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Set-valued mappings in partially ordered fuzzy metric spaces

Abstract

In this paper, we provide coincidence point and fixed point theorems satisfying an implicit relation, which extends and generalizes the result of Gregori and Sapena, for set-valued mappings in complete partially ordered fuzzy metric spaces. Also we prove a fixed point theorem for set-valued mappings on complete partially ordered fuzzy metric spaces which generalizes results of Mihet and Tirado.

MSC:54E40, 54E35, 54H25.

1 Preliminaries

The concept of fuzzy metric space was introduced by Kramosil and Michalek [1] and the modified concept by George and Veeramani [2] (for other modifications see [3, 4]). Furthermore, the fixed point theory in this kind of spaces has been intensively studied (see [514]).

The applications of fixed point theorems are remarkable in different disciplines of mathematics, engineering, and economics in dealing with problems in approximation theory, game theory, and many others (see [15] and references therein).

In 2004 Rodríguez-López and Romaguera [16] introduced the Hausdorff fuzzy metric of a given fuzzy metric space in the sense of George and Veeramani on the set of non-empty compact subsets.

Some fixed point results for set-valued mappings on fuzzy metric space can be found in [17, 18] and references therein.

The aim of this paper is to prove a coincidence point and fixed point theorem on a partially ordered fuzzy metric space satisfying an implicit relation and another fixed point theorem. Our result substantially generalizes and extends the result of Gregori and Sapena [8] and results of Miheţ [19] and Tirado [20] and also the result of Latif and Beg [21] for set-valued mappings in complete partially ordered fuzzy metric spaces. Implicit relations have been considered by several authors in connection with solving nonlinear functional equations (see [2225]).

For the sake of completeness, we briefly recall some basic concepts used in the following.

Definition 1.1 [26]

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

  1. (1)

    is associative and commutative,

  2. (2)

    is continuous,

  3. (3)

    a1=a for all a[0,1],

  4. (4)

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

The three basic continuous t-norms are: (i) The minimum t-norm is defined by ab=min{a,b}. (ii) The product t-norm is defined by ab=ab. (iii) The Łukasiewicz t-norm is defined by ab=max{a+b1,0}.

Definition 1.2 [27, 28]

  1. (i)

    A t-norm is said to be Hadžić-type t-norm, if the family { n } n 0 of its iterates defined for each s[0,1] by 0 (s)=1, n (s)=( n 1 (s))s, for all n0, are equi-continuous at s=1, that is, given λ>0, there exists η(λ)(0,1) such that for all n0

    1s>η(λ) n (s)>1λ.

The t-norm , defined by ab=min{a,b} is a trivial example of the t-norm of Hadžić-type, but there are other t-norms of Hadžić-type (see [27]).

  1. (ii)

    If be a t-norm and { x n } n 1 is a sequence of numbers in [0,1], one defines recurrently i = 1 n x i by i = 1 1 x i = x 1 and i = 1 n x i =( i = 1 n 1 x i , x n ), n2. i = 1 x i is defined as lim n i = 1 n x i and i = n x i as i = 1 x n + i .

If q(0,1) is given, we say that the t-norm is geometrically convergent (g-convergent) if lim n i = n (1 q i )=1.

The Łukasiewicz t-norm and t-norms of Hadžić-type are examples of g-convergent t-norms. Other examples be found in [28]. Also note that if the t-norm is g-convergent, then sup t < 1 tt=1.

Proposition 1.3 [28]

  1. (i)

    For abmax{a+b1,0} the following implication holds:

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

    If is of Hadžić-type, then lim n i = 1 x n + i =1, for every sequence { x n } n N in [0,1] such that lim n x n =1.

Definition 1.4 [2]

A 3-tuple (X,M,) is called a fuzzy metric space if X is an arbitrary non-empty set, is a continuous t-norm and M is a fuzzy set on X 2 ×(0,), satisfying the following conditions for each x,y,zX and t,s>0:

(FM-1) M(x,y,t)>0,

(FM-2) M(x,y,t)=1 for all t>0 if and only if x=y,

(FM-3) M(x,y,t)=M(y,x,t),

(FM-4) M(x,z,t+s)M(x,y,t)M(y,z,s),

(FM-5) M(x,y,):(0,)[0,1] is continuous.

Definition 1.5 [2]

Let (X,M,) be a fuzzy metric space. A sequence { x n } in X is called a Cauchy sequence, if, for each ϵ(0,1) and t>0, there exists n 0 N such that M( x n , x m ,t)>1ϵ for all n,m n 0 . A sequence { x n } in a fuzzy metric space (X,M,) is said to be convergent to xX if lim n M( x n ,x,t)=1 for all t>0. A 3-tuple (X,M,) is complete if every Cauchy sequence is convergent in X.

Lemma 1.6 [7]

Let (X,M,) be a fuzzy metric space. Then M(x,y,t) is non-decreasing with respect to t for all x,yX.

