Generalized Common Fixed Point Results via Greatest Lower Bound Property

Abstract

The aim of this paper is to unify the concept of greatest lower bound (g.l.b) property and establish some generalized common fixed results. We support our results by a nontrivial example.

1. Introduction and Preliminaries

The Banach fixed point theorem is used to establish the existence of a unique solution for a nonlinear integral equation [1]. Moreover this theorem plays an important role in several branches of mathematics. For instance, it has been used to show the existence of solutions of nonlinear Volterra integral equations and nonlinear integrodifferential equations in Banach spaces and to show the convergence of algorithms in computational mathematics. Because of its importance and usefulness for mathematical theory, it has become a very popular tool of mathematical analysis in many directions. Nadler [2] introduced the concept of multivalued contraction mappings and obtained the fixed points results for multivalued mappings. Huang and Zhang [3] introduced the notion of cone metric space which is a generalization of metric space. They extended Banach contraction principle to cone metric spaces. Since then, Arshad et al. [4], Azam and Arshad [5], Cho and Bae [6], and many others obtained fixed point theorems in cone metric spaces.

Azam et al. in [7] introduced the notion of complex-valued metric space and obtained some common fixed points of a pair of mappings satisfying rational expressions contractive condition. Although complex-valued metric spaces form a special class of cone metric space, yet this idea is intended to define rational expressions which are not meaningful in cone metric spaces. Subsequently, Rouzkard and Imdad [8] and Abbas et al. [9, 10] established some common fixed point theorems satisfying certain rational expressions in complex-valued metric spaces which generalize, unify, and complement the results of Azam et al. [7]. Sintunavarat et al. [11, 12] obtained common fixed point results by replacing constant of contractive condition with control functions. Klin-eam and Suanoom [13] established a common fixed point result for two single valued mappings in complex-valued metric spaces. Abbas et al. [14] introduced complex-valued generalized metric space and obtained common fixed point results in this space. For more details in fixed point theory, we refer the reader to [15–23]. Very recently, Ahmad et al. [24] introduced the notion of greatest lower bound (g.l.b.) property of the multivalued mappings and obtained some common fixed point results in the context of complex-valued metric spaces. Then Azam et al. [25] extend the concept of greatest lower bound (g.l.b.) property and proved some new common fixed point theorems in the setting of complex-valued metric spaces. In this paper, we present some new common fixed results and generalized the results of [24, 25].

Let denote the set of complex numbers. Let . 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 a mapping satisfies the following: (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 set whenever each element of is an interior point of and a subset is called closed set whenever each limit point of belongs to . The family is a subbase 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 , , 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 [7]). Let be a complex-valued metric space and let be a sequence in . Then converges to if and only if   as .

Lemma 3 (see [7]). Let be a complex-valued metric space and let be a sequence in . Then is a Cauchy sequence if and only if as .

2. Main Result

Let be a complex-valued metric space. In the sequel of [24], we denote nonempty, closed, and bounded subsets of by , respectively.

Throughout this paper, we denote for and for and .

For we denote

Remark 4. Let be a complex-valued metric space. If , then is a metric space. Moreover for , is the Hausdorff distance induced by .

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 5 (see [24]). Let be a complex-valued metric space. A nonempty subset of is called bounded from below if there exists some , such that for all .

Definition 6 (see [24]). Let be acomplex-valued metric space. A multivalued mapping is called bounded from below if for there exists such that for all .

Definition 7 (see [24]). Let be a complex-valued metric space. The multivalued mapping is said to have lower bound property (l.b. property) on , if for any 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 8 (see [24]). Let be a complex-valued metric space. The multivalued mapping is said to have greatest lower bound property (g.l.b. property) on , if greatest lower bound of exists in for all . We denote by the g.l.b. of . That is

Theorem 9. 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 have a common fixed point.

