Hybrid Coupled Fixed Point Theorems in Metric Spaces with Applications

Shoaib, Muhammad; Sarwar, Muhammad; Abdeljawad, Thabet

doi:https://doi.org/10.1155/2019/6870308

Journal of Function Spaces

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Abstract Introduction and Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 6870308 | https://doi.org/10.1155/2019/6870308

Hybrid Coupled Fixed Point Theorems in Metric Spaces with Applications

Muhammad Shoaib,¹Muhammad Sarwar,¹and Thabet Abdeljawad²

Academic Editor: Raúl E. Curto

Received10 Oct 2018

Revised17 Feb 2019

Accepted07 Mar 2019

Published01 Apr 2019

Abstract

In this manuscript, using CLR property, coupled coincidence and common coupled fixed point results for two-hybrid pairs satisfying - contraction are demonstrated. Using the established results existence of solution to the coupled system of functional and nonlinear matrix equations is also discussed. We provide examples where the main theorem is applicable but most current relevant results in literature fail to have a common coupled fixed point.

1. Introduction and Preliminaries

The existence and uniqueness of solution of a nonlinear matrix equations and functional equations are very interesting research area. Metric fixed point theory provides beneficial and best techniques for the existence of the above-mention equations. In [1–6] the authors worked on matrix equations and demonstrated the existence and uniqueness in the form of their positive definite solutions. Matrix equations and functional equations often arise from various areas, such as ladder networks [7, 8], control theory [9, 10], and dynamic programming [11–14].

Banach [15] has sorted out successful and well-known result; such consequence was later on named as Banach contraction principle (BCP). The Banach principle has been generalized in various spaces. Nadler [16] in 1969 further modified and elaborated the Banach contraction principle (BCP) to set-valued mapping using the Hausdorff metric known as Nadler contraction principle (NCP). Later, thousands of articles appeared in literature to generalize the BCP. Very recently, some authors proved the contraction principle in metric in controlled metric type spaces where the triangle inequality possess control functions (see [17–19] and the references therein).

Aamri and Moutawakil [20] defined (E.A) property for self-mappings which contained the class of compatible and noncompatible mappings and proved common fixed point results under strict contractive conditions. Kamran [21] demonstrated the (E.A) property for hybrid pair and established fixed point and coincidence points results with hybrid strict contractions. Liu et al. [22] introduced common (E.A) property for hybrid pairs of single and multivalued mappings and presented new common fixed point theorems using hybrid contractive conditions. Sintunavarat and Kumam [23] brought together the idea of common limit range (CLR) property for single-valued mappings and displayed its superiority over the property (E.A). Imdad et al. [24] defined common limit range property for a hybrid pair of mappings and demonstrated fixed point results in the symmetric (semimetric) spaces. These concepts were converted by Abbas et al. [25] to multivalued mappings and formulated coupled coincidence point and common coupled fixed point theorems linking hybrid pair of mappings satisfying generalized contractive conditions. Deshpande and Handa [26, 27] defined (E.A) property and occasional w-compatibility for hybrid (pair) coupled maps and also presented common (E.A) property for two hybrid coupled mappings.

In 2012, Wardowski [28] introduced a new type of contraction called F-contraction. In this way Wardowski generalized the Banach contraction principle (BCP) in different manner from the known results of literature. Following this direction Sgroi and Vetro [14] studied multivalued F-contractions and discussed their application on certain functional and integral equations. Recently, Nashine et al.[29] introduced generalized -contractions and studied common fixed point results for a hybrid pair under common limit range property with applications to certain system of functional equations and Volterra integral inclusion.

Coupled fixed points for several type contraction mappings were studied by many authors in different type metric spaces [30–32]. For more details see [25, 33–37]. For other types of common fixed point results we refer to [38–40] and the references therein.

Motivated by the above results, we studied common coupled fixed point results by defining the concept of CLR property for two hybrid pairs of mapping via generalized type contraction. Using the established results we also studied the existence of solution for the coupled system of functional and coupled system of nonlinear matrix equations. All over the paper , , and represent the set of all positive real numbers, the set of positive integers, and the set of nonnegative integers, respectively.

Definition 1. Suppose is nonempty set and let be a function satisfying the conditions (1) if and only if for all ;(2), where ;(3) for all .

Then is a metric on and the pair is called metric space.

Definition 2 (see [23]). Functions are said to satisfy the common limit range property of w.r.t (shortly, the -property w.r.t ) if there exists a sequence in such that, for some , .

