Some New Fixed Point Theorems in Complex Valued 𝐺-Metric Spaces

Abstract

Some new fixed point theorems are established in the setting of complex valued -metric spaces. These new results improve and generalize Kang et al.’s results, the Banach contraction principle, and some well-known results in the literature.

1. Introduction and Preliminaries

It is well known that Banach contraction principle [1] plays an important role in various fields of applied mathematical analysis and scientific applications and has been generalized and improved in many various different directions; see [2–16] and references therein. In 2011, Azam et al. [2] introduced so-called complex valued metric spaces and proved the existence of fixed points under some contraction conditions. In 2006, Mustafa and Sims [3] introduced the concept of -metric spaces to extend and generalize the notion of metric spaces. In 2013, Kang et al. [8] introduced the concept of complex valued -metric spaces to generalize and improve the notion of -metric spaces. In [8], the authors gave a complex valued -metric version of Banach contraction principle.

In what follows we will give some definitions and known results which will be needed in the sequel. Throughout the present paper, the symbols , , and are used to denote the sets of positive integers, real numbers, and complex numbers, respectively.

In 2006, Mustafa and Sims [3] introduced a new class of metric spaces called generalized metric spaces or -metric spaces as follows.

Definition 1 (see [3]). Let be a nonempty set and let be a function satisfying the following:(G1) if ,(G2) for all with ,(G3) for all with ,(G4) (symmetry in all three variables),(G5) for all (rectangle inequality).Then the function is called a generalized metric or a -metric on and the pair is called a -metric space.

Example 2 (see [3]). Let be a usual metric space. Then and are all -metric spaces, where for all .
For any , we can define a partial order on as follows: So, it is easy to see that holds if one of the following conditions is satisfied:(C1) and ,(C2) and ,(C3) and ,(C4) and .
In particular, we will write if and one of (C2), (C3), and (C4) is satisfied and we will write if only (C4) is satisfied.

Remark 3. It is obvious that the following statements hold.(1)If , then .(2)If and , then .
The idea of complex metric space was initialed by Azam et al. [2].

Definition 4. Let be a nonempty set. Suppose that the mapping satisfies(C1) for all and if and only if ,(C2) for all ,(C3) for all .Then is called a complex valued metric on and the pair is called a complex valued metric space.

Example 5 (see [6, Example 2]). Let where Define as follows: where , . Then is a complete complex valued metric space.

The notion of complex valued -metric space was introduced by Kang et al. [8] to generalize the notion of complex valued metric space and -metric space as follows.

Definition 6 (see [8]). Let be a nonempty set and let be a function satisfying the following:(CG1) if ,(CG2) for all with ,(CG3) for all with ,(CG4) (symmetry in all three variables),(CG5) for all (rectangle inequality). Then the function is called a complex valued generalized metric or a complex valued -metric on . We call the pair a complex valued -metric space.

Remark 7. In fact, condition (CG2) defined in [8] was stated as follows:(CG2) for all with . In this paper, we use the weak version of (CG2) as in Definition 6.

Example 8. Let and be defined by where for any . Then is a complex valued -metric space.

Definition 9. Let be a complex valued -metric space. A point in is a fixed point of a mapping if . The set of fixed points of is denoted by .

Definition 10 (see [8]). Let be a complex valued -metric space and let be a sequence in . We say that is complex value  -convergent to if, for every with , there exists such that for all . We refer to as the limit of the sequence and we write as .

Definition 11 (see [8]). Let be a complex valued -metric space.(i)A sequence in is said to be complex valued -Cauchy if, for every with , there exists such that for all .(ii) is said to be complete if every complex valued -Cauchy sequence in is complex valued -convergent in .

Some crucial facts in complex valued -metric spaces are listed as follows. First, the following proposition follows easily due to (CG5).

Proposition 12 (see [8]). Let be a complex valued -metric space. Then, for any , the following hold:(1),(2).

Proposition 13 (see [8]). Let be a complex valued -metric space. Then, for a sequence in and point , the following are equivalent.(1)  is complex valued  -convergent to  .(2)  as  .(3)  as  .(4)  as  .