Definition 1.7 [16]

Let (X,M,) be a fuzzy metric space. M is said to be continuous on X 2 ×(0,), if

lim n M( x n , y n , t n )=M(x,y,t)

whenever a sequence {( x n , y n , t n )} in X 2 ×(0,) converges to a point (x,y,t) X 2 ×(0,), that is,

lim n x n =x, lim n y n =y,and lim n M(x,y, t n )=M(x,y,t).

Lemma 1.8 [16]

Let (X,M,) be a fuzzy metric space. Then M is continuous function on X 2 ×(0,).

Definition 1.9 [6]

Let (X,M,) be a fuzzy metric space. The fuzzy metric M is triangular if it satisfies the condition

( 1 M ( x , y , t ) 1 ) ( 1 M ( x , z , t ) 1 ) + ( 1 M ( z , y , t ) 1 ) ,

for every x,y,zX and every t>0.

Example 1.10 [2]

Let (X,d) be a metric space. Define ab=ab (or ab=min{a,b}) and for all x,yX and t>0,

M d (x,y,t)= t t + d ( x , y ) .

Then (X, M d ,) is a fuzzy metric space. We call the fuzzy metric M d induced by the metric d the standard fuzzy metric. Note that every standard fuzzy metric is triangular.

Definition 1.11 Let (X,M,) is a fuzzy metric space and t>0. (i): A subset AX is said to be closed if for each convergent sequence { x n } with x n A and x n x as n, we have xA.

(ii): AX is said to be compact if each sequence in A has a convergent subsequence.

Throughout the article, let P(X), C(X), and K(X) denote the set of all non-empty subsets, the set of all non-empty closed subsets, and the set of all non-empty compact subsets of X, respectively.

Definition 1.12 Let X be a non-empty set. A point xX is called a coincidence point of the mappings F:XP(X) and f:XX if fxFx. Point xX is called a fixed point of the mappings F:XP(X) if fxFx.

Theorem 1.13 [16]

Let (X,M,) be a fuzzy metric space. For each A,BK(X) and t>0 define

H M (A,B,t)=min { inf a A M ( a , B , t ) , inf b B M ( A , b , t ) } ,

where M(a,B,t):=sup{M(a,b,t):bB}. Then the 3-tuple (K(X), H M ,) is a fuzzy metric space.

The fuzzy metric ( H M ,) will be called the Hausdorff fuzzy metric of (M,) on K(X).

Lemma 1.14 [16]

Let (X,M,) be a fuzzy metric space. Then, for each aX, BK(X) and t>0, there is b 0 B such that

M(a,B,t)=M(a, b 0 ,t).

2 Main results

Throughout this section, denotes a continuous t-norm and the set of all continuous real-valued mappings T: [ 0 , 1 ] 6 R satisfying the following properties:

T 1 : T( t 1 , t 2 ,, t 6 ) is non-increasing in t 2 ,, t 6 .

T 2 : If there exists k(0,1) such that for each t>0, we have

T ( w ( k t ) , v ( t ) , v ( t ) , u ( t ) , u ( t 2 ) v ( t 2 ) , 1 ) 1,

where u,v,w:(0,)[0,1] are non-decreasing functions with u(t),v(t),w(t)(0,1], then w(kt)v(t).

T 3 : For each t>0 and some k(0,1), the condition

T ( w ( k t ) , 1 , 1 , v ( t ) , v ( t ) , 1 ) 1,

implies w(kt)v(t).

Now we give our main result.

Theorem 2.1 Let (X,M,) be a complete fuzzy metric space with Hadžić-type t-norm such that M(x,y,t)1 as t, for all x,yX. Let be a partial order defined on X. Let F:XK(X) be a set-valued mapping with non-empty compact values and f:XX a mapping such that f(X) is closed and for some TT and all comparable elements x,yX, and t>0, we have

T ( H M ( F x , F y , k t ) , M ( f x , f y , t ) , M ( f x , F x , t ) , M ( f y , F y , t ) , M ( f x , F y , t ) , M ( f y , F x , t ) ) 1 .
(2.1)

Also suppose that the following conditions are satisfied:

  1. (i)

    F(X)f(X),

  2. (ii)

    fyF(x) implies xy,

  3. (iii)

    if y n F( x n ) is a sequence such that y n y=fx, then x n x for all n.

Then F and f have a coincidence point, that is, there exists xX such that fxF(x).

Proof Let t>0 be fixed and x 0 X. By using (i) and (ii), there exists x 1 X such that x 0 x 1 and y 0 =f x 1 F x 0 . Now from (i), (ii), and by Lemma 1.14, for x 1 X there is x 2 X such that x 1 x 2 and y 1 =f x 2 F x 1 with

M( y 0 ,F x 1 ,t)=M( y 0 , y 1 ,t),

thus

H M (F x 0 ,F x 1 ,t) sup y 1 F x 1 M( y 0 , y 1 ,t)=M( y 0 ,F x 1 ,t)=M( y 0 , y 1 ,t).
(2.2)

On the other hand by x= x 0 and y= x 1 in (2.1), we have

T ( H M ( F x 0 , F x 1 , k t ) , M ( f x 0 , f x 1 , t ) , M ( f x 0 , F x 0 , t ) , M ( f x 1 , F x 1 , t ) , M ( f x 0 , F x 1 , t ) , M ( f x 1 , F x 0 , t ) ) 1 .

