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

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.

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,BK ( 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 , ) ,

andd ( 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 = thnis 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 → 2Xbe 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 letF:XK ( 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 ̄ .

References

  1. Zadeh LA: Fuzzy sets. Inf. Control 1965, 8: 338–353.

    MathSciNet  Article  Google Scholar 

  2. Kaleva O, Seikkala S: On fuzzy metric spaces. Fuzzy Sets Syst 1984, 12: 215–229.

    MathSciNet  Article  Google Scholar 

  3. Schweizer B, Sklar A: Statistical metric spaces. Pac J Math 1960, 10: 313–334.

    MathSciNet  Article  Google Scholar 

  4. Erceg MA: Metric spaces in fuzzy set theory. J Math Anal Appl 1979, 69: 205–230.

    MathSciNet  Article  Google Scholar 

  5. Kramosil I, Michalek J: Fuzzy metric and statistical metric spaces. Kyber-netica 1975, 11: 326–334.

    MathSciNet  Google Scholar 

  6. George A, Veeramani P: On some results in fuzzy metric spaces. Fuzzy Sets Syst 1994, 64: 395–399.

    MathSciNet  Article  Google Scholar 

  7. George A, Veeramani P: On some results of analysis for fuzzy metric spaces. Fuzzy Sets Syst 1997, 90: 365–368.

    MathSciNet  Article  Google Scholar 

  8. Sedghi S, Altun I, Shobe N: Coupled fixed point theorems for contractions in fuzzy metric spaces. Nonlinear Anal 2010, 72: 1298–1304.

    MathSciNet  Article  Google Scholar 

  9. Fang JX: On fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst 1992, 46: 107–113.

    Article  Google Scholar 

  10. Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst 1983, 27: 385–389.

    MathSciNet  Article  Google Scholar 

  11. Gregori V, Sapena A: On fixed-point theorems in fuzzy metric spaces. Fuzzy Sets Syst 2002, 125: 245–252.

    MathSciNet  Article  Google Scholar 

  12. Miheţ D: On the existence and the uniqueness of fixed points of Sehgal contractions. Fuzzy Sets Syst 2005, 156: 135–141.

    Article  Google Scholar 

  13. Miheţ D: On fuzzy contractive mappings in fuzzy metric spaces. Fuzzy Sets Syst 2007, 158: 915–921.

    Article  Google Scholar 

  14. Liu Y, Li Z: Coincidence point theorems in probabilistic and fuzzy metric spaces. Fuzzy Sets Syst 2007, 158: 58–70.

    Article  Google Scholar 

  15. Žikić T: On fixed point theorems of Gregori and Sapena. Fuzzy Sets Syst 2004, 144: 421–429.

    Article  Google Scholar 

  16. Razani A: A contraction theorem in fuzzy metric space. Fixed Point Theory Appl 2005,2005(3):257–265.

    MathSciNet  Article  Google Scholar 

  17. Deb Ray A, Saha PK: Fixed point theorems on generalized fuzzy metric spaces. Hacettepe J Math Stat 39(2010):

    Google Scholar 

  18. Som T, Mukherjee RN: Some fixed point theorems for fuzzy mappings. Fuzzy Sets Syst 1989, 33: 213–219.

    MathSciNet  Article  Google Scholar 

  19. Rodríguez-López J, Romaguera S: The Hausdorff fuzzy metric on compact sets. Fuzzy Sets Syst 2004, 147: 273–283.

    Article  Google Scholar 

  20. Mizoguchi N, Takahashi W: fixed point theorems for multivalued mappings on complete metric spaces. J Math Anal Appl 1989, 141: 177–188.

    MathSciNet  Article  Google Scholar 

  21. Aubin JP, Siegel J: Fixed points and stationary points of dissipative multivalued maps. Proc Am Math Soc 1980, 78: 391–398.

    MathSciNet  Article  Google Scholar 

  22. Amini-Harandi A: Endpoints of set-valued contractions in metric spaces. Nonlin Anal 2010, 72: 132–134.

    MathSciNet  Article  Google Scholar 

  23. Hussain N, Amini-Harandi A, Cho YJ: Approximate endpoints for set-valued contractions in metric spaces. Fixed Point Theory Appl 2010., 2010: Article ID 614867

    Google Scholar 

  24. Fakhar M: Endpoints of set-valued asymptotic contractions in metric spaces. Appl Math Lett 2010, 24: 428–431.

    MathSciNet  Article  Google Scholar 

  25. Jachymski J: A stationary point theorem characterizing metric completeness. Appl Math Lett 2011, 24: 169–171.

    MathSciNet  Article  Google Scholar 

  26. Wlodarczyk K, Klim D, Plebaniak R: Existence and uniqueness of endpoints of closed set-valued asymptotic contractions in metric spaces. J Math Anal Appl 2007, 328: 46–57.

    MathSciNet  Article  Google Scholar 

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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.

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Kiany, F., Amini-Harandi, A. Fixed point and endpoint theorems for set-valued fuzzy contraction maps in fuzzy metric spaces. Fixed Point Theory Appl 2011, 94 (2011). https://doi.org/10.1186/1687-1812-2011-94

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Keywords

  • Fixed point
  • Endpoint
  • Set-valued fuzzy contraction map
  • Fuzzy metric space
  • Topology