Proposition 14 (see [8]). Let be a complex valued -metric space and let be a sequence in . Then is complex valued -Cauchy sequence if and only if as .

Proposition 15 (see [8]). Let be a complex valued -metric space. Then the function is jointly continuous in all three of its variables.

The main aim of this paper is to establish some new fixed point theorems which extend and generalize Kang et al.’s results in [8], the Banach contraction principle, and some well-known results in the literature.

2. Main Results

Recall that a function is said to be an -function (or -function) [11–16] if

It is obvious that if is a nondecreasing function or a nonincreasing function, then is an -function. So the set of -functions is a rich class.

Recently, Du [13] first proved the following characterizations of -functions.

Theorem 16 (see [13]). Let be a function. Then the following statements are equivalent.(a) is an -function.(b)For each , there exist and such that for all .(c)For each , there exist and such that for all .(d)For each , there exist and such that for all .(e)For each , there exist and such that for all .(f)For any nonincreasing sequence in , we have .(g) is a function of contractive factor; for any strictly decreasing sequence in , we have .

The following new fixed point theorem is one of the main results of this paper. It can be considered as a complex valued -metric version of Banach contraction principle and will generalize and improve [8, Theorem 2.5] and some well-known results in the literature.

Theorem 17. Let be a complete complex valued -metric space and let be a mapping on . Suppose that there exists a -function such that Then has a unique fixed point on .

Proof . Let be given. Define the sequence by For each , by (7), we have which implies Let for . Then, by (10), we have So we know that is a strictly decreasing sequence in . Applying (g) of Theorem 16, we obtain That is, Let Then . For each , by (10) again, we have For any with , by the last inequality and repeated use of (CG5), we get Since , . Hence, by the last inequality, we obtain For any , by Proposition 12, we obtain which implies From (17) and (19), we get as . Applying Proposition 14, is a complex valued -Cauchy sequence in . By the completeness of , there exists such that is complex valued -convergent to .
Next, we prove that . Assume that . For each , by (7), we have which deduces Since as and is continuous in all three of its variables, from Proposition 15 and by taking limit from both sides of (21), we get Since , by Remark 3, we know Hence, taking into account (22) and (23), we have which is a contradiction. Therefore or .
Finally, we want to show the uniqueness of fixed point of (i.e., is a singleton set). We have shown , so it suffices to show that . Let . Suppose . By (7), we obtain which implies By (26), we have Since , we have which deduces and hence . This contradicts (CG2). Therefore, it must be and so . The proof is completed.

Here, we give a simple example illustrating Theorem 17.

Example 18. Let and be defined by where for any . Then is a complex valued -metric space. Define and by Then is a -function. For any , where , we have By a routine calculation, one can verify that So all the hypotheses of Theorem 17 are fulfilled. It is therefore possible to apply Theorem 17 to get the fact that has a unique fixed point on (precisely speaking, is the unique fixed point of ).

The following fixed point theorem established in -metric space is immediate from Theorem 17.

Theorem 19. Let be a complete -metric space and let : be a mapping on . Suppose that there exists a -function such that Then has a unique fixed point on .

Since any nondecreasing function or any nonincreasing function is an -function, by applying Theorem 17, we have the following results.

Corollary 20. Let be a complete complex valued -metric space and let be a mapping on . Suppose that there exists a nondecreasing function such that Then has a unique fixed point on .

Corollary 21. Let be a complete -metric space and let be a mapping on . Suppose that there exists a nondecreasing function such that Then has a unique fixed point on .

Corollary 22. Let be a complete complex valued -metric space and let be a mapping on . Suppose that there exists a nonincreasing function such that Then has a unique fixed point on .

Corollary 23. Let be a complete -metric space and let be a mapping on . Suppose that there exists a nonincreasing function such that Then has a unique fixed point on .

Corollary 24 . (see [8, Theorem 2.5]). Let be a complete complex valued -metric space and let be a contraction mapping on ; that is, for all , where . Then has a unique fixed point.

Conflict of Interests

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

Acknowledgment

Wei-Shih Du was supported by Grant no. NSC 102-2115-M-017-001 of the National Science Council of the Republic of China.