Journal of Function Spaces
Volume 2019 (2019), Article ID 1812461, 14 pages
https://doi.org/10.1155/2019/1812461

Research Article

New Types of Contraction for Multivalued Mappings and Related Fixed Point Results in Abstract Spaces

Shaif Saleh Alshoraify,¹ Abdullah Shoaib,² and Muhammad Arshad¹

¹Department of Mathematics and Statistics, International Islamic University, H-10, Islamabad 44000, Pakistan
²Department of Mathematics and Statistics, Riphah International University, Islamabad, Pakistan

Correspondence should be addressed to Abdullah Shoaib; abdullahshoaib15@yahoo.com

Received 26 April 2019; Revised 7 June 2019; Accepted 23 June 2019; Published 16 July 2019

Academic Editor: Gestur Ólafsson

Copyright © 2019 Shaif Saleh Alshoraify 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.

In this article, we establish fixed point results for a pair of multivalued mappings satisfying generalized contraction on a sequence in dislocated -quasi metric spaces and Khan type contraction on a sequence in -quasi metric spaces. An example and an application have been discussed. Our results modify and generalize many existing results in literature.

1. Introduction and Preliminaries

A point is said to be a fixed point of a multivalued/self-mapping , if Fixed point theory has a large number of applications, for example, [1–4]. Czewick [5] initiated the study of fixed point in b-metric spaces. Many authors used the concept of b-metric spaces to prove the existence and the uniqueness of a fixed point for several contraction mappings [6–9]. Furthermore, dislocated quasi-metric spaces [10–13] generalized abstract spaces such as dislocated metric spaces [14] and quasi-metric spaces [15–17]. Recently, Klin-eam and Suanoom [18] introduced the concept of dislocated -quasi metric spaces. Fixed point results in complete dislocated -quasi metric spaces can be seen in [19, 20].

Wardowski [21] generalized many fixed point results in a beautiful way by introducing contraction (see also [6, 22–30]). Nadler [31] extended Banach’s contraction mapping principle to a fundamental fixed point theorem for multivalued mappings. Since then, an interesting and rich fixed point theory for such mappings was developed in many directions; see [32–36]. The results of single valued mappings can be generalized by using multivalued mappings. Results for multivalued mappings have applications in engineering, Nash equilibria, and game theory [37–40]. Rasham et al. [41] obtained fixed point results for a pair of multivalued contractive mappings, which are extensions of some multivalued fixed point results.

This paper introduces new types of contractions on a sequence and generalizes many recent results. An example has been given to show how our results are valid when the others fail. An application has been given to obtain a solution of a system of integral equations.

Definition 1 (see [18]). Let be a nonempty set and a real number. A mapping is called a dislocated quasi -metric (or simply -metric), if the following conditions hold for any

(a)If , then ;(b)

The pair is called a dislocated quasi -metric space (in short dislocated -quasi-metric space).
The following remarks can be observed
(a) If , then a dislocated -quasi-metric space becomes a dislocated quasi-metric space [12];
(b) if and implies , then becomes a quasi-metric space [17];
(c) if and implies , then becomes a -metric space [9].

Example 2 (see [20], let and ). Define by for Then is a -metric space with . But it is not a quasi -metric space. Also it is not a dislocated -metric space. It is obvious that is neither -metric space nor dislocated quasi-metric space.

Definition 3 (see [11]). Let be a dislocated -quasi-metric space. Let be a sequence in , and then
(a) is called Cauchy if , such that (respectively ,
(b) dislocated quasi -converges (for short -converges) to , if or for any , there exists , such that for all , and . In this case is called a -limit of
(c) is called complete if every Cauchy sequence in converges to a point .

Definition 4 (see [12]). Let be a dislocated -quasi metric space. Let be a nonempty subset of 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.

It is clear that if , then But if , then or may not equal zero. We denote by the set of all proximinal subsets of

Definition 5 (see [12]). The function , defined by

is called dislocated quasi Hausdorff metric on Also is known as dislocated quasi Hausdorff -metric space, where is the proximinal subset of

Ali et al. [6] extended the family of mapping defined by [21] to the family of all functions such that

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

(F2) for each sequence of positive numbers, if and only if ;

(F3) there exists such that .

(F4) For each sequence of positive real numbers and such that for each , and some , we have , for each .

Lemma 6. Let be a dislocated -quasi-metric space. Let be the dislocated quasi Hausdorff b-metric space on Then, for all and for each , there exists , such that and , where and

Lemma 7 (see [6]). Let be a b-metric space and let be any sequence in for which there exist and such that , Then is a Cauchy sequence in .

Lemma 8. Let be a dislocated -quasi metric space, and let be any sequence in for which there exist and such that

for each Then is a Cauchy sequence in

Proof. Let , for each . Thus, by (3) and property (F4), we get

Following similar arguments as given in [6], we obtain is a Cauchy sequence in

2. Main Result

Let be a dislocated -quasi metric space, and be multifunctions on . Let be an element such that , . Let be such that , Let be such that and so on. Thus, we construct a sequence of points in such that and , with , , and , , where We denote this iterative sequence by We say that is a sequence in generated by If , then we say that is a sequence in generated by

Let us introduce the following definition:

Definition 9. Let be a dislocated -quasi-metric space and be two multivalued mappings. The pair is called a contraction, if there exists and such that for every two consecutive points belonging to the range of an iterative sequence with , we have

where

And we now prove the following main result.