Definition 3 (see [41]). Suppose , are defined on a metric space . Then and are said to satisfy the common limit range property of w.r.t (shortly, -property w.r.t ) if there exists a sequence in and such that, for some ,

Definition 4 (see [41]). Functions and defined on a metric space , are to satisfy the common limit in the range of w.r.t (shortly, -property w.r.t to ) if there exist sequences and in and such that, for some , we have , , and

Remark 5. Clearly, if and in Definition 4 then we reobtain Definition 3.

Definition 6 (see [25]). Let and be mappings. (1)A point is called a coupled coincidence point of and if and (2)A point is called a coupled common point of and if and .

Definition 7 (see [25]). Let and be mappings. The mapping is called weakly commuting at some point, point if and

Definition 8 (see [27]). Mappings and on metric space are said to have the E.A property if there exist sequences and in and such that for some ,

Now, we recall some definitions for multivalued mappings defined in a metric space . Recall the Hausdorff metric for by where and

Lemma 9 (see [42]). Let be a metric space. For any We have , for all .

Lemma 10 (see [16]). Assume is a metric space and . Then for every and for each there exists such that

In [16] it was shown that the above lemma is also true for . In fact we have the following.

Lemma 11. Assume is a metric space and . Then for every and for each there exists such that

Definition 12 (see [28]). Let represent the family of all functions , with the following conditions (1) is continuous and strictly increasing;(2) ;(3)for , , there exists such that

Theorem 13 (see [27]). Let be a metric space. Assume and to be a mapping satisfying the following.(a) and satisfy the common (EA) property.(b)For all , there exist some and some such that (c) and are closed subsets of . Then have coupled coincidence point. have coupled coincidence point.If is weakly commuting at and , for , then and have a common coupled fixed point.If is weakly commuting at and , for , then and have a common coupled fixed point., and have common coupled fixed point if and are true.

Theorem 14 (see [27]). Let be a metric space. Assume and to be mappings satisfying and of Theorem 13 and(1) and are w-compatible.(2)Suppose that either is closed subset of or or is closed subset of and Then have a common coupled fixed point.

2. Main Results

We define the CLR property for the study of common coupled fixed point in the following way in metric space.

Definition 15. Mappings and on metric space are to satisfy the common limit in the range of with respect to (shortly, the -property with respect to S) if there exist sequences and in and such that, for some , we have ,

Definition 16. Let and be mappings on metric space Then and have -property, if there exist sequences , and and such that for some .

Example 17. Let with the usual metric. Define and by , , , ,
Consider the sequences , , ,
Clearly , , , Further, , , , Therefore and satisfy CLR property.

Example 18. Assume to be endowed through usual metric and , , define by Let , where , , where , where , and , where .
Clearly , , , Further, , , , Therefore and satisfy CLR property.
Throughout the paper denote the set of all closed and bounded subsets of and

Theorem 19. Let and be maps on metric space . Suppose that and have -property and furthermore assume that where and Here, , , , and . Then the following holds. have coupled coincidence point. have coupled coincidence point.If is weakly commuting at and , for , then and have a common coupled fixed point.If is weakly commuting at and , for . Then and have a common coupled fixed point., and have common coupled fixed point if and are true.

Proof. Since and have -property, therefore there exist sequences , and and such that Putting , , in inequality (8), we have where Applying limit to , we have which implies that Applying limit to (11) and using (14), we have which implies that Using definitions of and , we have But and using Lemma 11which is contradiction. Hence, . Therefore Putting , , in inequality (8), we have where Applying limit to , we have Applying limit to (20) and using (23), we havePutting , , in inequality (8), we have where Applying limit to , we have which implies that and we get Applying limit to (25) and using (29), we get which implies thatUsing definitions of and and using Lemma 9, we have which implies that Putting , , in inequality (8), we havewhere Applying limit to , we have which implies that Applying limit to (25) and using (38), we get which implies thatUsing definitions of and and using Lemma 9, we have and we obtained Similarly by putting and and and we can obtained and Since and -weakly are commuting then , . Since , . Thus is a common fixed point. A similar argument proves . Then using and , hold immediately.

Theorem 20. Let and are mapping on metric space . Furthermore assume that and have -property and where and Here, , , , and . Then the following holds. have coupled coincidence point. have coupled coincidence point.If is weakly commuting at and , for , then and have a common coupled fixed point.If is weakly commuting at and , for , then and have a common coupled fixed point., and have common coupled fixed point if and are true.

