Open Access

Fixed point and endpoint theorems for set-valued fuzzy contraction maps in fuzzy metric spaces

Fixed Point Theory and Applications20112011:94

https://doi.org/10.1186/1687-1812-2011-94

Received: 16 April 2011

Accepted: 6 December 2011

Published: 6 December 2011

Abstract

In this paper, we first present a fixed point theorem for set-valued fuzzy contraction type maps in complete fuzzy metric spaces which extends and improves some well-know results in literature. Then by presenting an endpoint result we initiate endpoint theory for fuzzy contraction maps in fuzzy metric spaces.

02000 Mathematics Subject Classification: 47H10, 54H25.

Keywords

Fixed pointEndpointSet-valued fuzzy contraction mapFuzzy metric spaceTopology

1. Introduction and preliminaries

Many authors have introduced the concept of fuzzy metric spaces in different ways [14]. Kramosil and Michalek [5] introduced the fuzzy metric space by generalizing the concept of the probabilistic metric space to fuzzy situation. George and Veeramani [6, 7] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [5] and obtained a Hausdorff topology for this kind of fuzzy metric spaces. Recently, the fixed point theory in fuzzy metric spaces has been studied by many authors [818]. In [11], the following definition is given.

Definition 1.1. A sequence (t n ) of positive real numbers is said to be an s-increasing sequence if there exists m0 such that t m + 1 ≤ tm+1, for all mm0.

Gregori and Sapena [11] proved the following fixed point theorem.

Theorem 1.2. Let (X, M, *) be a complete fuzzy metric space such that for every s-increasing sequence (t n ) and every x, y X
lim n * i = n M ( x , y , t n ) = 1 .
Suppose f : XX is a map such that for each x, y X and t > 0, we have
M ( f x , f y , k t ) M ( x , y , t ) ,

where 0 < k < 1. Then, f has a unique fixed point.

In this article, we first give a fixed point theorem for set-valued contraction maps which improve and generalize the above-mentioned result of Gregori and Sapena. Then, in Section 2, we initiate endpoint theory in fuzzy metric spaces by presenting an endpoint result for set-valued maps.

To set up our results in the next section we recall some definitions and facts.

Definition 1.3 (3). A binary operation * : [0, 1] × [0, 1] → [0, 1] is called a continuous t-norm if ([0,1], *) is an abelian topological monoid with unit 1 such that a * bc * d whenever ac and bd for all a, b, c, [0, 1]. Examples of t-norm are a * b = ab and a * b = min{a, b}.

Definition 1.4 (6). The 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 X2 × [0, ∞) satisfying the following conditions, for each x, y, z X and t, s > 0,
  1. (1)

    M(x, y, t) > 0,

     
  2. (2)

    M(x, y, t) = 1 if and only if x = y,

     
  3. (3)

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

     
  4. (4)

    M(x, y, t) * M(y, z, s) ≤ M(x, z, t + s),

     
  5. (5)

    M(x, y, t) : (0, ∞) → [0,1] is continuous.

     
Example 1.5. [6] Let (X, d) be a metric space. Define a * b = ab (or a * b = min{a, b}) and for all x, y X and t > 0,
M ( x , y , t ) = t t + d ( x , y ) .

Then (X, M, *) is a fuzzy metric space. We call this fuzzy metric M induced by the metric d the standard fuzzy metric.

Definition 1.6. Let (X, M, *) be a fuzzy metric space.
  1. (1)

    A sequence {x n } is said to be convergent to a point x X if limn→∞ M(x n , x, t) = 1 for all t > 0.

     
  2. (2)
    A sequence {x n } is called a Cauchy sequence if
    lim m , n M ( x m , x n , t ) = 1 ,
     
     for all t > 0.
  1. (3)

    A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.

     
  2. (4)

    A subset A X is said to be closed if for each convergent sequence {x n } with x n A and x n x, we have x A.

     
  3. (5)

    A subset A X is said to be compact if each sequence in A has a convergent subsequence.