Now since M(f x 0 ,F x 0 ,t)M(f x 0 , y 0 ,t), M(f x 1 ,F x 1 ,t)M( y 0 , y 1 ,t), also

M(f x 0 ,F x 1 ,t)M(f x 0 , y 1 ,t)M ( f x 0 , y 0 , t 2 ) M ( y 0 , y 1 , t 2 ) ,

and M(f x 1 ,F x 0 ,t)M( y 0 , y 0 ,t)=1, and by using T 1 , we get

T ( H M ( F x 0 , F x 1 , k t ) , M ( f x 0 , f x 1 , t ) , M ( f x 0 , y 0 , t ) , M ( f x 1 , y 1 , t ) , M ( f x 0 , y 0 , t 2 ) M ( y 0 , y 1 , t 2 ) , 1 ) 1 .

This means that

T ( w ( k t ) , v ( t ) , v ( t ) , u ( t ) , u ( t 2 ) v ( t 2 ) , 1 ) 1,

where w(t)= H M (F x 0 ,F x 1 ,t), v(t)=M(f x 0 ,f x 1 ,t), u(t)=M(f x 1 , y 1 ,t), then from T 2 , we have (w(kt)v(t))

H M (F x 0 ,F x 1 ,kt)M(f x 0 ,f x 1 ,t)=M(f x 0 , y 0 ,t),

hence by (2.2), we obtain

M( y 0 , y 1 ,t)M ( f x 0 , y 0 , t k ) .

Again by (i), (ii), and by Lemma 1.14, there exists x 3 X such that x 2 x 3 with y 2 =f x 3 F x 2 that satisfying in

H M (F x 1 ,F x 2 ,t)M( y 1 ,F x 2 ,t)=M( y 1 , y 2 ,t).
(2.3)

Since x 1 x 2 thus by replacing x= x 1 and y= x 2 in (2.1) and from T 1 , we obtain

T ( H M ( F x 1 , F x 2 , k t ) , M ( y 0 , y 1 , t ) , M ( y 0 , y 1 , t ) , M ( y 1 , y 2 , t ) , M ( y 0 , y 1 , t 2 ) M ( y 1 , y 2 , t 2 ) , 1 ) 1 .

Now by w(t)= H M (F x 1 ,F x 2 ,t), v(t)=M( y 0 , y 1 ,t), u(t)=M( y 1 , y 2 ,t), the property T 2 implies

H M (F x 1 ,F x 2 ,kt)M( y 0 , y 1 ,t),

thus from (2.3), we get

M( y 1 , y 2 ,t)M ( y 0 , y 1 , t k ) .

Repeatedly, there exists x 4 X with x 3 x 4 such that y 3 =f x 4 F x 3 and H M (F x 2 ,F x 3 ,t)M( y 2 , y 3 ,t), and

H M (F x 2 ,F x 3 ,kt)M( y 1 , y 2 ,t),

therefore

M( y 2 , y 3 ,t)M ( y 1 , y 2 , t k ) M ( y 0 , y 1 , t k 2 ) M ( f x 0 , y 0 , t k 3 ) .

Continuing the process, we can have a sequence { x n } in X with x n x n + 1 such that, for n0, y n =f x n + 1 F x n , and

M( y n , y n + 1 ,t)M ( y n 1 , y n , t k ) ,
(2.4)

and

M( y n , y n + 1 ,t)M ( y 0 , y 1 , t k n ) M ( f x 0 , y 0 , t k n + 1 ) .
(2.5)

From (2.4), we conclude that, for each i1,

M( y n + i , y n + i + 1 ,t)M ( y n , y n + 1 , t k i ) .
(2.6)

Next, we prove that the sequence y n is Cauchy. Suppose that δ>0 and ϵ(0,1) are given. Then, by Lemma 1.6 and (FM-4), for all m>n,

M ( y n , y m , δ ) M ( y n , y m , δ ( 1 k ) ( 1 + k + + k m n 1 ) ) M ( y n , y n + 1 , δ ( 1 k ) ) M ( y n + 1 , y n + 2 , δ k ( 1 k ) ) M ( y m 1 , y m , δ k m n 1 ( 1 k ) ) .
(2.7)

On the other hand, putting t=δ k i (1k) in (2.6), for all n0, i1, we get

M ( y n + i , y n + i + 1 , δ k i ( 1 k ) ) M ( y n , y n + 1 , δ ( 1 k ) ) .

Then by replacing the above inequality in (2.7), we obtain, for all m>n,

M ( y n , y m , δ ) M ( y n , y n + 1 , δ ( 1 k ) ) M ( y n + 1 , y n + 2 , δ ( 1 k ) ) M ( y m 1 , y m , δ ( 1 k ) ) = ( m n ) M ( y n , y n + 1 , δ ( 1 k ) ) .
(2.8)

By hypothesis, is a t-norm of Hadžić-type, and there exists η(0,1) such that for all m>n,

1s>η ( m n ) (s)>1ϵ.
(2.9)

By M(f x 0 , y 0 ,t)1 as t, there exists n 0 such that, for all n n 0 ,

M ( f x 0 , y 0 , δ ( 1 k ) k n + 1 ) >η.

