Fixed Points for Multivalued Suzuki Type (<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.081pt" id="M1" height="14.3416pt" version="1.1" viewBox="-0.0657574 -12.2606 31.7726 14.3416" width="31.7726pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-230" d="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z"/></g><g transform="matrix(.017,0,0,-0.017,8.547,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.017,0,0,-0.017,15.336,0)"><path id="g198-19" d="M852 669C820 657 792 643 768 628C705 667 627 687 547 687C310 687 154 548 154 412C154 332 212 282 296 282C434 282 515 426 523 557H503C474 402 410 312 298 312C241 312 195 352 195 408C195 535 341 657 546 657C613 657 683 644 738 609C640 538 574 431 486 271C374 71 314 16 189 16C118 16 69 60 69 95H71C78 84 93 83 101 83C129 83 144 106 144 131C144 160 121 179 93 179C60 179 34 151 34 108C34 42 100 -15 207 -15C364 -15 491 73 603 304C614 297 639 293 650 293C681 293 683 254 683 245C683 181 599 150 599 70C599 21 645 -12 692 -12C747 -12 804 30 868 103L851 119C789 52 741 18 706 18C685 18 674 29 674 52C674 92 750 157 750 226C750 257 727 287 696 310V312C812 315 876 394 876 473C876 525 849 577 793 612C812 627 832 639 859 650L852 669ZM814 486C814 417 765 345 686 337C669 348 649 354 626 354C673 451 708 533 766 588C799 561 814 525 814 486Z"/></g></svg>)-</span>Contraction Mapping with Applications

Abbas, Mujahid; Iqbal, Hira; Petrusel, Adrian

doi:https://doi.org/10.1155/2019/9565804

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Volume 2019 | Article ID 9565804 | https://doi.org/10.1155/2019/9565804

Fixed Points for Multivalued Suzuki Type ()-Contraction Mapping with Applications

Mujahid Abbas,^1,2Hira Iqbal,³and Adrian Petrusel^4,5

Academic Editor: Tomonari Suzuki

Received07 Sept 2018

Revised09 Jan 2019

Accepted03 Feb 2019

Published17 Feb 2019

Abstract

In this paper, we will introduce the concept of Suzuki type multivalued -contraction and we will prove some fixed point results in the setting of a metric space equipped with a binary relation. Our results generalize and extend various comparable results in the existing literature. Examples are provided to support the results proved here. As an application of our results, we obtain a homotopy result, proving the existence of a solution for a second-order differential equation and for a first-order fractional differential equation.

1. Introduction and Preliminaries

Let be a metric space and be a mapping on . An element is called a fixed point of if it remains invariant under the action of ; that is, A mapping on a metric space is said to be a Banach contraction if holds for all , where . A Banach contraction mapping defined on a complete metric space has a unique fixed point. This result is known as Banach contraction principle. Several authors have extended and generalized Banach contraction principle in different directions.

Jleli and Samet [1] suggested a modification in the contraction condition and introduced a -contraction mapping. Consistent with [1], the following notations, definitions, and results will be needed in the sequel.

Suppose that where

() is nondecreasing;

( for each sequence , if and only if ;

() there exists and such that .

Example 1. Define for by Then, .

Let be a metric space and . A mapping is called a -contraction if for any , we have whenever and . Jleli and Samet [1] proved the following fixed point theorem in the framework of a generalized metric space in the sense of Branciar; i.e., the triangle inequality is replaced by the inequality , for all pairwise distinct points .

Theorem 2. Let be a complete generalized metric space in the sense of Branciari and be a -contraction. Then has a unique fixed point.

Hussain et al. [2] considered the following class of mappings: where

() is nondecreasing;

() if and only if ;

() for each sequence , if and only if ;

() there exists and such that ;

() .

Example 3. Let be defined by and , respectively, then and .

Hussain et al. [2] proved the following result.

Theorem 4. Let be a complete metric space and be a continuous mapping. Suppose there exist and with , such that for any , we have Then has a unique fixed point.

For other results in this direction, we refer to [3, 4] and references mentioned therein.

Let be a metric space and (respectively ) be the family of all nonempty closed and bounded (nonempty compact, respectively) subsets of . For and , define is the Hausdorff-Pompeiu metric on (or on ) induced by . Let be a given multivalued mapping. An element is called a fixed point of if A mapping is said to be a Nadler contraction if there exists such that for any Nadler [5] obtained the following multivalued version of Banach contraction principle.

Theorem 5. Let be a complete metric space and be a Nadler contraction. Then has at least one fixed point.