     

Throughout the article, let K ( X ) denote the class of all compact subsets of X.

Lemma 1.7. [10]For all x, y X, M(x, y,.) is non-decreasing.

Definition 1.8. Let (X, M, *) be a fuzzy metric space. M is said to be continuous on X2 × (0, ∞) if
lim n M ( x n , y n , t n ) = M ( x , y , t ) ,
whenever {(x n , y n , t n )} is a sequence in X2 × (0, ∞) which converges to a point (x, y, t) X2 × (0, ∞); i.e.,
lim n M ( x n , x , t ) = lim n M ( y n , y , t ) = 1 and lim n M ( x , y , t n ) = M ( x , y , t ) .

Lemma 1.9. [10]M is a continuous function on X2 × (0, ∞).

2. Fixed point theory

The following lemma is essential in proving our main result.

Lemma 2.1. Let (X, M, *) be a fuzzy metric space such that for every x, y X, t > 0 and h > 1
lim n * i = n M ( x , y , t h i ) = 1 .
(2.1)
Suppose {x n } is a sequence in X such that for all n ,
M ( x n , x n + 1 , α t ) M ( x n - 1 , x n , t ) ,

where 0 < α < 1. Then {x n } is a Cauchy sequence.

Proof. For each n and t > 0, we have
M ( x n , x n + 1 , t ) M x n - 1 , x n , 1 α t M x n - 2 , x n - 1 , 1 α 2 t M x 0 , x 1 , 1 α n - 1 t .
Thus for each n , we get
M x n , x n + 1 , t M x 0 , x 1 , 1 α n - 1 t .
Pick the constants h > 1 and l such that
h α < 1 and i = l 1 h i = 1 h l 1 - 1 h < 1 .
Hence, for mn, we get
M ( x n , x m , t ) M x n , x m , 1 h l + 1 h l + 1 + + 1 h l + m t M x n , x n + 1 , 1 h l t * M x n + 1 , x n + 2 , 1 h l + 1 t * * M x m - 1 , x m , 1 h l + m t M x 0 , x 1 , 1 α n - 1 h l t * M x 0 , x 1 , 1 α n h l + 1 t * * M x 0 , x 1 , 1 α m - 2 h l + m - n - 2 t M x 0 , x 1 , 1 α h n - 1 t * M x 0 , x 1 , 1 α h n t * * M x 0 , x 1 , 1 α h m - 2 t * i = n M x 0 , x 1 , 1 α h i - 1 t
Then, from the above, we have
lim m , n M ( x n , x m , t ) lim n * i = n M x 0 , x 1 , 1 ( α h ) i - 1 t = 1 ,
for each t > 0. Therefore, we get
lim m , n M ( x n , x m , t ) = 1 ,

for each t > 0 and so {x n } is a Cauchy sequence.

In 2004, Rodríguez-López and Romaguera [19] introduced Hausdorff fuzzy metric on the set of the non-empty compact subsets of a given fuzzy metric space.

Definition 2.2. ([19]) Let (X, M, *) be a fuzzy metric space. For each A , B K ( X ) and t > 0, set
H M ( A , B , t ) = min { inf x A sup y B M ( x , y , t ) , inf y B sup x A M ( x , y , t ) } .

Lemma 2.3. [19]Let (X, M, *) be a fuzzy metric space. Then, the 3-tuple ( K ( X ) , H M , * ) is a fuzzy metric space.

Now we are ready to prove our first main result.

Theorem 2.4. Let (X, M, *) be a complete fuzzy metric. Suppose F : XX is a set-valued map with non-empty compact values such that for each x, y X and t > 0, we have
H M ( F x , F y , α ( d ( x , y , t ) ) t ) M ( x , y , t ) ,
(2.2)
where α : [0, ∞) → [0,1) satisfying
limsup r t + α ( r ) < 1 , t [ 0 , ) ,

and d ( x , y , t ) = t M ( x , y , t ) - t . Furthermore, assume that (X, M, *) satisfies (2.1) for some x0 X and x1 Fx0. Then F has a fixed point.

