Relational Quasicontractions and Related Fixed Point Theorems

Eljaneid, Nidal H. E.; Khan, Faizan A.; Mohammed, Hamid I. A.; Alam, Aftab

doi:https://doi.org/10.1155/2022/4477660

Journal of Mathematics

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

Research Article | Open Access

Volume 2022 | Article ID 4477660 | https://doi.org/10.1155/2022/4477660

Relational Quasicontractions and Related Fixed Point Theorems

Nidal H. E. Eljaneid,¹Faizan A. Khan,¹Hamid I. A. Mohammed,^1,2and Aftab Alam³

Academic Editor: Sumit Chandok

Received25 Apr 2022

Revised12 Jun 2022

Accepted29 Jun 2022

Published21 Aug 2022

Abstract

In this article, some results on fixed points under quasicontractions in the framework of metric space endowed with binary relation are proved. Our newly proved results improve and extend several noted fixed point theorems of the existing literature besides their relation-theoretic analogues. We conclude this article by constructing an example to affirm the efficacy of our results.

1. Introduction

Given a metric space , a mapping is said to be contraction if it satisfies

Banach contraction principle [1] plays a key role in the area of metrical fixed point theory. This core result guarantees of the existence and uniqueness of fixed point under the hypotheses that the ambient space remains a complete metric space, whereas the underlying mapping should be a contraction mapping. Many authors extended this classical result employing relatively more general contractive conditions. One of the interesting extensions of contraction mapping was given by Ćirić [2], often referred as quasicontraction. We say that a self-mapping defined on a metric space is a quasicontraction if for some and for all , we have

Indeed quasicontraction mappings subsume several noted generalized contractions due to Kannan [3], Chatterjea [4], Reich [5], Hardy and Rogers [6], Bianchini [7], Rhoades [8], and Zamfirescu [9]. For further details regarding quasicontractions, we refer some recent works due to Aydi et al. [10], Karapınar et al. [11], Bachar and Khamsi [12], Alfuraidan [13], Fallahi and Aghanians [14], Darko et al. [15], and Karapınar et al. [16].

A variant of Banach contraction principle under binary relation is proved by Alam and Imdad [17], wherein the ambient metric space and other involved notions are compatible with an amorphous relation. Under universal relation, the result reduces to classical Banach contraction principle, while under partially ordered relation, the result is transformed into classical order-theoretic metrical fixed point theorems of Ran and Reurings [18] and Nieto and Rodríguez-López [19]. In recent years, various metrical fixed theorems are proved under different types of contractive conditions employing certain binary relations (e.g., [20–34]).

The main intent of this article is to prove a fixed point theorem in the setting of metric space endowed with a binary relation under relation-preserving quasicontraction condition. We also deduce some consequences from our newly proved results. An example is also furnished to demonstrate our main results.

2. Preliminaries

This section deals with certain relevant notions and basic results which are utilized in our subsequent discussions. For any set , a subset of is termed as a binary relation on . We sometimes write instead of , e.g., in case of the relations of “less than equals to” and “greater than equals to” on , the set of real numbers is expressed, respectively, as and .

Definition 1. (see [35]). If is any binary relation on a set , then the setremains again a relation on , which is termed as an inverse relation of . Also, the setforms again a relation on , which is called the symmetric-closure of . Clearly, remains the least symmetric relation on among those binary relations which contain .

Definition 2. (see [17]). Any two elements and of a set are called -comparative, whereas remains a relation on if either or . Usually means that “ and are -comparative.”

Proposition 1 (see [17]). If remains a relation on , then

Definition 3. (see [35]). If remains a relation on a set , then is said to be complete if each pair of elements of is -comparative, i.e.,

Definition 4. (see [35]). If and stands for a relation on , then the setis termed as the restriction of to . It is clear that remains a relation on induced by .

Definition 5. (see [17]). By an -preserving sequence, whereas remains a relation on a set , we meant that the sequence , which satisfy

Definition 6. (see [17]). Given any set and a mapping from into itself, a relation on is termed as -closed if for any ,

Proposition 2 (see [20]). Given any set , suppose that remains a relation on while remains a function from into itself. If is -closed, then must be -closed.

Proposition 3 (see [21]). Given any set , suppose that remains a relation on while remains a function from into itself. If is -closed, then must be -closed (where ).

Definition 7. (see [36, 37]). Given any set , suppose that remains a function from into itself and is a mapping. We say that is -admissible if for all ,If we define the function , then the -admissibility of is equivalent to -closedness of .

Definition 8. (see [20]). A metric space is termed as -complete (whereas remains a relation on ) if each -preserving Cauchy sequence in converges.
Obviously, given any metric space, completeness implies -completeness whatever the relation . In particular, if remains the universal relation, then the concepts of -completeness and usual completeness are equivalent.

Definition 9. (see [20]). A mapping , whereas remains a metric space while remains a relation on , is termed -continuous at a point if for any -preserving sequence such that , we have . Moreover, is termed as -continuous if it is -continuous at every point of .
Obviously, for any mapping on a metric space endowed with the relation , continuity implies -continuity. In particular, if remains the universal relation, then the concepts of -continuity and usual continuity are equivalent.