Later on, many researchers have obtained fixed point results for multivalued mappings satisfying generalized contraction type conditions. For example, recently, Hançer et al. [6] proved the following fixed point result for multivalued -contractions.

Theorem 6. Let be a complete metric space and be a multivalued mapping. Suppose that there exist and such that for any , provided that . Then has at least one fixed point.

Durmaz [7] introduced a new type of generalized multivalued -contraction and proved some interesting fixed point results (see also [8]). Kikkawa and Suzuki [9] refined Nadler’ result by proving the following theorem.

Theorem 7. Let be defined as . Let be a complete metric space and . Assume there exists such that Then has at least one fixed point.

We denote and define the following class of mappings, which was considered in [10].

Example 8. Let be defined by where are given by and . Obviously .

Example 9. Let be defined by where are defined by and for all . Note that .

Many results, dealing with existence of fixed points of mappings satisfying certain contraction type conditions in the framework of complete metric spaces endowed with a partial ordering, have appeared in the last decade. Ran and Reurings [11] proved an analogue of Banach’s fixed point theorem in a metric space endowed with partial ordering and gave an application of their results to solve matrix equations. Alam and Imdad [12] proved another variant of Banach’s fixed point theorem in a metric space equipped with a binary relation which generalized many comparable results, including Ran and Reuring’s result in [11]. Senapati and Dey [13] proved Banach’s fixed point theorem in metric spaces equipped with an arbitrary binary relation using -distance. They employed their results to prove the existence of solutions of nonlinear fractional differential equations and fractional thermostat model involving the Caputo fractional derivative. A very nice Ph.D. thesis was written on the same subject; see Dobrican [14].

Let us first recall the following definitions.

Definition 10. Let be a nonempty set and be a binary relation defined on . Then, is -related to if and only if .

We denote and .

Definition 11. Let be a nonempty set and a binary relation on . A sequence is called -preserving if

Definition 12. Let be a metric space. A binary relation defined on is called -self closed if whenever is an -preserving sequence and converges to , then there exists a subsequence of with either or for all .

Definition 13. Let be a metric space and a binary relation defined on . A mapping is -closed if for any , implies that for any and .

If is a multivalued map, then we set In particular, if is single-valued, then we denote Motivated by the results in [2, 10, 12], we introduce the concept of a Suzuki type multivalued -contraction mapping and present some fixed point results in metric spaces equipped with a binary relation. Our results extend and generalize several results given in [2, 15–19]. We also provide applications of our results to homotopy theory proving the existence of a solution of second-order differential equations and first-order fractional differential equations.

2. Multivalued Suzuki Type -Contraction

In this section, we obtain a fixed point result for multivalued Suzuki type -contraction in a metric space equipped with a binary relation .

Throughout this paper satisfies the following additional property: where .

We will denote We start with the following definition.

Definition 14. Let be a metric space and a binary relation on . Assume that and . A mapping is a multivalued Suzuki type -contraction if for any with where with

Our first main result is the following.

Theorem 15. Let be a complete metric space, a binary relation on , and a multivalued Suzuki type -contraction. Suppose that following conditions hold: (1) is nonempty,(2) is -closed,(3) is -self closed or has closed graph. Then has at least one fixed point.

Proof. Since is nonempty, if we choose , then there exists some such that . If , the result follows. Assume that . As and we have Now and imply that Then by and , it follows that and hence Hence, we obtain and, in conclusion, we get that By we have Thus, We can choose such thatAs , , and is - closed, we have that . If , the result follows. Assume that . Also, Hence, By and , we have As and , we obtain that Hence, By , we have Hence, We can choose such that By (27), we get Since , and is -closed, we have .
Continuing this way, we can obtain a sequence such that and is -preserving. Obviously, we have , for all natural numbers . Hence,Letting , we have It follows from () that Now, we show that is a Cauchy sequence. If we set , then from (37), we obtainFurther, from (), there exist and such that Suppose . Let . Then there exists such that for all , we obtain that Hence, for all , we have This implies that If , then for there exists such that for all , we have which implies that Hence, for each case, we obtain that Thus, using (40) we have Therefore, So, there exists such that for all , we have which implies that Let . Then By the convergence of the series we get as . Hence, is Cauchy. Since is complete, there exists in such that . We show that has a fixed point. Assume on the contrary that does not have a fixed point. Then, for all natural numbers . As , we have which implies that Furthermore, gives thatThus which further implies thatHenceIf is -self closed, there exists a subsequence of such that either or . Assume that . If , then we have From (57), we obtain that a contradiction. Hence, for all .
By our assumption . Thus which implies thatConsequently, Also, From (61), it follows thatLetting in (64) we obtain that a contradiction. Hence, .
If has closed graph, since for each and , we get that .

