Journal of Function Spaces
Volume 2018 (2018), Article ID 4808764, 12 pages
https://doi.org/10.1155/2018/4808764

Research Article

Multivalued Fixed Point Results for New Generalized -Dominated Contractive Mappings on Dislocated Metric Space with Application

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 19 October 2017; Accepted 27 November 2017; Published 1 February 2018

Academic Editor: Manuel De la Sen

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 purpose of this paper is to find out fixed point results for semi--dominated multivalued mappings fulfilling a new generalized locally dominated multivalued contractive condition on a closed ball in complete dislocated metric space. Example and application both are given to show the novelty of our results. Our results merge, extend, and infer many results.

1. Introduction and Preliminaries

Fixed point theory has a foundational role in functional analysis. Banach [1] established the fundamental fixed point theorem, which has played a significant role in different fields of applied mathematics. Due to its significance, a large number of authors have proven many interesting multiplications of his result (see [1–33]). Many authors introduced fixed point theorems in complete dislocated metric space. The idea of dislocated topology has been applied in the field of logic programming semantics (see [14]).

Wardowski [33] introduced new type of contraction called contraction and showed a new generalized fixed point theorem. He generalized many previous fixed point results in different directions. Many other useful results on contractions can be seen in [3–5, 11, 12, 15, 18, 22, 24–27]. In this paper, we recalled the concept of contraction to obtain some common fixed point results for semi--dominated multivalued mappings on proximinal sets satisfying a new type rational -contraction in the context of complete dislocated metric spaces. We have also given an example in which the mapping is not -admissible but it fulfills the condition of -dominated.

Definition 1 (see [14]). Let be a nonempty set. A mapping is called a dislocated metric (or simply -metric) if the following conditions hold, for any :

(i)If , then (ii)(iii)

Then is called a dislocated metric on , and the pair is called dislocated metric space or metric space. It is clear that if , then, from (i), . But if , may not be We use DMS instead of dislocated metric space.

Definition 2 (see [14]). Let be a DMS:

(i)A sequence in is called a Cauchy sequence if given ; there corresponds such that for all one has or (ii)A sequence is said to be 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 3 (see [29]). Let be a nonempty subset of DMS 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 as the set of all closed proximinal subsets of

Definition 4 (see [29]). The function , defined by

is called dislocated Hausdorff metric on

Definition 5 (see [29]). Let be a multivalued mapping and . Let ; we say that is semi--admissible on , whenever implies that , for all ,, where If , then we say that is -admissible on .

Definition 6. Let be a DMS. Let be multivalued mapping and . Let , and we say that is semi--dominated on , whenever for all , where If , then we say that is -dominated on If is a self-mapping, then is semi--dominated on , whenever for all

Definition 7 (see [33]). Let be a metric space. A mapping is said to be an -contraction if there exists such that

where is a mapping satisfying the following conditions:

(F1) is strictly increasing; that is, for all , such that , .

(F2) For each sequence of positive numbers, if and only if .

(F3) There exists such that .

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

Example 9 (see [14]). If , then defines a dislocated metric on .

Example 10 (see [33]). The family of is not empty:

(1)(2)(3)

Example 11. Let Define the mapping by

Define the multivalued mappings by

Suppose that and As , then . Now, , and this means that the pair is not -admissible. Also, and This implies that and are not -admissible individually. Now, , for all Hence is -dominated mapping. Similarly, Hence it is clear that and are -dominated but not -admissible.

2. Main Result

Let be a DMS, let , and let be the multifunctions on . Let be an element such that Let be such that Let be such that Continuing this method, we get a sequence of points in such that and , where . Also , and We denote this iterative sequence by We say that is a sequence in generated by

Theorem 12. Let be a complete DMS. Suppose that there exists a function Let and let be the semi-dominated mappings on Assume that, for some ,

for all with either or , where and and

where and Then is a sequence in , for all , and Also if inequality (6) holds for and either or for all , then and have common fixed point in .

Proof. Consider a sequence From (7), we get

It follows that

Let for some . If , where , then since are semi-dominated mappings on , and As , this implies that Also , so Now, by using Lemma 8, we have

this implies that

for all As is strictly increasing, we have

which implies that

Here Hence

Now,

Thus, Hence , for all Continuing this process, we get

this implies that

for all As is strictly increasing, we have

which implies that

Here Hence

Consequently,

which implies that

This implies that

And so By , we find that

We shall prove that is Cauchy in So it suffices to show that We argue by contradiction. Suppose that there exist and sequences and of natural numbers such that

By triangular inequality, we have

From (24), there exist such that, for all ,

Combining (26) with (27) yields

As are semi-dominated mappings on , and , for all Now, by using Lemma 8 and condition (6), we get

which implies that

As , we get

we deduce that

which is a contradiction. Thus, is a Cauchy sequence in Since is a complete metric space, so there exist such that as ; then

Since and , by using Lemma 8 and inequality (6), we have

By using (33), we get

which implies that

which is a contradiction; hence or Similarly, by using Lemma 8, inequality (6), and the inequality

we can show that Hence, and have a common fixed point in Now,

This implies that

Example 13. Let and let be the complete DMS on defined by

Define the multivalued mapping, , by

Considering that , So we obtain a sequence in generated by Also, Let

, , , , , and .

Now, if , then we have the following cases.

Case 1. If , then we consider only

Thus,

which implies that

That is, if , then

Case 2. If , then by using the similar arguments of Case 1 we can get the same results. Now, if , , then

And consequently condition (6) does not hold on Thus the mappings and satisfy all the conditions of Theorem 12 on closed ball rather than on whole space.

