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Journal of Mathematics
Volume 2019, Article ID 8509494, 15 pages
https://doi.org/10.1155/2019/8509494
Research Article

Common Fixed Points of Rational Contractions with Applications

1Department of Mathematics, Government College University, Lahore 54000, Pakistan
2Department of Mathematics and Applied Mathematics, University of Pretoria Hatfield 002, Pretoria, South Africa
3Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, Pakistan
4Institute of Research and Development of Processes, University of The Basque Country, Campus of Leioa (Bizkaia), 48080 Leioa, Spain

Correspondence should be addressed to Manuel De La Sen; sue.uhe@nesaled.leunam

Received 3 December 2018; Revised 6 February 2019; Accepted 17 February 2019; Published 26 March 2019

Academic Editor: Basil K. Papadopoulos

Copyright © 2019 Mujahid Abbas 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 present the notion of set valued rational contraction mappings and then some common fixed point results of such mappings in the setting of metric spaces are established. Some examples are presented to support the concepts introduced and the results proved in this paper. These results unify, extend, and refine various results in the literature. Some fixed point results for both single and multivalued rational contractions are also obtained in the framework of a space endowed with partial order. As application, we establish the existence of solutions of nonlinear elastic beam equations and first-order periodic problem.

1. Introduction and Preliminaries

Let be a metric space. A mapping is called a contraction if there exists a constant such that, for any , we have The widely known Banach contraction theorem [1] states that a contraction mapping on a complete metric space has a unique fixed point; that is, there exists a point in such that .

In the last few decades, several authors have extended and generalized this principle in various directions.

Jleli and Samet [2] presented a new type of contractive mapping, namely -contraction mapping and established an interesting fixed point theorem for such mappings in a generalized metric space. The concept of generalized metric spaces was introduced by Branciari [3], where the triangle inequality is replaced by the inequality for all pairwise distinct points .

Jleli and Samet [2] considered the set of real valued functions which satisfy the following conditions: is nondecreasing;for each sequence , if and only if ;there exist and such that .

Example 1. Define where by Then .

A mapping on a metric space is called a contraction if for any and , we have whenever and .

Theorem 2 (see [2]). Let be a complete generalized metric space and . If there exist and such that holds for any whenever . Then has a fixed point.

Ahmad et al. [4] modified the class of mappings as follows: where is continuous.

Example 3. Define for by Then, .

Authors in [4] considered the following result of Jleli and Samet [2] with the function instead of :

Theorem 4. Let be a complete metric space and a -contraction, where . Then has a unique fixed point and for any , the sequence converges to .

Note that the Banach contraction theorem immediately follows from the above theorem.

Let be a nonempty set endowed with a metric . Let be the set of all nonempty subsets of , denotes the set of all nonempty compact subsets of , and denotes the set all nonempty closed and bounded subsets. For and , define distance of a point from the set by A mapping defined by is called the generalized Pompeiu-Hausdorff distance induced by .

Let . A point is called a fixed point of if

Nadler [5] obtained the following multivalued version of Banach contraction principle.

Theorem 5. Let be a complete metric space. If satisfies for any and , then has a fixed point.

Afterwards, many researchers have obtained fixed point results for multivalued mappings satisfying certain generalized contractive conditions. Hançer et al. [6] introduced multivalued contraction mappings as follows.

Let be a metric space, , and . Then, is called a multivalued -contraction if, for any , holds whenever where .

They established the following fixed point results for multivalued -contraction mappings on complete metric spaces.

Theorem 6. Let be a complete metric space and a multivalued -contraction. Then has a fixed point.

For further results in this direction, we refer to [710].

Another variation of contraction mapping that can be found in literature is -contraction mapping.

Asl et al. [11] initiated the concept of -admissibility in case of multivalued mappings, whereas Mohammadi et al. [12] presented the notion of -admissibility in case of multifunctions.

Karapinar et al. [13] presented the idea of a triangular -admissible mapping.

Definition 7. Let . A pair is called triangular if for any , and imply that .

Recently, Abbas et al. [14] proposed a concept of -closed mappings for set valued mappings. We present the following generalization of the definition.