If we take in Theorem 15, we obtain a Suzuki type generalization of the result in [6] in the framework of a complete metric space equipped with a binary relation .

Corollary 16. Let be a complete metric space, a binary relation on and . Assume that and . Suppose that there exists such that for any with , implies that If conditions (1)-(3) in Theorem 15 are satisfied, then has a fixed point.

If we take n, Theorem 15, then we have the following multivalued extension of Cirić result in [16].

Corollary 17. Let be a complete metric space, a binary relation on , and . Suppose that and there exist with , such that for any with , implies that Assume that conditions (1)-(3) in Theorem 15 are satisfied. Then has a fixed point.

Remark 18. Note that the conclusion of Corollary 17 can be written as

Notice that if we take in Theorem 15, using Remark 18, we obtain the following multivalued Suzuki type generalization of Chatterjea’s result in [15].

Corollary 19. Let be a complete metric space, a binary relation on , and . Suppose that and there exists such that for any with , implies that Assume that conditions (1)-(3) in Theorem 15 are satisfied, then has a fixed point.

If we take in Theorem 15, using Remark 18, we obtain the following multivalued Kannan type result in [17].

Corollary 20. Let be a complete metric space, a binary relation on , and . Suppose that and there exist with such that for any with , implies that Assume that conditions (1)-(3) in Theorem 15 are satisfied. Then has a fixed point.

Taking in Theorem 15, we have a multivalued extension and generalization of Reich’s result in [20].

Corollary 21. Let be a complete metric space, a binary relation on , and . Suppose that and there exist with , such that for any with , the following implication is true implying that Assume that conditions (1)-(3) in Theorem 15 are satisfied. Then has a fixed point.

Similarly, if we take , then we obtain the following corollary.

Corollary 22. Let be a complete metric space, a binary relation on , and . Suppose that and there exists with , such that for any with , implies that Assume that conditions (1)-(3) in Theorem 15 are satisfied. Then has a fixed point.

We now give an example of a multivalued Suzuki type -contraction which is neither a multivalued Banach contraction nor a multivalued -contraction.

Example 23. Let Define the binary relation on as follows: Let be defined byDefine the mapping by Clearly, is -closed and is nonempty. Indeed, if , , then . Take an -preserving sequence such that converges to and for all . Then, , for all . Thus, or . Since both and are closed, either or .
Let and . We consider following cases: (1)If , then .(2)If , then clearly . For and we have which implies that All other cases are trivial. Thus all the conditions of Theorem 15 are satisfied. Moreover, has two fixed points and .
Note that, for , for any . Thus is not a Nadler’s contraction. Also, for any Thus, is not multivalued -contraction.
If is a single-valued map, then we have the following result.

Theorem 24. Let be a complete metric space, a binary relation on , and . Suppose that , and there exist with , such that for any with , implies that In addition, assume that the following conditions also hold: (1) is nonempty,(2)if such that , then ,(3) is -self closed or has closed graph. Then has at least one fixed point.

Remark 25. The above theorem generalizes various existing results in literature. If in Theorem 24 we take , then we have the following: (1)We obtain a Suzuki type generalization of -contraction result in [2] in the setup of metric spaces endowed with binary relation. If , the above theorem also generalizes the result in [16].(2)If , , and , we obtain a Suzuki type result in the setting of metric spaces endowed with binary relation (see [19]).(3)If , , and , we obtain a Suzuki type version of Chatterjea result [15] in the setup of metric spaces endowed with binary relation.(4)If , , and , we obtain a Suzuki type result for generalized Kannan mappings [17] in the setup of metric spaces endowed with binary relation.

Example 26. Let be the usual metric space with binary relation defined as follows: Define the mapping by Let and . Note that is not a -contraction because Consider the following cases: (1)If , then does not hold.(2)If , then clearly is a Suzuki type -contraction.(3)If , then we have For and , we obtain that which implies that Clearly, is -closed and is nonempty, since . Take an -preserving sequence such that converges to and we have for all . Note that . Then, for . Thus, is a subset of , , or . Since all of these sets are closed, either or . Thus all the conditions of Theorem 24 are satisfied. Moreover, is a fixed point of .

3. Application to Homotopy Results

In this section, as an application of our above fixed point result for Suzuki type -contractions, we obtain a homotopy result for this class of multivalued mappings. For the beginning, we give a local fixed point result for multivalued Suzuki type -contraction. Let be an open ball and the closure of .