Proof. Since and have -property, therefore there exist sequences , and and such that Putting , , in inequality (45), we get where Applying limit to , we have Applying limit to (48) and using (51) we can deduce that which implies that Using definitions of and , we have Using Lemma 11, we have Thus, we have Similarly by taking , , in inequality (45) we get where Applying limit to we have Applying limit to (57) and using (60) we can deduce that By taking , , in inequality (45), we get where Applying limit to , we have Applying limit to (62) using (65), we have By taking , , in inequality (45) we get whereApplying limit to , we have Applying limit to (67) using (70), we have Following the similar line of Theorem 19 we can obtain that , and have common coupled fixed point.

Example 21. Let with the usual metric. Define , , and by , , , , , and .
Consider the sequences , , ,
Now, Therefore and satisfy CLR property. Now, Taking logarithm on both sides and , we conclude that all the other conditions of Theorem 19 are satisfied. Therefore and have common coupled fixed point.

Example 22. Let with the usual metric. Define , , , by , , , , , and .
Consider the sequences , , ,
Now, Therefore and satisfy CLR property. Now, Taking logarithm on both sides and , we conclude that all the other conditions of Theorem 19 are satisfied. Therefore, and have common coupled fixed point.

Remark 23. From the above examples the following is clear. (i)Theorem 13 is not applicable to Example 21 because nor are closed.(ii)Theorem 14 is not applicable to Example 22, because neither nor (iii)Similarly the main results of [43] Theorem 2.1 and Theorem 2.6 are not applicable to the above examples

Next, we explain Example 27 of [27] to which our Theorem 19 is also applicable.

Example 24. Let , equipped with the metric define by Define , , and by And Consider the sequences , , ,
Now, Therefore and satisfy CLR property.
Now for , we discuss the following cases.

Case 1. If , then

Case 2. If and , then

Case 3. If , then Similarly it is easy to show the same result for and and for Taking logarithm on both sides and we conclude that all conditions of our Theorem 19 are satisfied. Therefore and have common coupled fixed point.

Remark 25. (i) From Example 24 it is clear that all conditions of our Theorem 19 are satisfied for Example 27 of [27] and hence the corresponding conclusions hold.

3. Applications to System of Functional Equations

In this section, we discuss common solution for two coupled functional equations with the help of Theorem 19. Throughout this unit and stand for Banach spaces, the state space is , the decision space is , and the space of all bounded real-valued functions on is which is Banach space.

Define , by Here,

Consider the following system where , , for and , denote the state vectors and decision vectors, respectively, signify the transformations of the process, and symbolized the sup return functions under the initial state .

Let , defined by

Theorem 26. Assume to be maps given by (86) which holds the following conditions. (1) and , for , are bounded.(2)For , and Here, Then, system (85) has a common solution in .

Proof. Let be an arbitrary positive real number and there exist , for arbitrary , such that From definition of and we have Next, from (89) and (92) we have Similarly from (90) and (91) we get Combining (93) and (94) we conclude that By taking , and in Theorem 19. Then we deduce that the mappings have a common coupled fixed point in ; that is the system (85) has a solution.

4. Applications to Matrix Equations

In this section, we study the nonlinear matrix equations with the help of Theorem 20. Here is a positive definite matrix, are arbitrary matrices, and continuous order preserving maps are defined from into such that

In this unit we will use the following notations:

symbolizes the set of all complex matrices, the set of all Hermitian matrices, and is the set of all positive definite matrices. As a replacement for of we will also write . Similarly, positive semidefinite matrix is denoted by . We also signify by the spectral norm, i.e., where the biggest eigenvalue of is . We will use the metric induced by the trace norm defined by , where are the singular values of . The set is a complete metric space endowed with this norm.

The following lemma which is taken from [6] will be useful in the study of the matrix equations.

Lemma 27. Let and be matrices; then .

In this section, we define the mapping byHere , , and are continuous order-preserving maps. In the following theorem we first discuss the existence of common coupled fixed point of and in .

Theorem 28. Let such that (1)for every , and and(2),(3) and Then, there exist such that and . and .

Proof. Let ; then Thus, the contractive condition of Theorem 20 is satisfied for all . By taking , , and in Theorem 20. From Theorem 20, and have a common coupled fixed point.

Data Availability

Data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The third author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

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Copyright © 2019 Muhammad Shoaib 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.

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