Journal of Function Spaces
Volume 2017 (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

Muhammad Nazam,¹ Ma Zhenhua,2,3 Sami Ullah Khan,1,4 and Muhammad Arshad¹

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; mazhenghua_1981@163.com

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., [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 have

We 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 , therefore

Thus, 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 have

We 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

a contradiction; hence, implies . Again since is continuous, therefore,

Since, the pair is compatible,

Now put and in (62) and suppose on the contrary that ; we obtain

where

Taking the upper limit in (87), we have

a contradiction; hence, which then implies . Assume that and from (62) we get

where

Taking the upper limit in (90) and using the fact that , we have

a contradiction; hence, which then implies . Finally, assume that . From (62) and the equation , 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 (62), we have

where

Thus, from (96), we have

which is a contradiction. Hence, and is a unique common fixed point of the four maps , , , and .

The following example explains Theorem 13.

Example 14. Let and define by , so, is a complete -metric space. Define the mappings , for all , by

Clearly, , , , and are continuous self-mappings complying with . We note that the pair is compatible. Indeed, let be a sequence in satisfying

Then

and equivalently

implies

The uniqueness of limit gives ; thus, is only possible solution. 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 (62) is satisfied for all . Hence, all the hypotheses of Theorem 13 are satisfied. Note that , , , and have a unique common fixed point .

Theorem 15 generalizes the result presented by Reich [30] and can be proved by following the proof of Theorem 13.

Theorem 15 (Reich type). 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 such that

then , , , and have a unique common fixed point provided , are continuous and pairs , are compatible.

Theorem 16 generalizes the result presented by Chatterjea [31] and can be proved by following the proof of Theorem 13.

Theorem 16 (Chatterjea type). Let be a complete -metric space and , , , and are self-mappings on such that , . If for all for some the inequality

holds for all such that , then , , , and have a unique common fixed point provided , are continuous and pairs , are compatible.

Similarly, Theorems 17 and 18 can be proved by following the proof of Theorem 13.

Theorem 17. Let be a complete -metric space and , , , and are self-mappings on such that , . If for all for some , the inequality

holds, where

with are nonnegative functions such that

then , , , and have a unique common fixed point provided , are continuous and pairs , are compatible.

Theorem 18. 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.

4. Application of Theorem 6 to a System of Integral Equations

Let be the space of all continuous real valued functions defined on , . Let the function be defined by

for all . Obviously, is a complete -metric space. We shall apply Theorem 6 to show the existence of a common solution of the system of Volterra type integral equations given by

for all , where is a continuous function and are lower semicontinuous operators. Now we prove the following theorem to ensure the existence of a solution of the system of integral equations (117).

Theorem 19. Let and define the mappings by

where is a continuous function and are lower semicontinuous operators. Assume the following conditions are satisfied:

(i)There exists a continuous function such that

 for each and .(ii)There exists and, for each , one has

(iii)There exists a sequence in such that and , whenever

Then the system of integral equations given in (117) has a solution.

Proof. Following assumptions (i) and (ii), we have

Consequently, we have

As the natural logarithm belongs to , applying it on above inequality and after some simplification, we get

So taking , all hypotheses of Theorem 6 are satisfied. Hence the system of integral equations given in (117) has a unique common solution.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Ma Zhenhua acknowledges the support of Scientific Research Foundations of Education Burrean of Hebei Province, Grant no. QN2016191.

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