From (2.5) and the above inequality, we have

M ( y n , y n + 1 , δ ( 1 k ) ) >η,

therefore, (2.8) and (2.9) imply that, for all n n 0 and each m>n,

M( y n , y m ,δ)>1ϵ.

This shows that { y n } is a Cauchy sequence. Since X is complete, there exists some yX such that

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

Now, since f(X) is closed, there exists x ¯ X such that y=f x ¯ f(X). Also (ii) implies that x n x ¯ for any n. Thus from (2.1), we have

T ( H M ( F x n , F x ¯ , k t ) , M ( f x n , f x ¯ , t ) , M ( f x n , F x n , t ) , M ( f x ¯ , F x ¯ , t ) , M ( f x n , F x ¯ , t ) , M ( f x ¯ , F x n , t ) ) 1 .

By taking the limit as n, by the continuity of T, and from Lemma 1.8, we get

T ( lim n H M ( F x n , F x ¯ , k t ) , 1 , 1 , M ( y , F x ¯ , t ) , M ( y , F x ¯ , t ) , 1 ) 1.

Now by using T 3 , we have

lim n H M (F x n ,F x ¯ ,kt)M(y,F x ¯ ,t),

on the other hand H M (F x n ,F x ¯ ,kt)M( y n ,F x ¯ ,kt), so

M(y,F x ¯ ,kt) lim n H M (F x n ,F x ¯ ,kt)M(y,F x ¯ ,t).

It follows that M(y,F x ¯ ,t)=1 for each t>0. Now since F x ¯ is closed (note that F x ¯ is compact), we get f x ¯ =yF x ¯ , thus x ¯ is a coincidence point of F and f. The proof is complete. □

Remark 2.2 In Theorem 2.1, we proved that the sequence y n is Cauchy; one can replace the condition ‘ is Hadžić-type t-norm and M(x,y,t)1 as t, for all x,yX’ with the following: ‘ lim n i = n M(x,y,t h i )=1 for each h>1’. To see this, choose some q>1 and nN such that kq<1 and i = n 1 1 q i 1. Then from (FM-4) and (2.5), for every m>n n 1 , we have

M ( y n , y m , t ) M ( y n , y m , i = n m 1 1 q i t ) M ( y n , y n + 1 , 1 q n t ) M ( y n + 1 , y n + 2 , 1 q n + 1 t ) M ( y m 1 , y m , 1 q m 1 t ) M ( f x 0 , y 0 , 1 k n + 1 q n t ) M ( f x 0 , y 0 , 1 k n + 2 q n + 1 t ) M ( f x 0 , y 0 , 1 k m q m 1 t ) M ( f x 0 , y 0 , 1 ( k q ) n + 1 t ) M ( f x 0 , y 0 , 1 ( k q ) n + 2 t ) M ( f x 0 , y 0 , 1 ( k q ) m t ) i = n M ( f x 0 , y 0 , 1 ( k q ) i + 1 t ) > 1 ϵ .

Thus, { y n } is a Cauchy sequence. Then we have the following theorem.

Theorem 2.3 Let (X,M,) be a complete fuzzy metric space and suppose for each h>1, lim n i = n M(x,y,t h i )=1. Let be a partial order defined on X. Let F:XK(X) be a set-valued mapping with non-empty compact values and f:XX a mapping such that f(X) is closed and for some TT and all comparable elements x,yX, and t>0, we have

T ( H M ( F x , F y , k t ) , M ( f x , f y , t ) , M ( f x , F x , t ) , M ( f y , F y , t ) , M ( f x , F y , t ) , M ( f y , F x , t ) ) 1 .

Also suppose that the following conditions are satisfied:

  1. (i)

    F(X)f(X),

  2. (ii)

    fyF(x) implies xy,

  3. (iii)

    if y n F( x n ) is a sequence such that y n y=fx, then x n x for all n.

Then F and f have a coincidence point, that is, there exists xX such that fxF(x).

If in Theorem 2.1 and 2.3 we put T( u 1 ,, u 6 ):= u 1 ( k t ) u 2 ( t ) , where k(0,1), then we have the following corollaries.

Corollary 2.4 Let (X,M,) be a complete fuzzy metric space with Hadžić-type t-norm such that M(x,y,t)1 as t, for all x,yX. Let be a partial order defined on X. Let F:XK(X) be a set-valued mapping with non-empty compact values and f:XX a mapping such that f(X) be closed and for all comparable elements x,yX, and t>0, we have

H M (Fx,Fy,kt)M(fx,fy,t).

Also suppose that the following conditions are satisfied:

  1. (i)

    F(X)f(X),

  2. (ii)

    fyF(x) implies xy,

  3. (iii)

    if y n F( x n ) is a sequence such that y n y=fx, then x n x for all n.

Then there exists xX such that fxF(x).

