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Journal of Function Spaces
Volume 2017, Article ID 9389768, 11 pages
https://doi.org/10.1155/2017/9389768
Research Article

Common Fixed Points of Four Maps Satisfying -Contraction on -Metric Spaces

1Department of Mathematics and Statistics, International Islamic University, H-10, Islamabad, Pakistan
2Department of Mathematics and Physics, Hebei University of Architecture, Zhangjiakou 075024, China
3School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 10001, China
4Department of Mathematics, Gomal University D. I. Khan, Khyber Pakhtunkhwa 29050, Pakistan

Correspondence should be addressed to Ma Zhenhua; moc.361@1891_auhgnehzam

Received 16 October 2017; Accepted 7 December 2017; Published 28 December 2017

Academic Editor: Ahmad S. Al-Rawashdeh

Copyright © 2017 Muhammad Nazam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We manifest some common fixed point theorems for four maps satisfying Círíc type -contraction and Hardy-Rogers type -contraction defined on complete -metric spaces. We apply these results to infer several new and old corresponding results. These results also generalize some results obtained previously. We dispense examples to validate our results.

1. Introduction and Preliminaries

After the famous Banach Contraction Principle, a large number of researchers revealed many fruitful generalizations of Banach’s fixed point theorem. One of these generalizations is known as -contraction presented by Wardowski [1]. Wardowski [1] evinced the fact that every -contraction defined on complete -metric space has a unique fixed point. The concept of -contraction proved to be another milestone in fixed point theory and numerous research papers on -contraction have been published (see, e.g., [28]). Recently, Cosentino and Vetro [9] established a fixed point result for Hardy-Rogers type -contraction and Mınak et al. [10] presented a fixed point result for Círíc type generalized -contraction.

In 1989, Bakhtin [11] investigated the concept of -metric spaces; however, Czerwik [12] initiated study of fixed point of self-mappings in -metric spaces and proved an analogue of Banach’s fixed point theorem. Since then, numerous research articles have been published comprising fixed point theorems for various classes of single-valued and multivalued operators in -metric spaces (see, e.g., [1319]).

We shall bring into use the idea of Círíc type -contraction and Hardy-Rogers type -contraction comprising four self-mappings defined on -metric space. We present common fixed point results for four self-maps satisfying Círíc type and Hardy-Rogers type -contraction on -metric space. We apply our results to infer several new and old results.

We denote by , by , by , and set of natural numbers by .

We bring back into reader’s mind some definitions and properties of -metric.

Definition 1 (see [12]). Let be a nonempty set and be a real number. A function is said to be a -metric if, for all , one has if and only if ,, In this case, the pair is called a -metric space (with coefficient ).

Definition 1 allows us to remark that the class of -metric spaces is effectually more general than that of metric spaces because a -metric is a metric when . The following example describes the significance of a -metric.

Example 2. Let be a metric space and , is a real number. Then is -metric space with . Apparently, and of Definition 1 are satisfied. If , then the convexity of the function implies that gives . Thus, for all , one has Therefore, , where , which shows that is a -metric space. Nevertheless, if is a metric space, then may not be a metric space. Indeed, if and (a usual metric), then does not define a metric on .

For the notions like convergence, completeness, and Cauchy sequence in the setting of -metric spaces, the reader is referred to Aghajani et al. [20], Czerwik [12], Amini-Harandi [21], Huang et al. [13], Hussain et al. [14], Radenović et al. [15], Khamsi and Hussain [16], Latif et al. [22], Parvaneh et al. [17], Roshan et al. [18], and Zabihi and Razani [23].

In line with Wardowski [1], Cosentino et al. [24] investigated a nonlinear function complying with the following axioms: is strictly increasing.For each sequence of positive numbers, the following equation holds: if and only if .There exists such that . implies , for each and some .

We denote by the set of all functions satisfying the conditions .

Example 3. Let be defined by (a),(b). It is easy to check that (a) and (b) are members of .

Definition 4 (see [25]). Let be a -metric space. The pair is said to be compatible if and only if

Lemma 5. Let be a -metric space. If there exist two sequences , such that Then .

Proof. Due to the triangular inequality we have and the result follows after applying limit as .

2. Círíc Type Fixed Point Theorem

In this section we set up a fixed point theorem for four self-maps satisfying Círíc type -contraction. We explain this theorem through an example and discuss its consequences.

Theorem 6. Let be a complete -metric space and , , , and are self-mappings on such that , . If, for all , for some and , the inequality holds, where then , , , and have a unique common fixed point provided , are continuous and pairs , are compatible.