Proof. Let be an arbitrary point in and . From (14), we have This implies that for all . Since , we have So there exists some , such that That is By the greatest lower bound property (g.l.b. property) of and , we get which implies that Then Similarly, we get This implies that for all . Since , we have So there exists some , such that That is, By the greatest lower bound property (g.l.b. property) of and , we get which implies that Then Putting , we obtain a sequence in such that for and Now for , we get and so This implies that is a Cauchy sequence in . As is complete space, there exists such that as . We now show that and . From (14), we have This implies that for all . Since , we have By definition There exists some such that that is, By the greatest lower bound property (g.l.b. property) of and , we have Since we get which implies that Taking the limit as , we get as . By lemma 1 [7], we have as . Since is closed, . Similarly, it follows that . Hence and have a common fixed point and our theorem follows.

Consequently, we have the following corollaries.

By setting in Theorem 9, we get the following corollary.

Corollary 10. Let be a complete complex-valued metric space and let be multivalued mapping with g.l.b. property such that for all , where , , , , and are nonnegative real numbers with . Then has a fixed point.

By choosing in Theorem 9, we get the following corollary.

Corollary 11. 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 have a common fixed point.

By setting in Corollary 11, we get the following corollary.

Corollary 12. Let be a complete complex-valued metric space and let be multivalued mapping with g.l.b. property such that for all , where , , , and are nonnegative real numbers with . Then has a fixed point.

By choosing in Theorem 9, we get the following corollary.

Corollary 13. 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 have a common fixed point.

By setting in Corollary 13, we get the following corollary.

Corollary 14. Let be a complete complex-valued metric space and let be multivalued mapping with g.l.b. property such that for all , where , , , and are nonnegative real numbers with . Then has a fixed point.

By choosing in Theorem 9, we get the following corollary.

Corollary 15 (see [24]). 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 have a common fixed point.

By setting in Corollary 15, we get the following corollary.

Corollary 16 (see [24]). Let be a complete complex-valued metric space and let be multivalued mapping with g.l.b. property such that for all , where , , and are nonnegative real numbers with . Then has a fixed point.

By choosing in Theorem 9, we get the following corollary.

Corollary 17 (see [24]). Let be a complete complex-valued metric space and let be multivalued mappings with g.l.b. property such that for all , where are nonnegative real numbers with . Then have a common fixed point.

By setting in Corollary 17, we get the following corollary.

Corollary 18. Let be a complete complex-valued metric space and let be multivalued mapping with g.l.b. property such that for all , where are nonnegative real numbers with . Then has a fixed point.

By Remark 4, we get the following corollaries.

Corollary 19. Let be a complete metric space and let be multivalued mappings such that for all and , , , , and are nonnegative real numbers with . Then have a common fixed point.

Corollary 20. Let be a complete metric space and let be multivalued mappings such that for all and , , , , and are nonnegative real numbers with . Then has a fixed point.

Remark 21. By equating , , , , and to in all possible combinations, one can derive a host of corollaries which include Banach fixed point theorem for multivalued mappings in complete metric space.

Theorem 22. 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 have a common fixed point.

Proof. Let be an arbitrary point in and . From (55), we have This implies that for all . Since , we have So there exists some , such that That is, By using the greatest lower bound property (g.l.b. property) of and , we get which implies that As earlier, by the triangular inequality we get Similarly we can prove that By putting and continuing in a similar way to the proof of Theorem 9, we obtain that is a Cauchy sequence in and as . We now show that and . From (55), we have This implies that for all . Since , we have By definition There exists some such that that is, By the greatest lower bound property (g.l.b. property) of and , we have Since we get which implies that Taking the limit as , we get as . By lemma 1 [7], we have as . Since is closed, so . Similarly, it follows that . Hence and have a common fixed point and our theorem follows.

Corollary 23. Let be a complete complex-valued metric space and let be multivalued mapping with g.l.b. property such that for all , where , , and are nonnegative real numbers with . Then has fixed point.

Now, let us consider the following example.

Example 24. Let ; define by Then is a complex-valued metric space. Let be the mappings defined by The contractive condition of main theorem is trivial for the case when . Suppose without any loss of generality that all are nonzero and . Then

It is clear that, for any value of , , , , and , we have Thus Hence all the conditions of our main Theorem 9 are satisfied and is a common fixed point of and .

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All the authors contributed equally and significantly to writing this paper. All the authors read and approved the final paper.