Proof. Let t > 0 be fixed. Notice first that if A and B are non-empty compact subsets of X and x A then by [19, Lemma 1], there exists a y B such that
H M ( A , B , t ) sup b B M ( x , b , t ) = M ( x , B , t ) = M ( x , y , t ) .
Thus given αH M (A, B, t) there exists a point y B such that
M ( x , y , t ) α .
Let x0 X and x1 Fx0. If Fx0 = Fx1 then x1 Fx1 and x1 is a fixed point of F and we are finished. So, we may assume that Fx0Fx1. From (2.2), we get
H M ( F x 0 , F x 1 , α ( d ( x 0 , x 1 , t ) ) t ) M ( x 0 , x 1 , t ) .
Since x1 Fx0 and F is compact valued then by Rodríguez-López and Romaguera [19, Lemma 1] there exists a x2 Fx1 satisfying
M ( x 1 , x 2 , t ) M ( x 1 , x 2 , α ( d ( x 0 , x 1 , t ) ) t ) = sup y F x 1 M ( x 1 , y , α ( d ( x 0 , x 1 , t ) ) t ) H M ( F x 0 , F x 1 , α ( d ( x 0 , x 1 , t ) ) t ) M ( x 0 , x 1 , t ) .
Continuing this process, we can choose a sequence {x n }n ≥ 0in X such that xn+1 Fx n satisfying
M ( x n + 1 , x n + 2 , t ) M ( x n + 1 , x n + 2 , α ( d ( x n , x n + 1 , t ) ) t ) = sup y F x n + 1 M ( x n + 1 , y , α ( d ( x n , x n + 1 , t ) ) t ) H M ( F x n , F x n + 1 , α ( d ( x n , x n + 1 , t ) ) t ) M ( x n , x n + 1 , t ) .

Then, the sequence {M(xn+1, xn+2, t)} n is non-decreasing.

Thus {d(xn+1, xn+2, t)} n is a non-negative non-increasing sequence and so is convergent, say to, l ≥ 0. Since by the assumption
limsup n α ( d ( x n + 1 , x n + 2 , t ) ) limsup r t + α ( r ) < 1 ,
then there exists k < 1 and N such that
α ( d ( x n + 1 , x n + 2 , t ) ) < k , n > N .
(2.4)
Since M(x, y,.) is non-decreasing then (2.3) together with (2.4) yield
M ( x n + 1 , x n + 2 , k t ) M ( x n + 1 , x n + 2 , α ( d ( x n , x n + 1 , t ) ) t ) M ( x n , x n + 1 , t ) .
Then from the above, we get
M ( x n + 1 , x n + 2 , k t ) M ( x n , x n + 1 , t ) .
Hence by Lemma 2.1, we get {x n }, which is a Cauchy sequence. Since (X, M, *) is a complete fuzzy metric space, then there exists x ̄ X such that lim n x n = x ̄ , that means lim n M ( x n , x ̄ , t ) = 1 , for each t > 0. Thus, lim n d ( x n , x ̄ , t ) = 0 , for each t > 0. Since
limsup n α ( d ( x n , x ̄ , t ) ) limsup r 0 + α ( r ) < 1 ,
then there exists k < l < 1 such that
limsup n α ( d ( x n , x ̄ , t ) ) < l .
Now we claim that x ̄ F x ̄ . To prove the claim notice first that since H M ( F x n , F x ̄ , l t ) H M ( F x n , F x ̄ , k t ) H M ( F x n , F x ̄ , α ( d ( x n , x ̄ , t ) ) t ) M ( x n , x ̄ , t ) , and lim n M ( x n , x ̄ , t ) = 1 then for each t > 0, we get
lim n H M ( F x n , F x ̄ , t ) = 1 .
(2.5)
Since xn+1 Fx n then from (2.5), we have
lim n sup y F x ̄ M ( x n + 1 , y , t ) = 1 .
Thus there exists a sequence y n F x ̄ such that
lim n M ( x n , y n , t ) = 1 ,
for each t > 0. For each n , we have
M ( y n , x ̄ , s + t ) M ( y n , x n , s ) * M ( x n , x ̄ , t ) .
Hence, from the above, we get
lim n M ( y n , x ̄ , t ) = 1 ,

which means lim n y n = x ̄ . Since F x ̄ is closed (note that F x ̄ is compact), y n x ̄ and y n F x ̄ then, we get x ̄ F x ̄ .