Definition 8. Let and . We say that a pair is triangular -closed if the pair is triangular and for any with we have for all and .

If then a mapping which is triangular closed is referred to simply as a triangular -closed mapping.

Example 9. Let . Let be defined by Define the mappings by and It is obvious that the pair is triangular -closed.

The following lemma is crucial in our results.

Lemma 10. Let Suppose that the pairs and are a triangular -closed. Assume that there exists with where . Define sequences and , then for all with .

Proof. By assumption, there exist and such that . Since is -closed and we obtain where and . As is -closed, we have in such that . Continuing this way, we have sequences and with and for all .
Since the pair is triangular we obtain that Thus by induction, we have for any with .

Parvaneh et al. [15] introduced the concept of -contraction with respect to a family of functions and obtained some contraction fixed point results in metric and ordered metric spaces.

They introduced the following family of functions:

Let denote the set of functions satisfying condition :

For all with there exists such that

Following are some examples of such functions [15].

Example 11. , where and .

Example 12. , where and .

The following definition which is a generalization of continuity [16] is needed in the sequel.

Definition 13. Let be a metric space, and . A pair is -continuous at the point if, for any sequence in , and for all implies that . We say that pair is -continuous on if the pair is -continuous on each .

In this paper, we introduce multivalued - rational contraction pair of multivalued mappings and prove the existence of common fixed points of the pair in a metric space. We also obtain some fixed point results for both single and multivalued rational contraction mappings in a space endowed with a partial order. As application, we establish the existence of solutions of nonlinear elastic beam equations and first-order periodic problem.

2. Common Fixed Point Results

Throughout this section we assume that is a metric space and where satisfies , and . Let be a family of continuous and nondecreasing functions where for .

We now present the following definitions:

Definition 14. Let , , , and (1)A pair is called a multivalued rational contraction pair if, for any with and , the following condition holds: where and (2) A pair is called a multivalued rational contraction if, for any with and , the following condition holds: where and(3) A mapping is called a multivalued rational contraction if, for any with and , the following condition holds: where and

Remark 15. (1)If are defined as for all in Definition 14, then the pairs of mappings and are multivalued generalized rational contractions.(2)If are defined as for all in Definition 14, then is a multivalued generalized rational contraction mapping.

Theorem 16. Let . Suppose that the pairs and are multivalued rational contractions such that(C1)the pairs and are triangular -closed;(C2)there exist and with ;(C3) and are -continuous.Then there exists such that .

Proof. If , for some , then we have our conclusion. Assume that , for all . By assumption there exist and such that . If or then the result follows. Assume that and then This implies Since and is multivalued rational contraction, we obtain thatwhereSince , by there exists such that Therefore, from (29) and using the fact that is nondecreasing we obtainAlso, since Thus, Replacing (31) and (34) in (28) we get If , then a contradiction. Hence . Therefore,By we have Thus, there exists such that Then from (37) we haveSince , , , and is -closed, we have . If or then the result follows. Suppose that and . Thus As , we have whereSince , by there exists such that Therefore, from (43) and using the fact that is nondecreasing we haveFurther, Since we getReplacing (45) and (48) in (42) we have If , then a contradiction. Hence . Therefore,By we have Thus, there exists such that Therefore, from (51) we have Furthermore, from (40) we have Proceeding in the same manner, we obtain a sequence in such that , , , , and with and it satisfiesfor each . As , , and , we have . Then, Therefore, Thus, Since by there exists such that . Thus, from definition of and (59) we obtainAlso, Then Therefore, replacing (60) and (62) in (58) we obtain If then a contradiction. Further, Again, using we have Therefore, there exists such that Thus, we have From (56), we have Hence, we have a sequence in and such that for all . On taking limit as , we obtain which implies that . Then by we obtainNow, we show that is a Cauchy sequence. If is not Cauchy, then there exist and for all such that Thus, Therefore, from the above inequality and (72), we obtainAlso,andOn taking limit as in (76) and using (75) (77), we have Therefore,Similarly, we obtain thatBy Lemma 10, we have If , we have Taking limit we have a contradiction to our assumption. Thus, assume that