Journal of Mathematics
Volume 2018 (2018), Article ID 5816364, 8 pages
https://doi.org/10.1155/2018/5816364

Research Article

Fixed Point Results for Multivalued Contractive Mappings Endowed with Graphic Structure

Tahair Rasham,¹ Abdullah Shoaib,² Badriah A. S. Alamri,³ and Muhammad Arshad¹

¹Department of Mathematics, International Islamic University, H-10, Islamabad 44000, Pakistan
²Department of Mathematics and Statistics, Riphah International University, Islamabad 44000, Pakistan
³Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Tahair Rasham; tahir_resham@yahoo.com

Received 9 August 2018; Accepted 27 November 2018; Published 19 December 2018

Academic Editor: Ali Jaballah

Copyright © 2018 Tahair Rasham 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.

The aim of this paper is to introduce the notion of admissible multivalued mappings and to set up fixed point results for such mappings fulfilling generalized locally Ćirić type rational-contractive conditions on a closed ball in complete dislocated -metric space. Example and application have been given to demonstrate the novelty of our results. Our results combine, extend, and infer several comparable results in the existing literature.

1. Introduction and Preliminaries

FP theory plays a fundamental role in functional analysis. Banach proved significant result for contraction mappings. After that, a huge number of FP theorems have been established by various authors and they made different generalizations of the Banach’s result. Shoaib et al. [1], discussed the result related to --Ćirić type multivalued mappings on an intersection of a closed ball and a sequence with graph. Further FP results on a closed ball can be seen ([2–12]).

Boriceanu [13] proved FP results for multivalued generalized contraction on a set with two -metrics. After this Aydi et al. [14] established FP theorem for set-valued quasi contraction in -metric spaces. Nawab et al. [15] established the new idea of dislocated -metric space as a conception of metric space and proved some common FP results for four mappings fulfilling the generalized weak contractive conditions in partially ordered dislocated -metric space.

Nadler [16] initiated the study of FP theorems for the multivalued mappings (see also [17]). Several results on multivalued mappings have been observed (see [18–20]). Asl et al. [21] gave the idea of - contractive multifunctions and -admissible mapping and got some FP conclusions for these multifunctions (see also [22, 23]). In 1974, Ćirić [24] introduced quasi contraction. Reference [25] established some new common fixed points of generalized rational-contractive mappings in dislocated metric spaces with applications. In this paper, the concept of new Ćirić type rational multivalued contraction has been introduced. Now we prove new type of result for a different multivalued rational expression studied by Rasham et al. [6]. Common FP results for such contraction on a closed ball in complete dislocated -metric space have been established. Example and application have been given. We give the following definitions and results which will be needed in the sequel.

Definition 1 (see [15]). Let be a nonempty set and let be a function, called a dislocated -metric (or simply -metric), if for any and the following conditions hold:

(i)if , then ;(ii);(iii)

The pair is called a dislocated -metric space. It should be noted that the class of metric spaces is effectively larger than that of metric spaces, since is a metric when
It is clear that if , then from (i), . But if , may not be For and is a closed ball in We use space instead of dislocated -metric space.

Example 2. If , then defines a on .

Definition 3 (see [15]). Let be a space.
(i) A sequence in is called Cauchy sequence if, given , there corresponds such that for all we have or
(ii) A sequence dislocated -converges (for short -converges) to if In this case is called a -limit of
(iii) is called complete if every Cauchy sequence in converges to a point such that .

Definition 4. Let be a nonempty subset of space and let An element is called a best approximation in if

If each has at least one best approximation in , then is called a proximinal set.
We denote by the set of all proximinal subsets of Let , where , denote the family of all nondecreasing functions such that and for all , where is the iterate of Also

Definition 5 (see [26]). Let be the closed valued multifunctions and be a function. We say that the pair is -admissible if for all

where When , then we obtain the definition of -admissible mapping given in [21].

Definition 6. Let be a space, be multivalued mapping, and . Let , and we say that the is semi--admissible on , whenever implies that for all , where If , then we say that the is -admissible on

Definition 7. The function , defined by

is called dislocated Hausdorff -metric on

Lemma 8. Let be a space. Let is a dislocated Hausdorff -metric space on Then for all and for each there exists satisfying ; then .

2. Main Result

Let be a space, and let and be the multifunctions on . Then there exist such that Let be such that Continuing this process, we construct a sequence of points in such that We denote this iterative sequence by We say that is a sequence in generated by

Theorem 9. Let be a complete space, , and be a semi-admissible multifunction on ; is a sequence in generated by Assume that, for some and

where , the following hold:

Then, is a sequence in , , and Also if or , for all and the inequality (5) holds for all , then has a common fixed point in .