Corollary 2.5 Let (X,M,) be a complete fuzzy metric space and suppose for each h>1, lim n i = n M(x,y,t h i )=1. Let be a partial order defined on X. Let F:XK(X) be a set-valued mapping with non-empty compact values and f:XX a mapping such that f(X) be closed and for all comparable elements x,yX, and t>0, we have

H M (Fx,Fy,kt)M(fx,fy,t).

Also suppose that the following conditions are satisfied:

  1. (i)

    F(X)f(X),

  2. (ii)

    fyF(x) implies xy,

  3. (iii)

    if y n F( x n ) is a sequence such that y n y=fx, then x n x for all n.

Then there exists xX such that fxF(x).

Putting f=I (the identity mapping) in Corollary 2.4 and 2.5, we get the following corollaries.

Corollary 2.6 Let (X,M,) be a complete fuzzy metric space with Hadžić-type t-norm such that M(x,y,t)1 as t, for some x 0 X and x 1 F x 0 . Let be a partial order defined on X. Let F:XK(X) be a set-valued mapping with non-empty compact values for all comparable elements x,yX, and t>0, we have

H M (Fx,Fy,kt)M(x,y,t).

Also suppose that the following conditions are satisfied:

  1. (i)

    yF(x) implies xy,

  2. (ii)

    if y n F( x n ) is a sequence such that y n x, then x n x for all n.

Then F has a fixed point.

Corollary 2.7 Let (X,M,) be a complete fuzzy metric space and suppose for each h>1, lim n i = n M(x,y,t h i )=1 for some x 0 X and x 1 F x 0 . Let be a partial order defined on X. Let F:XK(X) be a set-valued mapping with non-empty compact values for all comparable elements x,yX, and t>0, we have

H M (Fx,Fy,kt)M(x,y,t).

Also suppose that the following conditions are satisfied:

  1. (i)

    yF(x) implies xy,

  2. (ii)

    if y n F( x n ) is a sequence such that y n x, then x n x for all n.

Then F has a fixed point.

Remark 2.8 Note that we assumed the implicit relation (2.1) only for the comparable elements of the partially ordered fuzzy metric space.

Remark 2.9 Corollary 2.7 improves and generalizes the mentioned result of Gregori and Sapena (see Theorem 4.8 of [8]) for set-valued mappings in complete partially ordered fuzzy metric spaces.

In continuation, in the spirit of Miheţ [19], we introduce the notion of a set-valued fuzzy order ψ-contraction of (ϵ,λ)-type mappings and give a fixed point theorem in partially ordered fuzzy metric spaces.

Definition 2.10 Let (X,M,) be a fuzzy metric space and ψ:(0,1)(0,1). A mapping F:XC(X) is a set-valued fuzzy order ψ-contraction of (ϵ,λ)-type if the following implication holds:

M(x,y,ϵ)>1λpFxqFy;M(p,q,ϵ)>1ψ(λ),
(2.10)

for every ϵ>0, λ(0,1) and all comparable elements x,yX.

If ψ(t)=αt (t(0,1)) for some α(0,1), then F will be called a set-valued fuzzy order α-contraction of (ϵ,λ)-type.

Also note that if ψ(t)<t for all t(0,1), then every set-valued fuzzy order ψ-contraction of (ϵ,λ)-type satisfies the relation

pFxqFy;M(p,q,t)M(x,y,t),

for all comparable elements x,yX and t>0. Indeed, if for some comparable x,yX and t>0 there exists pFx such that for all qFy, we have M(p,q,t)<M(x,y,t); then there is λ(0,1) such that M(p,q,t)<1λ<M(x,y,t), that is, M(x,y,t)>1λ and M(p,q,t)<1λ<1ψ(λ), which is a contradiction.

Example 2.11 Let (X,M,) be a fuzzy metric space. Let F:YC(Y) be a set-valued mapping, where YC(X). If there is α(0,1) such that

pFxqFy;1M(p,q,t)α ( 1 M ( x , y , t ) ) ,

for all comparable elements x,yX and t>0, then F is a set-valued fuzzy order α-contraction of (ϵ,λ)-type. Indeed, if M(x,y,ϵ)>1λ, then for every comparable elements x,yX and some α(0,1), we have

pFxqFy;1M(p,q,ϵ)α ( 1 M ( x , y , ϵ ) ) <αλ,

thus M(p,q,ϵ)>1αλ.

Now we state our main theorem.

Theorem 2.12 Let (X,M,) be a complete fuzzy metric space with sup t < 1 tt=1. YC(X) and F:YC(Y) be a set-valued fuzzy order ψ-contraction of (ϵ,λ)-type, where lim n ψ ( n ) (t)=0 for all t(0,1). Let ‘’ be a partial order defined on X, and lim n i = n (1 ψ ( i ) (ξ))=1 for all ξ(0,1). Suppose that there exist x 0 Y and x 1 F x 0 such that M( x 0 , x 1 , 0 + )>0 and the following two conditions hold:

  1. (i)

    yF(x) implies xy,

  2. (ii)

    if x n is a sequence with x n + 1 F x n and x n x, then x n x for all n.