Definition 10. (see [17]). A relation on a metric space is termed as -self-closed if each -preserving sequence with has a subsequence whose terms are -comparative with , i.e.,In the sequel, for a metric space , a relation and a function from into itself, the following notations will be adopted:(i):= the collection of the fixed points of (ii)(iii)

3. Main Results

In the following lines, we prove a fixed point theorem for quasicontractions employing a binary relation.

Theorem 1. Let be a metric space while is a relation on and is a function from into itself. Also, suppose that(1) is -complete.(2) is -closed.(3) is -continuous or is -self-closed.(4) is nonempty.(5) for some and for all with , the following assumption is satisfied:

Then, admits a fixed point. Further, the completeness of implies the uniqueness of fixed point.

Proof. In view of assumption , we can choose . Define a sequence of Picard iteration based at the initial point so thatAs , using -closedness of and Proposition 3, we getso thatIt follows that the sequence is -preserving.
Denote . Applying the contractive condition and using (13) and (15), we obtain for all thatwhich gives rise toso thatAs , we have . By induction, equation (18) reduces toso thatFor , using triangular inequality and (20), we obtainyielding thereby as Cauchy. Hence, is an -preserving Cauchy sequence. By -completeness of , one can find satisfyingNow, we use assumption to show that is a fixed point of . Suppose that is -continuous. Since is -preserving with , therefore -continuity of asserts that . The uniqueness of limit ensures that so that is a fixed point of . Next, we assume that is -self-closed, then since remains an -preserving sequence converging to , therefore one can a subsequence of satisfying for all . Obviously, we haveOn using , symmetry of and assumption , we haveMaking use of the triangular inequality , equation (24) reduces toUsing (25) and triangular inequality, we getthereby yielding(owing to (20) and (23)) so that . Thus, is a fixed point.
Finally, to prove the uniqueness of fixed point, we take , then we haveAs is complete, we have . Applying contractive condition on these points, we getso thatwhich gives rise toIt follows that has a unique fixed point. This completes the proof.
Now, we present the following results which ensure that Theorem 1 is true for under some restricted hypotheses.

Theorem 2. Let be a metric space while is a relation on and is a function from into itself. Also, suppose that(1) is -complete.(2) is -closed and transitive.(3) is -continuous or is -self-closed.(4) there exists such that .(5) for some and for all with , the following assumption is satisfied:

Then, admits a fixed point. Further, the completeness of implies the uniqueness of fixed point.

Proof. Following the lines of the proof of Theorem 1, one can show that the sequence defined by is -preserving. Using (15) and transitivity of , we obtainUsing (33) and assumption , we getwhich implies thatHence, we havewhich yields thatAs , the sequence is Cauchy. Rest of the proof can be completed by proceeding the lines of the proof of Theorem 1.

4. Applications

As applications, we deduce the following consequences of Theorem 1, which are indeed relation-theoretic versions of some well-known theorems existing in the literature.

Corollary 1 (Hardy-Rogers type). The conclusion of Theorem 1 holds under the hypotheses (a)–(d) along with the following:(1) for some and for all with , the following assumption is satisfied:

Proof. Take with and write , and then, we haveThus, the result follows from Theorem 1.

Corollary 2 (Reich type). The conclusion of Theorem 1 holds under the hypotheses (a)–(d) along with the following:(1) for some and for all with , the following assumption is satisfied:

Proof. Choosing in Corollary 1, the assumption reduces to . Consequently, the result follows from Corollary 1.

Corollary 3 (Kannan type). The conclusion of Theorem 1 holds under the hypotheses (a)–(d) along with the following:(1) for some and for all with , the following assumption is satisfied:

Proof. Choosing in Corollary 1, the assumption reduces to . Consequently, the result follows from Corollary 1.

Corollary 4 (Chatterjea type). The conclusion of Theorem 1 holds under the hypotheses (a)–(d) along with the following:(1) for some and for all with , the following assumption is satisfied:

Proof. Choosing in Corollary 1, the assumption reduces to . Consequently, the result follows from Corollary 1.

Corollary 5 (Bianchini type). The conclusion of Theorem 1 holds under the hypotheses (a)–(d) along with the following:(1) for some and for all with , the following assumption is satisfied:

Proof. Take with and , then we haveThus, the result follows from Theorem 1.

Corollary 6 (Zamfirescu type). The conclusion of Theorem 1 holds under the hypotheses (a)–(d) along with the following:(1) for some and for all with , at least one of the following is true:(i)(ii)(iii)

5. An Example

In this section, we furnish the following example which demonstrates the importance of Theorem 1.

Example 1. Consider with the standard (usual) metric , whereOn , define a relation as follows:Define the mapping byTake with . Then, is -closed. Also, by routine calculation, it can easily verify that satisfies the contractive condition with . Consequently, has a unique fixed point. Indeed, here is complete and so . Indeed, (unique) fixed point of is .

6. Conclusion

In this paper, we have proved the fixed point theorems in the setting of relational metric space under relation-preserving quasicontractions. For possible works, one can prove the variants of the results in the context of relational quasimetric space, relational symmetric space, relational partial metric space, relational -metric space, relational generalized metric space, relational -metric space, relational cone metric space, relational complex-valued metric space, relational rectangular metric space, relational -metric space.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Conflicts of Interest

All the authors declare that there are no conflicts of interest.

Acknowledgments

This study was funded by the Computational and Analytical Mathematics and Their Applications Research Group, Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia.

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Copyright © 2022 Nidal H. E. Eljaneid 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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