If we take in Theorem 12, then we are left with the result.

Corollary 14. Let be a complete DMS. Suppose that there exists a function Let , let , and let be the semi-dominated mappings on Assume that, for some ,

for all with either or , where and and

where and Then is a sequence in , for all , and Also if inequality (48) holds for and either or for all , then and have common fixed point in .

If we take in Theorem 12, then we are left with the result.

Corollary 15. Let be a complete DMS. Suppose that there exists a function Let , let , and let be the semi-dominated mappings on Assume that, for some ,

for all with either or , where and and

where and Then is a sequence in , for all , and Also if inequality (50) holds for and either or for all , then and have common fixed point in .

If we take in Theorem 12, then we are left with the result.

Corollary 16. Let be a complete DMS. Suppose that there exists a function Let , let , and let be the semi-dominated mappings on Assume that, for some ,

for all with either or , where and and

where and Then is a sequence in , for all , and Also if inequality (52) holds for and either or for all , then and have common fixed point in .

If we take in Theorem 12, then we are left only with the result.

Corollary 17. Let be a complete DMS. Suppose that there exists a function Let , let , and let be the semi-dominated mappings on Assume that, for some ,

for all with either or , where and and

where and Then is a sequence in , for all , and Also if inequality (54) holds for and either or for all , then and have common fixed point in .

3. Fixed Point Results for Graphic Contractions

In this section, we present an application of Theorem 12 in graph theory. Jachymski [21],proved the result concerning for contraction mappings on metric space with a graph. Hussain et al. [16] introduced the fixed points theorem for graphic contraction and gave an application to system of integral equations. A graph is a connected graph if there must exist a path among any two different vertices (for details, see [13, 32]).

Definition 18. Let be a nonempty set and let be a graph such that , and is said to be multigraph-dominated if , for all .

Theorem 19. Let be a complete DMS endowed with a graph . Suppose that there exists a function Let , let , and assume a sequence in generated by , with Assume that the following hold:
(i) and are multigraph-dominated for all
(ii) There exist some , and

for all and or
(iii) for all

Then, is a sequence in , , and Also, if inequality (56) holds for and or for all , then and have common fixed point in and .

Proof. Define by

As and are semigraph-dominated on , then, for , for all and for all . So, for all and for all This implies that and Hence , for all So, are the semi--dominated mappings on Moreover, inequality (56) can be written as

for all elements in with either or Also, (iii) holds. Then, by Theorem 12, we have that is a sequence in and Now, and either or implies that either or So, all the conditions of Theorem 12 are satisfied. Hence, by Theorem 12, and have a common fixed point in and

4. Fixed Point Results for Single Valued Mapping

In this section, we discuss some new fixed point results for single valued mapping in complete DMS.

Theorem 20. Let be a complete DMS. Suppose that there exists a function Let , let , and let be the semi-dominated mappings on Assume that, for some ,

for all with either or , where and and

where and Then is a sequence in , for all , and Also if inequality (59) holds for and either or for all , then and have common fixed point in .

Proof. The proof of the above Theorem is the same as Theorem 12.

If we take in Theorem 20, then we have the following result.

Corollary 21. Let be a complete DMS. Suppose that there exists a function Let , let , and let be the semi-dominated mappings on Assume that, for some ,

for all with either or , where and and

where and Then is a sequence in , for all , and Also if inequality (61) holds for and either or for all , then and have common fixed point in .

If we take in Theorem 20, then we are left with the result.

Corollary 22. Let be a complete DMS. Suppose that there exists a function Let , let , and let be the semi-dominated mappings on Assume that, for some ,

for all with either or , where and and

where and Then is a sequence in , for all , and Also if inequality (63) holds for and either or for all , then and have common fixed point in .

If we take in Theorem 20, then we are left with the result.

Corollary 23. Let be a complete DMS. Suppose that there exists a function Let , let , and let be the semi-dominated mappings on Assume that, for some ,

for all with either or , where and and

where and Then is a sequence in , for all , and Also if inequality (65) holds for and either or for all , then and have common fixed point in .

If we take in Theorem 20, then we are left with the result.

Corollary 24. Let be a complete DMS. Suppose that there exists a function Let , let , and let be the semi-dominated mappings on Assume that, for some ,

for all with either or , where and and

where and Then is a sequence in , for all , and Also if inequality (67) holds for and either or for all , then and have common fixed point in .

5. Application to the Systems of Integral Equations

Theorem 25. Let be a complete DMS. Let , let , and let be the dominated mappings on Assume that, for some ,

for all with , where and

where and Then is a sequence in , for all and Then and have common fixed point in .

Proof. The proof of the above theorem is the same as the theorem proven in the previous section. In this section, we discuss the application of fixed point Theorem 25 in form of Volterra type integral equations.

for all We find the solution of (71). Let be the set of all continuous functions on , endowed with the complete dislocated metric. For , define supremum norm as , where is taken arbitrarily. Then define

for all , and, with these settings, becomes a complete DMS.

Now we prove the following theorem to ensure the existence of solution of integral equations.

Theorem 26. Assume that the following conditions are satisfied:

(i).(ii)Define

Suppose that there exists , such that

for all and , where

where , , and Then integral equations (71) have a solution.

Proof. By assumption (ii),

This implies that

which further implies that

So all the conditions of Theorem 25 are satisfied for ; and . Hence integral equations given in (71) have a unique common solution.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

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

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