Then F has a fixed point.

Proof Since there exist x 0 Y and x 1 F x 0 such that M( x 0 , x 1 , 0 + )>0, we have x 0 x 1 with M( x 0 , x 1 , 0 + )>0. We may suppose that M( x 0 , x 1 , 0 + )<1. For, if we assume the contrary, then M( x 0 , x 1 ,t)=1 for all t>0, that is, x 0 = x 1 F x 0 and we have finished the proof. Therefore, for some δ 1 (0,1) and every t>0, δ( δ 1 ,1), we have

M( x 0 , x 1 ,t)M( x 0 , x 1 , 0 + )=1 δ 1 >1δ.

Since F is a set-valued fuzzy order ψ-contraction of (ϵ,λ)-type mapping, there exists x 2 F x 1 with x 1 x 2 such that M( x 1 , x 2 ,t)>1ψ(δ). Repeating this argument, we get a sequence { x n } in Y such that x n + 1 F x n with x n x n + 1 and such that

M( x n , x n + 1 ,t)>1 ψ n (δ).
(2.11)

Suppose that ϵ>0 and λ(0,1) are given. Since lim n i = n (1 ψ ( i ) (ξ))=1 for all ξ(0,1), there exists n 0 N such that for all n n 0 and all ξ(0,1), and we have

i = n ( 1 ψ ( i ) ( ξ ) ) >1λ.
(2.12)

Now by using (FM-4) and from (2.11)-(2.12), for all m>n n 0 , we get

M ( x n , x m , ϵ ) M ( x n , x n + 1 , ϵ m n ) M ( x n + 1 , x n + 2 , ϵ m n ) M ( x m 1 , x m , ϵ m n ) ( 1 ψ ( n ) ( δ ) ) ( 1 ψ ( n + 1 ) ( δ ) ) ( 1 ψ ( m 1 ) ( δ ) ) i = n ( 1 ψ ( i ) ( δ ) ) > 1 λ .

This shows that { x n } is a Cauchy sequence. Since X is complete, { x n } converges to some x ¯ X, that is, lim n M( x n , x ¯ ,t)=1. Now we prove that x ¯ F x ¯ . But F x ¯ = F x ¯ ¯ ; then it is enough to show that for every ϵ >0 and λ (0,1) there exists zF x ¯ such that M( x ¯ ,z, ϵ )>1 λ .

Let ϵ >0 and λ (0,1) be arbitrary. From sup t < 1 tt=1, it follows that there exists λ 1 ( λ )(0,1) such that

(1 λ 1 )(1 λ 1 )>1 λ .
(2.13)

Also for λ 1 there are λ 2 (0,1) such that

(1 λ 2 )(1 λ 2 )>1 λ 1 .
(2.14)

Now put λ 3 =min{ λ 1 , λ 2 }. We prove that there exists μ(0,1) such that ψ(μ)< λ 3 . For, if ψ(t) λ 3 for every t(0,1), then ψ n (t) λ 3 for every nN and every t(0,1), therefore i = n (1 ψ ( i ) (ξ)) i = n (1 λ 3 )1 λ 3 for all nN, which means that 1= lim n i = n (1 ψ ( i ) (ξ))1 λ 3 <1, and this is a contradiction.

Since lim n M( x n , x ¯ ,t)=1 for all t>0, there exists n 1 N such that for all n n 1 , and we have M( x n , x ¯ , ϵ 3 )>1μ; thus, since x n x ¯ and by using (2.10), there exists zF x ¯ such that

M ( x n + 1 , z , ϵ 3 ) >1ψ(μ)>1 λ 3 .
(2.15)

On the other hand lim n ψ ( n ) (t)=0 for every t(0,1). Therefore (2.11) implies the existence of the element n 2 N such that for all n n 2 , we have

M ( x n , x n + 1 , ϵ 3 ) >1 λ 3 .
(2.16)

Also since lim n x n = x ¯ , there exists n 3 N such that for all n n 3 ,

M ( x n , x ¯ , ϵ 3 ) >1 λ 3 .
(2.17)

Now if nmax{ n 1 , n 2 , n 3 }, then by (2.13)-(2.17), we get

M ( x ¯ , z , ϵ ) M ( x ¯ , x n , ϵ 3 ) M ( x n , x n + 1 , ϵ 3 ) M ( x n + 1 , z , ϵ 3 ) > ( 1 λ 3 ) ( 1 λ 3 ) ( 1 λ 3 ) > 1 λ .

Hence x ¯ F x ¯ = F x ¯ ¯ , consequently x ¯ is a fixed point of F. The theorem is proved. □

Corollary 2.13 Let (X,M,) be a complete fuzzy metric space with Lukasiewicz t-norm and ‘’ be a partial order defined on X. Let YC(X) and F:YC(Y) be a set-valued mapping with the property that there is α(0,1) such that

pFxqFy;1M(p,q,t)α ( 1 M ( x , y , t ) ) ,

for all comparable elements x,yX and t>0, and the following conditions hold:

  1. (i)

    yF(x) implies xy,

  2. (ii)

    if x n is a sequence with x n + 1 F x n and x n x, then x n x for all n.

