On Fuzzy Fixed Points for Fuzzy Maps with Generalized Weak Property

Abstract

Let be a complex valued metric space and let be mappings from to a set of all fuzzy subsets of . We present sufficient conditions for the existence of a common -fuzzy fixed point of and . Our results improve and extend certain recent results in literature. Moreover, we discuss an illustrative example to highlight the realized improvements.

1. Introduction

In 1981, Heilpern [1] used the concept of fuzzy set to introduce a class of fuzzy mappings, which is a generalization of the set-valued mapping, and proved a fixed point theorem for fuzzy contraction mappings in a metric linear space. It is worth noting that the result announced by Heilpern [1] forms a fuzzy extension of the Banach contraction principle. Subsequently, several other authors have studied existence of fixed points of fuzzy mappings or in fuzzy metric spaces; for example, see the work of Azam et al. [2, 3], Bose et al. [4], Chang et al. [5], Cho and Petrot [6], Hussain et al. [7], Qiu and Shu [8], Rashwan and Ahmed [9], and Zhang [10].

Recently, Azam et al. [11] introduced the concept of complex valued metric space and obtained sufficient conditions for the existence of common fixed points of a pair of mappings satisfying contractive type condition involving rational expressions. For more details on complex valued metric space we refer the reader to [12–17].

In [18], Azam obtained some common fuzzy fixed points for fuzzy mappings under a rational contractive condition on a metric space in connection with the Hausdorff metric on the family of fuzzy sets.

The aim of this paper is to obtain a common -fuzzy fixed point of a pair of fuzzy mappings and on a complete complex valued metric space under a generalized rational contractive condition for -level sets. Our results generalize the results proved by Azam et al. [11, 18].

2. Preliminaries

Let be the set of complex numbersand. Define a partial order on as follows:

It follows that if one of the following conditions is satisfied:(i), (ii), (iii), (iv). In particular, we will write if and one of (i), (ii), and (iii) is satisfied and we will write if only (iii) is satisfied. Note that

Definition 1. Let be a nonempty set. Suppose that the mapping
satisfies(1)for all and if and only if ;(2) for all ;(3), for all .Then is called a complex valued metric on , and is called a complex valued metric space. A point is called interior point of a set whenever there exists such that

A point is called a limit point of whenever, for every ,

is called open whenever each element of is an interior point of . Moreover, a subset is called closed whenever each limit point of belongs to . The family is a subbasis for a Hausdorff topology on .

Let be a sequence in and . If for every with there is such that, for all , then is said to be convergent, converges to , and is the limit point of. We denote this by , or as . If for every with there is such that, for all , where , then is called a Cauchy sequence in . If every Cauchy sequence is convergent in , then is called a complete complex valued metric space. We require the following lemmas.

Lemma 2 (see [11]). Let be a complex valued metric space and let be a sequence in . Then converges to if and only if as .

Lemma 3 (see [11]). Let be a complex valued metric space and let be a sequence in . Then is a Cauchy sequence if and only if as where .

A fuzzy set in is a function with domain and values in ; is the collection of all fuzzy sets in . If is a fuzzy set and , then the function values are called the grade of membership of in . The -level set of is denoted by and is defined as follows:

Here denotes the closure of the set . Let be the collection of all fuzzy sets in a metric space . For means for each . We denote the fuzzy set by unless and until it is stated, where is the characteristic function of the crisp set . A fuzzy set in a metric linear space is said to be an approximate quantity if and only if is compact and convex in for each and. The collection of all approximate quantitiesinis denoted by .

Definition 4. Let be a nonempty set and let be a complex valued metric space. A mapping is called fuzzy mapping if is a mapping from into . A fuzzy mapping is a fuzzy subset on with membership function . The function is the grade of membership of in .

Definition 5. Let be a complex valued metric space and let be fuzzy mappings from into . A point is called a fuzzy fixed point of   if , for some . The point is called a common fuzzy fixed point of and if for some . When is called a common fixed point of fuzzy mappings.

3. Main Result

Let be a complex valued metric space. We denote the family of all nonempty, closed and bounded subsets of a complex valued metric space by .

From now on, we denote for and for and .

For , we denote

Lemma 6. Let be a complex valued metric space.(i)Let . If , then .(ii)Let and . If then .(iii)Let and let and . If , then for all or for all .

Remark 7. If is a metric space, for is the Hausdorff distance induced by the metric .