Theorem 10. Let be a complete dislocated -quasi-metric with and be a contraction. Then Also, if (5) holds for each , then and have a common fixed point in and

Proof. Let be the iterative sequence in generated by a point . If

for some , then

Clearly, if , then Also implies So, and Now, implies and implies So, and is a common fixed point of and . So the proof is completed in this case. Now, let

for all By Lemma 6, we have

and

From (11), (F1) and using condition (5), we get

From (6), we have

If , then

which is a contradiction due to (F1) and Therefore,

From (11), (F1) and using condition (5), we get

By using (15) and (F1), we get

Combining (16) and (19), we get

By using (10) and (5), we have

From (6), we have

If , then we obtain

which is a contradiction due to (F1). Therefore,

By using (10) and (5), we have

From (24), , so

Combining (24) and (26), we get

Combining (20) and (27), we get

By Lemma 8, is a Cauchy sequence in Since is a complete dislocated -quasi-metric space, so there exists such that ; that is,

Now, suppose , and then , so

By using Lemma 6 and (5), we have

Since is strictly increasing, we have

Taking on both sides, we get

From (6)

Taking limit as , and by using (29), we get

Using inequality (35) in (33), we get

Now,

Taking limit as ,

Using inequalities (29) and (36) in (38), we get

This is a contradiction, so Now, suppose , and then there exists such that for all By Lemma 6  , so

for all Following similar arguments as above, we get

Now,

Taking limit as , and using inequalities (29) and (41), we get

which is a contradiction, so Hence Similarly by using (29), Lemma 6, and the inequality

we can show that Similarly, Hence, the pair has a common fixed point in Now,

This implies that Hence the proof is completed.

Now, let us introduce the following example.

Example 11. Let and if , and , if Then is a dislocated -quasi-metric space with . Define the mappings as follows:

Case 1. If holds. Define the function by for all and As , and by taking , we define the sequence in generated by Also, Now, if , we have

Also

Case (i). If , and then we have

This implies that

Case (ii). Similarly, if , and then we have

Hence,

Case 2. If holds.

where

Case (i). If , and then we have

so

Case (ii). Similarly, if , and then we have

Hence,

Now, if , then the contraction does not hold. Hence all the hypotheses of Theorem 10 are satisfied so and have a common fixed point.

If we take in Theorem 10, then we obtain the following theorem.

Theorem 12. Let be a complete dislocated -quasi-metric space with and be a multivalued mapping such that for every two consecutive points belonging to the range of an iterative sequence with , ,

where

Then Moreover, if (59) also holds for , then has a fixed point in and

Remark 13. By setting the different values of in (6), we can obtain different results on multivalued contractions as corollaries of Theorem 10.

3. -Khan Type Contraction in Quasi b-Metric Spaces

Piri et al. [42] extended the results of Khan [43] and Fisher [44] by introducing a new general contractive condition with rational expressions. Recently, Piri et al. [30] improved some fixed point results of -Khan type self-mapping on complete metric spaces. In this section, we introduce a new type of contraction satisfying an inequality of rational expressions and prove a new fixed point theorem concerning this type of contraction. Our result is real generalization of Khan fixed point theorem; we introduced -Khan type multivalued for two mappings in -quasi-metric space. We start this section with the following definitions.

Definition 14. Let be a nonempty set, , and be a mapping such that and , implying . Let define = and Let be the multivalued mappings; then the pair is said to be Alt multivalued mapping; if , then

Definition 15 (see [30]). Let be a metric space. A mapping is said to be -Khan type contraction if there exists and such that

for all , and if , then and if , then

Definition 16. Let be a -quasi-metric space and be a pair of multivalued mappings. Then is called Khan type contraction, if there exists and such that for every two consecutive points belonging to the range of an iterative sequence with , , and , we have

where

Theorem 17. Let be a complete -quasi-metric space with Let and be a pair of Khan type contractions and the set is closed and contained Then Also, if (63) holds for each , then and have a common fixed point in and

Proof. As is an arbitrary element of , from condition of the theorem Let be the iterative sequence in generated by a point . Let be elements of this sequence. Clearly, if

for some , then

As , so and Now, implies and implies So, and is a common fixed point of and So the proof is done. In order to find common fixed point of both and , when

for all . Since , and . As is multivalued mapping, Now, , and implies that By induction we deduce that and , for all . Now, by Lemma 6, we have

and

As , then (69) implies

As , and , then by using the condition (63), we get

From (64), we get

Therefore,

and this implies

As , then (69) implies

As , and , then using condition (63), we get

Therefore,

Combining (74) and (77), we get

As , then (68) implies

As , and , then by using condition (63), we get

Therefore,

Similarly, by using (63), (64), and (68), we get

Combining (81) and (82), we get

Combining (78) and (83), we get

By Lemma 8, is a Cauchy sequence in for all So is a subsequence of contained in As is closed, there exists such that , that is,

