The existence theorems for fixed and periodic points of nonexpansive mappings in intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2008, 37: 781–791. 10.1016/j.chaos.2006.09.093

    MathSciNet  Article  Google Scholar 

  12. Abbas M, Babu GVR, Alemayehu GN: On common fixed points of weakly compatible mappings satisfying generalized condition. Filomat 2011, 25(2):9–19. 10.2298/FIL1102009A

    MathSciNet  Article  Google Scholar 

  13. Gordji ME, Baghani H, Cho YJ: Coupled fixed point theorem for contractions in intuitionistic fuzzy normed spaces. Math. Comput. Model. 2011, 54: 1897–1906. 10.1016/j.mcm.2011.04.014

    Article  Google Scholar 

  14. Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, Dordrecht; 2001.

    Google Scholar 

  15. Haghi RH, Rezapour SH, Shahzad N: Some fixed point generalizations are not real generalizations. Nonlinear Anal. 2011, 74: 1799–1803. 10.1016/j.na.2010.10.052

    MathSciNet  Article  Google Scholar 

  16. Lakshmikantham V, Ćirić LB: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020

    MathSciNet  Article  Google Scholar 

  17. Sintunavarat W, Cho YJ, Kumam P: Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl. 2011, 2011: 81. 10.1186/1687-1812-2011-81

    MathSciNet  Article  Google Scholar 

  18. Zhu XH, Xiao JZ: Note on ‘Coupled fixed point theorems for contractions in fuzzy metric spaces’. Nonlinear Anal. 2011, 74: 5475–5479. 10.1016/j.na.2011.05.034

    MathSciNet  Article  Google Scholar 

  19. George A, Veeramani P: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 1994, 64: 395–399. 10.1016/0165-0114(94)90162-7

    MathSciNet  Article  Google Scholar 

  20. Gregori V, Romaguera S, Veeramani P: A note on intuitionstic fuzzy spaces. Chaos Solitons Fractals 2006, 28: 902–905. 10.1016/j.chaos.2005.08.113

    MathSciNet  Article  Google Scholar 

  21. Mursaleen M, Mohiuddine SA: On stability of a cubic functional equation in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 2009, 42: 2997–3005. 10.1016/j.chaos.2009.04.041

    MathSciNet  Article  Google Scholar 

  22. Park JH: Intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2004, 22: 1039–1046. 10.1016/j.chaos.2004.02.051

    MathSciNet  Article  Google Scholar 

  23. Saadati R, Vaezpour S: Some results on fuzzy Banach spaces. J. Appl. Math. Comput. 2005, 17: 475–484. 10.1007/BF02936069

    MathSciNet  Article  Google Scholar 

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Acknowledgements

First, the authors are very grateful to the referees for their careful reading of the manuscript, valuable comments and suggestions. Next, this research was supported by the National Natural Science Foundation of China (11171286).

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Zhu, J., Wang, Y. & Zhu, CC. Fixed point theorems for contractions in fuzzy normed spaces and intuitionistic fuzzy normed spaces. Fixed Point Theory Appl 2013, 79 (2013). https://doi.org/10.1186/1687-1812-2013-79

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Keywords

  • coupled fixed point
  • coupled coincidence
  • intuitionistic fuzzy normed space
  • partially ordered set
  • mixed monotone mapping