Proof. Consider a sequence generated by Then, we have , and , for all By Lemma 8, we have for all If , then is a fixed point in of Let From (6), we have

It follows that

If , then is a fixed point in of Let Since and is semi-admissible multifunction on , then As and , so Let for some As , we have , which further implies Continuing this process, we have Now, by using Lemma 8

If , then This is contradiction to the fact that for all Hence, we obtain Therefore, we have

Now, by using triangular inequality and by (10), we have

Thus Hence, by induction, As , then we have Also is semi-admissible multifunction on , and therefore This further implies that Continuing this process, we have for all Now, inequality (10) can be written as

Fix and let , such that

Let with Now, we have

Thus, is a Cauchy sequence in As every closed ball in a complete space is complete, there exist such that , and

By assumption, we have for all Thus, Now, we have

Letting and by using inequality (15), we obtain So , and then Hence So has a fixed point in
Let be a nonempty set. Then is called a preordered space if is called on Let be a preordered space and We say that whenever for each there exist such that Also, we say that whenever, for each and , we have

Corollary 10. Let be a preordered complete space, , and be a multifunction on ; is a sequence generated by , with Assume that, for some and

where , the following hold:

If , such that implies Then is a sequence in , , and Also if or , for all , and inequality (18) holds for all Then is a fixed point of in .

Corollary 11. Let be a preordered complete space, , and be a multifunction on ; is a sequence generated by , with Assume that, for some and

where , the following hold:

If , such that implies Then is a sequence in , , and Also if or , for all , and inequality (21) holds for all Then is a fixed point of in .

Corollary 12. Let be a preordered space, , and be a multifunction on ; is a sequence generated by , with Assume that, for some and

where , the following hold:

If , such that implies Then is a sequence in , , and Also if or , for all , and inequality (24) holds for all Then is a fixed point of in .

Corollary 13. Let be a preordered complete space, , and be a multifunction on ; is a sequence generated by , with Assume that, for some and

where , the following hold:

If , such that implies Then is a sequence in , , and Also if or , for all , and inequality (27) holds for all Then is a fixed point of in .

Example 14. Let and let be the space on defined by

with parameter Define the multivalued mappings, by

Considering , and , then Now So we obtain a sequence in generated by Let , and then Define

Now,

So the contractive condition does not hold on Now, for all , we have

So the contractive condition holds on As , then

Hence, all the conditions of Theorem 9 are satisfied. Now, we have that is a sequence in and Moreover, is a fixed point of

3. Fixed Point Results For Graphic Contractions

In this section, we present an application of Theorem 9 in graph theory. Jachymski [27] proved the result concerning contraction mappings on metric space with a graph. Hussain et al. [28] discussed the fixed points theorem for graphic contraction and gave an application. A graph is affix if there is a way among any two vertices (see for details [29, 30]).

Definition 15. Let be a nonempty set and be a graph such that , and is said to be multigraph preserving if , and then for all and .

Theorem 16. Let be a complete space endowed with a graph . Suppose there exists a function Let, , , and let for a sequence in generated by , with Suppose that the following are satisfy:
(i) is a graph preserving for all ;
(ii) there exists and

where such that

for all and ;
(iii) for all and
Then, is a sequence in and Also, if and inequality (36) holds for and or for all , then has a fixed point in .

Proof. Define, by

As is a sequence in generated by with , we have Let , and then From (i), we have for all and This implies that for all and This implies that So, is a semi-admissible multifunction on Moreover, inequality (36) can be written as

for all elements in with either or Also, (iii) holds. Then, by Theorem 9, we have that is a sequence in and Now, and either or for all , and inequality (36) holds for all Then we have or for all and inequality (5) holds for all So, all the conditions of Theorem 9 are satisfied. Hence, by Theorem 9, has a common fixed point in and .

4. Fixed Point Results for Single-Valued Mapping

In this section we discussed some fixed point results for self-mapping in complete space. Let be a space, , and be a mapping. Let , Continuing this process, we construct a sequence of points in such that We denote this iterative sequence by We say that is a sequence in generated by

Theorem 17. Let be a complete space, , and be a semi-admissible function on ; is a sequence in generated by Assume that, for some and

where , the following hold:

Then, is a sequence in , , and Also if or , for all , and inequality (40) holds for all Then has a common fixed point in .

Proof. The proof of the above theorem is similar to Theorem 17.

Corollary 18. Let be a preordered complete space, , and be a self-mapping on ; is a sequence generated by , with Assume that, for some and

where , the following hold:

If , such that implies Then is a sequence in , , and Also if or , for all , and inequality (43) holds for all Then is a fixed point of in .

Corollary 19. Let be a preordered complete space, , and be a self-mapping on ; is a sequence generated by , with Assume that, for some and

where , the following hold:

If , such that implies Then is a sequence in , , and Also if or , for all , and inequality (46) holds for all Then is a fixed point of in .
Recall that if is a preordered set and is such that for , with implying , then the mapping is said to be nondecreasing.

Corollary 20. Let be a preordered complete space, be an arbitrary point in , be a self-mapping on , and be a Picard sequence in with initial guess , with For some and

where , the following hold:

Then, is a sequence in , such that and Also if or , for all , and inequality (49) holds for all Then is a fixed point of in .

Data Availability

The 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.

Authors’ Contributions

Each author equally contributed to this paper and read and approved the final manuscript.

Acknowledgments

The authors acknowledge with thanks the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, for financial support.

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