Theorem 27. Let be a complete metric space, a binary relation on , and . Suppose that be a multivalued Suzuki type -contraction with where . Assume that the following conditions are also satisfied: (1),(2) is -closed,(3) has closed graph or is -self closed. Then has a fixed point in .

Proof. Since , there exists such that . If , the result follows. Assume that . Let be such that and . Clearly, Now implies that Thus, we have Hence which implies that By the condition (, we get We may now choose such that and where . Furthermore,Note that Thus as Continuing this way, we obtain a sequence with following properties: (1),(2),(3),(4). By similar arguments to those in the proof of Theorem 15, we obtain that is a Cauchy sequence which converges to and .

Now we present the following homotopy result.

Theorem 28. Let be a complete metric space, an open set of , a -self closed binary relation on and be a multivalued mapping with closed values. Suppose that for each , satisfies the conditions (1)-(2) in Theorem 27. Assume that the following conditions are also satisfied: (1), for each and each ;(2) is a multivalued Suzuki type -contraction with closed graph, for each ;(3)there exists a continuous increasing function such that for all and each .

Then has a fixed point if and only if has a fixed point.

Proof. Suppose that has a fixed point , then by (1) . Define As , so . Define partial order on by and Let be a totally ordered set in and . Consider a sequence in such that and as . Then, we haveAs by , we have Thus (106) gives that and Hence is Cauchy sequence which converges to an element . As for and is closed, so . Also, from (1) we have . Hence, . Since is totally ordered, therefore for each . That is, is an upper bound of . By Zorn’s Lemma, the set admits a maximal element . We claim that . Assume, on the contrary, that . Choose and such that and Note that Thus satisfies all assumptions of Theorem 27. Hence, for all , there exists such that . Hence . Now, implies that which further implies that , a contradiction to the fact that is a maximal element.
Conversely, if has a fixed point, then by similar arguments to those given before, we obtain that has a fixed point.

4. Existence of a Solution of Second-Order Differential Equation

In this section, we study the existence of solution of a two-point boundary value problem associated with a second-order differential equation. Let be the space of all continuous functions defined on . The metric on is given byDefine the binary relation on by Note that the space is complete metric space. We consider the following two-point boundary value problem:where is a continuous function. Then, the problem (116) can be written in the following integral form:where the associated Green function is see [21] for details.

Theorem 29. Assume that the following conditions are satisfied: (1) is continuous,(2) is increasing for all ,(3)there exists such that the following condition holds for all , with , (4)there exists such that for all , we have Then the problem (116) has a solution in .

Proof. If we define the mapping by then our problem can be written as a fixed point equation .
Obviously, is continuous. As is increasing, for any with for all , we obtain that Thus, is -closed. If such that , then we haveAs , we get that . The inequality (123) becomes Hence, we have which implies that Taking square root on both sides and passing through exponential function, we obtain that where as . Hence, Thus, where and .
As above inequality is true for any with , so is for any , with . Thus By Theorem 24 we get that (116) has a solution in .

5. Existence of a Solution of Fractional Boundary Value Problem

In this section, we investigate the existence of solutions of a nonlinear fractional differential equation. Let the space and the metric be defined as in the above section.

Consider the following fractional differential equationwith boundary conditionsHere stands for the Caputo fractional derivative of order , defined by (where we consider and ) and denotes the Riemann-Liouville fractional integral of order of a continuous function , given by Senapati and Dey ([13]) showed that the problem (131)+(132) can be written in the following integral form:

Theorem 30. Suppose that following conditions are satisfied: (1) is a continuous function,(2) is an increasing function,(3)for every with , the following condition holds: where ,(4)there exists such that, for all , we have Then, (131)+(132) has at least one solution in .

Proof. Define the mapping by Then (135) can be written as a fixed point equation for ; i.e., . Consider on the binary relation defined as followsBy the given assumption (4), is nonempty. If are such that for every , then by assumption that is increasing we have which implies that . Therefore, is -closed. Nieto and López [22] have shown that if there exists a sequence in such that and converges to some , then . Hence, is -self closed. If such that , then where is the beta function. From the above inequality, we obtain that Taking square root on both sides and passing through exponential function, we have that is, where . Since the above inequality holds for any such that so is true for any , with Hence we have Thus is a Suzuki type -contractive mapping. Since all the conditions of Theorem 24 are satisfied, the problem (131) has at least one solution.

Data Availability

The authors declare that the data supporting the findings of this study are available within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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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.

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