## Nonlinear Operators in Fixed Point Theory with Applications to Fractional Differential and Integral Equations

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Muhammad Nazam, Ma Zhenhua, Sami Ullah Khan, Muhammad Arshad, "Common Fixed Points of Four Maps Satisfying -Contraction on -Metric Spaces", *Journal of Function Spaces*, vol. 2017, Article ID 9389768, 11 pages, 2017. https://doi.org/10.1155/2017/9389768

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

**Academic Editor:**Ahmad S. Al-Rawashdeh

#### 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., [2–8]). 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., [13–19]).

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