Proof. Let . As , there exists such that . Since , we can choose such that . In general, and are chosen in such that and . Define a sequence in such that and for all ; we show that is a Cauchy sequence. Assume that ; then from contractive condition (6), we get for all , where If , then which is a contradiction due to . Therefore, for all Similarly, for all Hence, from (11) and (12), we have for all . Let for each , from (13) and axiom we have Repeating the process, we obtain On taking limit in (15), we have ; due to axiom () we get and () implies that there exists such that From (15), for all , we obtain On taking limit in (17), we have This implies there exists , such that for all or To prove as a Cauchy sequence, we use (19) and, for , we consider The convergence of the series entails Hence is a Cauchy sequence in . Since is a complete -metric space, there exists such that . Also we haveWe show that is a common fixed point of , , , and . Since is continuous, therefore, Since, the pair is compatible, so, Now put and in (6) and suppose on the contrary that ; we obtain where Taking the upper limit in (24), we have a contradiction; hence, implies . Again since is continuous, therefore, Since the pair is compatible, so, Now put and in (6) and suppose on the contrary that ; we obtain where Taking the upper limit in (29), we have a contradiction; hence, which then implies . Now assuming on the contrary, from (6) we get where Taking the upper limit in (32) and using the fact that , we have a contradiction; hence, which then implies . Finally, assume on the contrary . From (6) and the fact that , we obtain where This implies that a contradiction; hence, which then implies . Hence, proves to be a common fixed point of the four maps , , , and . Also is unique; indeed if is another fixed point of , , , and , then from (6), we have where Thus, from (38), we have which is a contradiction. Hence, and is a unique common fixed point of the four maps , , , and .

The following example explains Theorem 6.

Example 7. Let and define by , so is a complete -metric space. Define the mappings , for all by Clearly, , , , and are self-mappings complying with , . We note that the pair is compatible. If is a sequence in satisfying then and equivalently implies Uniqueness of limit gives which implies . Due to the continuity of , we have Similarly, the pair is compatible. Now for each , consider The above inequality can be written as Define the function by , for all . Hence, for all such that , we obtain Thus, contractive condition (6) is satisfied for all . Hence, all the hypotheses of Theorem 6 are satisfied; note that have a unique common fixed point .

The following theorems can easily be obtained by chasing the proof of Theorem 6. Theorem 8 generalizes the result presented by Bianchini [26].

Theorem 8 (Bianchini type). Let be a complete -metric space and , , , are self-mappings on such that , . If, for all , for some , the inequality holds, where then , , , and have a unique common fixed point provided , are continuous and pairs , are compatible.

Theorem 9 generalizes the result presented by Sehgal [27].

Theorem 9 (Sehgal type). Let be a complete -metric space and , , , and are self-mappings on such that , . If, for all , for some , the inequality holds, where then , , , and have a unique common fixed point provided , are continuous and pairs , are compatible.

Theorem 10 generalizes the result presented by Círíc [28].

Theorem 10. Let be a complete -metric space and , , , and are self-mappings on such that , . If, for all , for some , the inequality holds, where for all then , , , and have a unique common fixed point provided , are continuous and pairs , are compatible.

Theorem 11. Let be a complete -metric space and , , , and are self-mappings on such that , . If, for all , for some , the inequality holds, where for all then , , , and have a unique common fixed point provided , are continuous and pairs , are compatible.

Theorem 12. Let be a complete -metric space and , , , and are self-mappings on such that , . If for all , the inequality holds, where then , , , and have a unique common fixed point provided , are continuous and pairs , are compatible.

Proof. Since, for all , we have thus, if , we have where ; thus, the contractive condition (58) reduces to (6) with . Hence, Theorem 6 is a generalization of [29, Theorem ].

3. Hardy and Rogers Type Fixed Point Theorem

In this section, we set up a fixed point theorem for four self-maps satisfying Hardy and Rogers type -contraction. We give an example to validate this theorem and discuss its consequences.

Theorem 13. Let be a complete -metric space and , , , and are self-mappings on such that , . If, for all , for some , the inequality holds, where then , , , and have a unique common fixed point provided , are continuous and pairs , are compatible.

Proof. Let . As , there exists such that . Since , we can choose such that . In general and are chosen in such that and . Define a sequence in such that and for all ; we show that is a Cauchy sequence. Assume that ; then from contractive condition (62), we get for all , where Now from (64) we have Since is strictly increasing, (66) implies Since , thereforeThus, from (66) we obtain for all Similarly, for all Hence, from (69) and (70), we have for all Let for each ; from (71) and axiom we have Repeating the process, we obtain On taking limit in (73), we have ; due to axiom () we get and () implies that there exists such that From (73), for all , we obtain On taking limit in (75), we have This implies there exists , such that for all or To prove as a Cauchy sequence, we use (77) and for ; we consider The convergence of the series entails Hence is a Cauchy sequence in . Since is a complete -metric space, there exists such that . Also we haveWe show that is a common fixed point of , , , and . Since is continuous, therefore, Since, the pair is compatible, so, Now put and in (62) and suppose on the contrary that ; we obtain where Taking the upper limit in (82) and considering the fact that , we have