Corollary 2.5. Let (X, M, *) be a complete fuzzy metric. Suppose F : XX is a set-valued map with non-empty compact values such that for each x, y X and t > 0, we have
H M ( F x , F y , k t ) M ( x , y , t ) ,

where 0 < k < 1. Furthermore, assume that (X, M, *) satisfies (2.1) for some x0 X and x1 Fx0. Then F has a fixed point.

From Corollary 2.5, we get the following improvement of the above mentioned result of Gregori and Sapena [11] (note that for each t > 0 and h > 1, the sequence t n = th n is s-increasing).

Theorem 2.6. Let (X, M, *) be a complete fuzzy metric space. Suppose f : XX is a map such that for each x, y X and t > 0, we have
M ( f x , f y , k t ) M ( x , y , t ) ,

where 0 < k < 1. Furthermore, assume that (X, M, *) satisfies (2.1) for some x0 X, each t > 0 and h > 1. Then f has a fixed point.

Let (X, d) be a metric space and A and B are non-empty closed bounded subsets of X. Now set
H ( A , B ) = max { sup x A inf y B d ( x , y ) , sup y B inf x A d ( x , y ) } .

Then H is called the Hausdorff metric. Now, we are ready to derive the following version of Mizoguchi-Takahashi fixed point theorem [20].

Corollary 2.7. Let (X, d) be a complete metric space. Suppose F : MM is a set-valued map with non-empty compact values such that for some k < 1
H ( F x , F y ) α ( d ( x , y ) ) d ( x , y ) ,
where α : [0, ∞) → [0,1) satisfying
limsup r t + α ( r ) < 1 , t [ 0 , ) .

Then F has a fixed point.

Proof. Let (X, M, *) be standard fuzzy metric space induced by the metric d with a * b = ab. Now we show that the conditions of Theorem 2.4 are satisfied. Since (X, d) is a complete metric space then (X, M, *) is complete. It is easy to see that (X, M, *) satisfies (2.1). For each non-empty closed bounded subsets of X, we have
H M ( A , B , t ) = min inf x A sup y B M ( x , y , t ) , inf y B sup x A M ( x , y , t ) = min inf x A sup y B t t + d ( x , y ) , inf y B sup x A t t + d ( x , y ) = min t t + sup x A inf y B d ( x , y ) , t t + sup y B inf x A d ( x , y ) = t t + max sup x A inf y B d ( x , y ) , sup y B inf x A d ( x , y ) = t t + H ( A , B ) .
By the above and our assumption, we have
H M ( F x , F y , α ( d ( x , y , t ) ) t ) = α ( d ( x , y ) ) t α ( d ( x , y ) ) t + H ( F x , F y ) α ( d ( x , y ) ) t α ( d ( x , y ) ) ( t + d ( x , y ) ) = t t + d ( x , y ) = M ( x , y , t ) ,

for each t > 0 and each x, y X. Therefore, the conclusion follows from Theorem 2.4.

3. Endpoint theory

Let X be a non-empty set and let F : X → 2 X be a set-valued map. An element x X is said to be an endpoint (invariant or stationary point) of F, if Fx = {x}. The investigation of the existence and uniqueness of endpoints of set-valued contraction maps in metric spaces have received much attention in recent years [2126].

Definition 3.1. Let (X, M, *) be a fuzzy metric space and let F : XX be a multi-valued mapping. We say that F is continuous if for any convergent sequence x n x0 we have H M (Fx n , Fx0, t) → 1 as n → ∞, for each t > 0.