Let be a complex valued metric space and be a collection of nonempty closed subsets of . Let be a multivalued map. For and , define Thus for

Definition 8. Let be a complex valued metric space. A subset of is called bounded from below if there exists some , such that for all .

Definition 9. Let be a complex valued metric space. A multivalued mapping is called bounded from below if for each there exists such that for all .

Definition 10. Let be a complex valued metric space. The fuzzy mapping is said to have lower bound property (l.b property) on , if, for any associated with some , the multivalued mapping defined by is bounded from below. That is, for there exists an element such that for all , where is called lower bound of associated with .

Definition 11. Let be a complex valued metric space. The fuzzy mapping is said to have greatest lower bound property (g.l.b property) on , if for any and any , greatest lower bound of exists in for all . One denotes by the g.l.b of . That is,

3.1. Banach Type Fuzzy Fixed Point Result

Theorem 12. Let be a complete complex valued metric space and let be fuzzy mappings from into . Assume that there exists some , such that, for each and are nonempty closed bounded subsets of ; greatest lower bound of exists in for all and for all , where are nonnegative real numbers with . Then there exists some .

Proof. Let be an arbitrary point in . By assumption, we can find . So, we have By Lemma 6(iii), we have By definition there exists some , such that That is, By the meaning of and for , we get which implies that where Inductively, we can construct a sequence in such that, for , with , for and .
Now for , we get and so This implies that is a Cauchy sequence in . Since is complete, so there exists such that as We now show that and . From (16), we have By Lemma 6(iii), we have By definition there exists some such that That is, By the meaning of and for , we get Since by triangle inequality, we get So using (31) in (32), we get Taking the limit as , we get as By Lemma 2 [11], we have as Since is closed, so . Similarly, it follows that . Thus and have a common fuzzy fixed point.

By setting in Theorem 12, we get the following corollary.

Corollary 13. Let be a complete complex valued metric space and let be fuzzy mappings from into . Assume that there exists some , such that, for each , and are nonempty closed bounded subsets of ; greatest lower bound of exists in for all and for all , where and are nonnegative real numbers with . Then there exists some .

By setting in Theorem 12, we get the following corollary.

Corollary 14. Let be a complete complex valued metric space and let be fuzzy mapping from into . Assume that there exists some , such that, for each , is nonempty closed bounded subset of ; greatest lower bound of exists in for all and for all , where , and are nonnegative real numbers with . Then there exists some .

By Definition 11, one can have the following corollaries easily from Theorem 12.

Corollary 15. Let be a complete complex valued metric space and let be fuzzy mappings from into with g.l.b property such that, for each and and are nonempty closed bounded subsets of and for all , and , and are nonnegative real numbers with . Then there exists some .

Corollary 16. Let be a complete complex valued metric space and let be fuzzy mappings from into with g.l.b property such that, for each and and are nonempty closed bounded subsets of and for all , and and are nonnegative real numbers with . Then there exists some .

Corollary 17. Let be a complete complex valued metric space and let be fuzzy mapping from into with g.l.b property such that, for each and is nonempty closed bounded subset of and for all , and , and are nonnegative real numbers with . Then there exists some .

Corollary 18 (see [19]). Let be a complete complex valued metric space and let be multivalued mappings with g.l.b property such that for all , where , and are nonnegative real numbers with . Then there exists some .

Proof. Consider a pair of fuzzy mappings defined by where . Then Thus, Theorem 12 can be applied to obtain such that

3.2. Kannan Type Fuzzy Fixed Point Result

Theorem 19. Let be a complete complex valued metric space and let be fuzzy mappings from into . Assume that there exists some , such that, for each and are nonempty closed bounded subsets of ; greatest lower bound of ,exists in for all and for all and nonnegative real numbers , and with . Then there exists some .

Proof. Let be an arbitrary point in . By assumption, we can find . So, we have By Lemma 6(iii), we have
By definition there exists some , such that
That is, By the meaning of and for , we get which implies that
Thus where. Inductively, we can construct a sequence in such that, for , with , for and . Now for , we get and so This implies that is a Cauchy sequence in . Since is complete, there exists such that as . We now show that and . From (43), we get
By Lemma 6 (iii), we have
By definition there exists some such that That is, By the meaning of and for , we get
Now by using (58) and the triangular inequality, we get which implies that By letting in above inequality, we get By Lemma 2 [11], we have as . Since is closed, so . Similarly, it follows that . Thus there exists some .