Then F has a fixed point.

Proof By using Definition 1.2, sup t < 1 tt=1. Also, from Example 2.11 it follows that F is a set-valued fuzzy order ψ-contraction of (ϵ,λ)-type with ψ(t)=αt. Since, for all λ(0,1), i = 1 ψ ( i ) (λ)= i = 1 α i λ<, from Proposition 1.3, we have lim n i = n (1 ψ ( i ) (λ))=1. Next, since

pFxqFy;M(p,q,t)1α+αM(x,y,t)1α>0,

for all comparable elements x,yX and t>0, there exist x 0 Y and x 1 F x 0 such that M( x 0 , x 1 , 0 + )>0. Consequently, by the preceding theorem, F has a fixed point. □

Corollary 2.14 Let (X,M,) be a complete fuzzy metric space with a continuous g-convergent t-norm and ‘’ be a partial order defined on X. Let YC(X) and F:YC(Y) be a set-valued fuzzy order α-contraction of (ϵ,λ)-type. If there exist x 0 Y and x 1 F x 0 such that M( x 0 , x 1 , 0 + )>0 and the following two conditions hold:

  1. (i)

    yF(x) implies xy,

  2. (ii)

    if x n is a sequence with x n + 1 F x n and x n x, then x n x for all n.

Then F has a fixed point.

Theorem 2.12 and Corollary 2.13 are, respectively, generalizations of the theorems of Mihet [19] and Tirado [20] to the set-valued case in partial ordered fuzzy metric spaces.

Now we introduce a definition and, by using it, we shall state fixed and common fixed point theorems in the partially ordered fuzzy metric space. Our results generalize and extend Theorems 4.1 and 4.2 of [21] to set-valued mappings in complete partially ordered fuzzy metric spaces.

Definition 2.15 Let Y be a non-empty subset of fuzzy metric space (X,M,). Mapping F:YP(X) is called fuzzy order K-set-valued mapping, if for all xY, u x Fx, there exists u y Fy with u x u y such that

1 M ( u x , u y , t ) 1k [ 1 M ( x , u x , t ) 1 + 1 M ( y , u y , t ) 1 ] ,
(2.18)

for every t>0 and yY with xy and some k(0, 1 2 ).

Theorem 2.16 Let (X,M,) be a complete fuzzy metric space, with M triangular, and ‘’ a partial order on X. Let YC(X) and F:YC(Y) be a fuzzy order K-set-valued mapping. Also let there for some x 0 Y exist x 1 F x 0 with x 0 x 1 , and the following condition is satisfied:

If x n x is a sequence in Y whose consecutive terms are comparable, then x n x, for all n.

Then F has a fixed point in X.

Proof By the hypothesis, for x 0 Y there exists x 1 F x 0 such that x 0 x 1 . Now because F is a fuzzy order K-set-valued mapping, there exists x 2 F x 1 such that x 1 x 2 and

1 M ( x 1 , x 2 , t ) 1k [ 1 M ( x 0 , x 1 , t ) 1 + 1 M ( x 1 , x 2 , t ) 1 ] ,

thus

1 M ( x 1 , x 2 , t ) 1 k 1 k [ 1 M ( x 0 , x 1 , t ) 1 ] .

Then it follows by induction that

1 M ( x n , x n + 1 , t ) 1 ( k 1 k ) n [ 1 M ( x 0 , x 1 , t ) 1 ] ,
(2.19)

where { x n } is a sequence whose consecutive terms are comparable, that is, x n + 1 F x n . Now we prove that { x n } is a Cauchy sequence. By putting λ= k 1 k , and by (2.19), and since M is triangular, we have for all m>n

1 M ( x n , x m , t ) 1 i = 0 m n 1 [ 1 M ( x n + i , x n + i + 1 , t ) 1 ] ( 1 M ( x 0 , x 1 , t ) 1 ) i = n m 1 λ i ( 1 M ( x 0 , x 1 , t ) 1 ) λ n 1 λ .
(2.20)

For each t>0 and each ϵ(0,1), we can choose a sufficiently large n 0 N such that

( 1 M ( x 0 , x 1 , t ) 1 ) λ n 0 1 λ < 1 1 ϵ 1.
(2.21)

Thus from (2.20) and (2.21), M( x n , x m ,t)>1ϵ, for all m,n> n 0 and t>0. This shows that the sequence { x n } is Cauchy, and, since X is complete, it converges to a point xX. But Y is closed, thus xY and also by using the hypothesis x n x. Now we show that xFx. From x n F x n 1 , and x n 1 x for all n, since F is a fuzzy order K-set-valued mapping, there exists u n Fx such that x n u n , and

1 M ( x n , u n , t ) 1k [ 1 M ( x n 1 , x n , t ) 1 + 1 M ( x , u n , t ) 1 ] .
(2.22)

Now since M is triangular, by using (2.22), we get

1 M ( x , u n , t ) 1 1 1 k [ 1 M ( x , x n , t ) 1 + 1 M ( x n 1 , x n , t ) 1 ] ,

and so, letting n, u n x. Consequently, since Fx is closed, we have xFx. Then F has a fixed point. □

From the above theorem we can immediately obtain the following generalization for getting a common fixed point.