As far as we know the following is the first endpoint result for set-valued contraction type maps in fuzzy metric spaces.

Theorem 3.2. Let (X, M, *) be a complete fuzzy metric space and let F : X K ( X ) be a continuous set-valued mapping. Suppose that for each x X there exists y Fx satisfying
H M ( y , F y , k t ) M ( x , y , t ) , t > 0 ,
(3.1)

where k [0,1). Then, F has an endpoint.

Proof. For each x X, define the function f : X → [0, ∞) by f(x, t) = H M (x, Fx, t) = infyFxM(x, y, t), x X. Suppose that {x n } converges to x; then for any y Fx and z Fx n , we have
M x , y , t M x , x n , t 3 * M x n , z , t 3 * M z , y , t 3 M x , x n , t 3 * H M x n , F x n , t 3 * H M z , F x , t 3 M x , x n , t 3 * f x n , t 3 * H M F x n , F x , t 3 .
Since y Fx is arbitrary then from the above, we get
f ( x , t ) = H M ( x , F x , t ) M x , x n , t 3 * f x n , t 3 * H M F x n , F x , t 3 .
It follows from the continuity of F that
f x , t limsup n M x , x n , t 3 * f x n * H M F x n , F x , t 3 = limsup n f x n , t 3 .
Hence,
f x , t limsup n f x n , t 3 ,
whenever x n x. Let x0 X. Then by (3.1) there exists a x1 Fx0 such that
H M x 1 , F x 1 , k t M x 0 , x 1 , t .
Continuing this process, we can choose a sequence {x n }n≥0in X such that xn+1 Fx n satisfying
H M x n + 1 , F x n + 1 , k t M x n , x n + 1 , t .
(3.2)
From the definition of H M (x n , Tx n ), we have
M x n , x n + 1 , t H M x n , F x n , t .
(3.3)
From (3.2) and (3.3), we get
H M x n + 1 , F x n + 1 , k t M x n , x n + 1 , t H M x n , F x n , t H M x n , F x n , k t M x n - 1 , x n , 1 k t ,
(3.4)
which implies that {H M (x n , Fx n , kt)} n is a non-negative non-decreasing sequence of real numbers and so is convergent. To find the limit of {H(x n , Fx n , kt)} n notice that
H M x n + 1 , F x n + 1 , k t H M x n , F x n , t H M x n - 1 , F x n - 1 , 1 k t H M x 0 , F x 0 , 1 k n t .
(3.5)
Since Fx0 is compact then there exists a y0 Fx0 such that
H M x 0 , F x 0 , 1 k n t = M x 0 , y 0 , 1 k n t .
(3.6)
(3.5) together with (3.6) imply that for each n
H M x n + 1 , F x n + 1 , k t M x 0 , y 0 , 1 k n t .
From (2.1) we have lim n M x 0 , y 0 , 1 k n t = 1 and so
lim n H M x n , F x n , t = 1 , t > 0 .
From (3.2), we get
M x n , x n + 1 , t M x n - 1 , x n , 1 k t ,
from which and Lemma (2.1), we get {x n } is a Cauchy sequence. Since (X, M, *) is a complete fuzzy metric space then there exists a x ̄ X such that lim n x n = x ̄ . By assumption the function f(x) = H M (x, Fx, t) is upper semicontinuous, then
H M x ̄ , F x ̄ , t lim n H M x n , F x n , t = 1 .
Thus
H M x ̄ , F x ̄ , t = 1 ,

and so F x ̄ = x ̄ .

Declarations

Acknowledgements

This research was in part supported by the grant from IPM (90470017). The second author was also partially supported by the Center of Excellence for Mathematics, University of Shahrekord.

Authors’ Affiliations

(1)
Department of Mathematics, Science and Research Branch, Islamic Azad University
(2)
Department of Mathematics, University of Shahrekord
(3)
School of Mathematics, Institute for Research in Fundamental Sciences (IPM)

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© Kiany and Amini-Harandi; licensee Springer. 2011

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