By setting and in Theorem 19, we get the following corollary.

Corollary 20. Let be a complete complex valued metric space and let be fuzzy mappings from into . Assume that there exists some , such that, for each and are nonempty closed bounded subsets of ; greatest lower bound of ,exists in for all and for all and . Then there exists some .

By setting in Theorem 19, we get the following corollary.

Corollary 21. Let be a complete complex valued metric space and let be fuzzy mapping from into . Assume that there exists some , such that, for each , is nonempty closed bounded subset of ; greatest lower bound of exists in for all and for all and nonnegative reals , and with . Then there exists some .

By Definition 11, one can have the following corollaries easily from Theorem 19.

Corollary 22. Let be a complete complex valued metric space and let be fuzzy mappings from into with g.l.b property such that, for each and and are nonempty closed bounded subsets of and for all and nonnegative real numbers , and with . Then there exists some .

Remark 23. By Definition 11, one can have a host of corollaries of Kannan type contractive fuzzy mappings with g.l.b property easily from Theorem 19.

Corollary 24 (see[20]). Let be a complete complex valued metric space and let be multivalued mappings with g.l.b property such that for all and nonnegative real numbers , and with . Then there exists some .

Proof. Consider a pair of fuzzy mappings defined by where . Then Thus, Theorem 19 can be applied to obtain such that

3.3. Chatterjea Type Fuzzy Fixed Point Result

Theorem 25. Let be a complete complex valued metric space and let be fuzzy mappings from into . Assume that there exists some , such that, for each , and are nonempty closed bounded subsets of ; greatest lower bound of ,exists in for all and for all and nonnegative reals , and with . Then there exists some .

Proof. Let be an arbitrary point in . By assumption, we can find . So, we have By Lemma 6(iii), we have By definition there exists some , such that That is, By the meaning of and for , we get which implies that Similarly from (69), we have By Lemma 6(iii), we have By definition there exists some , such that That is, By the meaning of and for , we get which implies that Inductively, we can construct a sequence in such that, for , with , and . Now for , we get and so This implies that is a Cauchy sequence in . Since is complete, there exists such that as . We now show that and . From (69), we get
By Lemma 6(iii), we have By definition there exists some , such that That is, By the meaning of and for , we get Now by using the triangular inequality, we get and it follows that By using again triangular inequality, we get and it follows that By letting in above inequality, we get as . By Lemma 2 [11], we have as . Since is closed, so . Similarly, it follows that . Thus there exists some .

Corollary 26. Let be a complete complex valued metric space and let be fuzzy mappings from into . Assume that there exists some , such that, for each and are nonempty closed bounded subsets of ; greatest lower bound of ,exists in for all and for all and . Then there exists some .

By taking in Theorem 25, we get the following corollary.

Corollary 27. Let be a complete complex valued metric space and let be fuzzy mapping from into . Assume that there exists some such that, for each , is nonempty closed bounded subset of ; greatest lower bound of exists in for all and for all and nonnegative reals , and with . Then there exists some .

Corollary 28. Let be a complete complex valued metric space and let be fuzzy mappings from into with g.l.b property such that, for each and and are nonempty closed bounded subsets of and for all and nonnegative reals , and with . Then there exists some .

Remark 29. By Definition 11, one can have a host of corollaries of Chatterjea type contractive fuzzy mappings with g.l.b property easily from Theorem 25.

Corollary 30 (see [20]). Let be a complete complex valued metric space and let be multivalued mappings with g.l.b property such that for all and nonnegative reals , and with . Then there exists some .

Proof. Consider a pair of fuzzy mappings defined by where . Then Thus, Theorem 25 can be applied to obtain such that

Remark 31. Consider the following.(i)By setting , and in Corollary 18, , and in Corollary 24, and , and in Corollary 30 with different combinations, one can get corresponding results in [19, 20] as corollaries.(ii)By Remark 7 and Corollaries 18, 24, and 30, one can easily get the results of [13, 18–21].

Example 32. Let and let be the set of complexnumbers; define as follows: Then is a complete complex valued metric space. Define a pair of mappings for as follows.
For with , we have such that and then
Denoteand bythe greatest lower boundsof and . Then and also Moreover, if such that then
For , we have also as It follows that, with , such that , we have and for with and , such that , we have Hence and satisfy all the conditions of our main Theorem 12 to obtain .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the authors acknowledge with thanks DSR, KAU, for financial support.