Also

Now, we show that is a fixed point for . We claim that . On the contrary, we assume that Now

So, there exists such that for all By Lemma 6, we have for all , so

for all By Lemma 6, and , we get

Now, and , and then by (64), we get

Since is strictly increasing, we have

Taking limit as , on both sides of inequality (91), we get

Since , taking limit as , on both sides, we get

By (64), we have

Taking limit as and using inequality (93), we have

Now, inequality (92) implies

Taking limit as on both sides of inequality (87) and using the above inequality, we have

So our assumption is wrong and Now assume that , and then there exists such that for all By Lemma 6  , so

for all Following similar arguments as above, we get

Now,

Taking limit as , on both sides of inequality (100) and using (85) and (99), we get

which is a contradiction, so Hence As and , then Definition 14 implies

Now, we show that is a fixed point for . We claim that . On the contrary, we assume that , and then there exists such that for all By Lemma 6, , so

for all By Lemma 6, and , we get

Now, and , and then by (64), we get

Since is strictly increasing, we have

Taking limit , on both sides of inequality (106), we get

Since , taking limit , on both sides, we get

By using (64), we get

Taking limit as and using inequality (108), we have

Now, inequality (107) implies

Now

Taking limit as ,

Using inequalities (85) and (111) in (113), we get

This is a contradiction, so Now assume that , and then there exists such that for all By Lemma 6  , so

for all Following similar arguments as above, we get

So Hence As and , then Definition 14 implies

Hence, the pair has a common fixed point in Hence the proof is completed.

Corollary 18 (see [30]). Let be a complete metric space and be an -Khan contraction. Then, has a unique fixed point and for every the sequence converges to .

4. Single Valued Result with Application to System of Integral Equations

Let be two self-mappings and . Let , , and so on. In this way, we construct a sequence in such that

We say that is a sequence in generated by

The following result is obtained by replacing the multivalued mappings with the single valued mappings in Theorem 10. Our result generalizes Theorem 24 in [41]. Also, we prove uniqueness of common fixed point in our result.

Theorem 19. Let be a complete dislocated -quasi-metric space with constant and be two self-mappings. If there exists and such that for every two consecutive points belonging to the range of an iterative sequence with , we have

where

then Also, if satisfies (119), then and have a unique common fixed point in and .

Proof. Now, we have to prove uniqueness only. Let be another common fixed point of . Suppose . Then, we have

which implies that

which is contradiction. Then Also

And then, we get So, Now, we deduce the following main result.

Corollary 20. Let be a complete dislocated metric space with constant and be two self-mappings. If there exists and such that for every two consecutive points belonging to the range of an iterative sequence with , we have

where

then Also, if satisfies (124), then and have a unique common fixed point in and .

Let be the set of all functions defined by [21]. Then, we have the following new result.

Corollary 21. Let be a complete dislocated quasi-metric space and be two self-mappings. If there exists and such that for every two consecutive points belonging to the range of an iterative sequence with , we have

where

then Also, if satisfies (126), then and have a unique common fixed point in and .

Now, as an application, we discuss the application of Theorem 19 to find solution of the system of Volterra type integral equations. Consider the following integral equations:

for all We find the solution of (128) and (129). Let be the set of all continuous functions on , endowed with the complete dislocated -quasi-metric. For , define supremum norm as , where is taken arbitrarily. Then define

for all , and with these settings, becomes a dislocated -quasi-metric space.

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

Theorem 22. Assume the following conditions are satisfied:
(i) and are continuous.
(ii) Define

Suppose there exist , such that

for all and , where

Then integral equations (128) and (129) have a unique solution.

Proof. By assumption (ii) and (132), we have

This implies

That is,

which further implies

So, all the conditions of Theorem 19 are satisfied for , , . Hence integral equations given in (128)and (129) have a common unique solution.

Remark 23. By setting different values of in (132), we can obtain different weak contractive inequalities and results as corollaries of Theorem 22.

5. Conclusion

In this work, we have discussed the notion of dislocated -quasi-metric space and given an application to find the solutions of the nonlinear integral equations in such spaces. New results in b-quasi-metric, quasi-metric, quasi dislocated metric, dislocated metric, and metric can be obtained as corollaries of our theorems, which are still not present in the literature. The notions of Alt multivalued mapping and Khan type contraction on a sequence have been introduced. Our observation is that the fixed points of mappings which are contractive only on a sequence can be ensured by the fixed point results. Our results extend the results given in [41, 45].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Acknowledgments

This article was partially supported by the Department of Mathematics and Statistics, Riphah International University, Islamabad, Pakistan. This is a part of research work done by PhD student, Shaif Saleh Alshoraify, under the supervision of second author. Article processing charges will be shared by the brother of Shaif Saleh Alshoraify from Yemen.

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