Theorem 2.17 Let (X,M,) be a complete fuzzy metric space, with M triangular, and ‘’ a partial order on X. Let YC(X) and, for every nN, F n :YC(Y) be a sequence of mappings such that, for every two mappings F i , F j and for all xY, u x F i (x), there exists u y F j (y) with u x u y such that

1 M ( u x , u y , t ) 1k [ 1 M ( x , u x , t ) 1 + 1 M ( y , u y , t ) 1 ] ,

for every t>0 and yY with xy and some k(0, 1 2 ). Also let there exist, for some x 0 Y, x 1 F 1 x 0 with x 0 x 1 , and the following condition be satisfied:

If x n x is a sequence in Y whose consecutive terms are comparable, then x n x, for all n.

Then there exists xY such that x F n x, that is, { F n } has a common fixed point.

Proof We can find x 2 F 2 x 1 such that x 1 x 2 and that

1 M ( x 1 , x 2 , t ) 1 k 1 k [ 1 M ( x 0 , x 1 , t ) 1 ] .

Also for x 2 there exists x 3 F 3 x 2 with x 2 x 3 and such that

1 M ( x 2 , x 3 , t ) 1 k 1 k [ 1 M ( x 0 , x 1 , t ) 1 ] .

By continuing this process, we get

1 M ( x n , x n + 1 , t ) 1 ( k 1 k ) n [ 1 M ( x 0 , x 1 , t ) 1 ] ,

where { x n } is a sequence with x n + 1 F n + 1 x n . Now similar to the proof of the preceding theorem, we can prove that { x n } is a Cauchy sequence and by the completeness of X it follows that { x n } converges to some xX. Furthermore, xY and x n x. Now suppose that F N is any arbitrary member of F n . Since x n F n x n 1 , x n 1 x for all n, and by the hypothesis, there exists u n F N x such that x n u n , and

1 M ( x n , u n , t ) 1k [ 1 M ( x n 1 , x n , t ) 1 + 1 M ( x , u n , t ) 1 ] ,

thus

1 M ( x , u n , t ) 1 1 1 k [ 1 M ( x , x n , t ) 1 + 1 M ( x n 1 , x n , t ) 1 ] .

Next by the letting n, we get u n x, and then x F N x. As F N is an arbitrary member of F n , x F n x, and x is a common fixed point of { F n }. The theorem is proved. □

Example 2.18 Let X=[0,) with t-norm defined ab=min{a,b} for all a,b[0,1] and M(x,y,t)= t t + x y , for all x,yX and t>0. Then (X,M,) is a complete fuzzy metric space. Let the natural ordering ≤ of the numbers as the partial ordering . Define Y=[0,1] and F:YC(Y) as Fx={z, x 5 } for each 0x< 1 2 , and {z, x 4 } for each 1 2 x1, where zY is an arbitrary. If x,yY such that xy and u x =zFx, then there exists u y =zFy such that u x u y and (2.18) is satisfied. Thus F is a fuzzy order K-set-valued mapping. But if u x zFx, then three cases arise.

Case (i). If 0xy< 1 2 , then for every t>0

1 M ( x 5 , y 5 , t ) 1= y x 5 t 4 9 [ 4 ( x + y ) 5 t ] = 4 9 [ 1 M ( x , x 5 , t ) 1 + 1 M ( y , y 5 , t ) 1 ] .

Case (ii). If 1 2 xy1, then for every t>0

1 M ( x 4 , y 4 , t ) 1= y x 4 t 4 9 [ 3 ( x + y ) 4 t ] = 4 9 [ 1 M ( x , x 4 , t ) 1 + 1 M ( y , y 4 , t ) 1 ] .

Case (iii). If 0x< 1 2 y1, then for every t>0

1 M ( x 5 , y 4 , t ) 1 4 9 [ 16 x + 15 y 20 t ] = 4 9 [ 1 M ( x , x 5 , t ) 1 + 1 M ( y , y 4 , t ) 1 ] .

Hence F is a fuzzy order K-set-valued mapping with k= 4 9 < 1 2 . Moreover, there exists x 0 =0 (or x 0 =z) with x 1 =0 ( x 1 =z) such that x 0 x 1 . Thus all the hypotheses of Theorem 2.16 are satisfied and x=0 (or x=z) is the fixed point of F.

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Acknowledgements

The authors would like to thank the referees for giving useful suggestions and comments for the improvement of this paper.

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Correspondence to S Mansour Vaezpour or Choonkil Park.

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All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.

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Sadeghi, Z., Vaezpour, S.M., Park, C. et al. Set-valued mappings in partially ordered fuzzy metric spaces. J Inequal Appl 2014, 157 (2014). https://doi.org/10.1186/1029-242X-2014-157

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Keywords

  • fixed point
  • coincidence point
  • set-valued mapping
  • partially ordered set
  • fuzzy metric space