Discrete Dynamics in Nature and Society

Volume 2015, Article ID 914158, 12 pages

http://dx.doi.org/10.1155/2015/914158

## On Some Fixed Point Results for -Type Contraction Mappings in Metric Spaces

^{1}School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, Chhattisgarh 492010, India^{2}Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain

Received 8 April 2015; Accepted 10 June 2015

Academic Editor: Guang Zhang

Copyright © 2015 Hemant Kumar Pathak and Rosana Rodríguez-López. 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 prove some fixed point theorems for -type multivalued contractive mappings in the setting of Banach spaces and metric spaces. The results provided allow recovering different well-known results.

#### 1. Introduction

The interest in the study of fixed point theory in the frame of multivalued mappings by using the Hausdorff metric has its origin in the contributions of Markin [1] and Nadler [2], which have lead to interesting achievements on this topic with reference to both theoretical results and applications (see the monograph by Singh et al. on fixed point theory [3] and the references therein).

Starting from the work of Nadler [2], many authors have contributed to the huge development of fixed point theory for set-valued contractions. We mention some interesting contributions, such as those made by Reich [4, 5], Lami Dozo [6], Singh [7], Lim [8], Kaneko [9], Mizoguchi and Takahashi [10], Dhompongsa et al. [11], Feng and Liu [12], Klim and Wardowski [13], Suzuki [14], and Pathak and Shahzad [15, 16] (see also the monographs by Goebel and Kirk [17], Petrusel [18], and so forth). We also refer to some recent works on this topic, for instance, those by Hasanzade Asl et al. [19], Samet et al. [20], and Kumam and Sintunavarat [21].

In this paper, we provide an extension of some fixed point results to the context of multivalued -type -contraction mappings, by establishing a common frame which allows obtaining as corollaries some well-known fixed point results.

#### 2. Preliminaries

For a normed linear space and a nonempty subset of , we denote by , , and the collection of all nonempty subsets of , the collection of all nonempty closed and bounded subsets of , and the collection of all compact subsets of , respectively.

For , we define the following mappings: where and . It is well known that the mapping is a metric on called the* Hausdorff metric* induced by the norm in . From Proposition 2.1 [16], we also know that is a metric on . Moreover, and are equivalent metrics [22], since By the results in the monograph by Kuratowski [22], we deduce that is complete provided that is complete ( denotes the metric induced by the norm ) and is a closed subspace of (see Theorem 2.6 [16]). The mapping is also continuous and satisfies the following properties (see Proposition 2.2 [16]): (i), for any and ;(ii), for any and .

Although we have introduced the definitions of and for normed linear spaces, we can also do it for metric spaces just by using the metric instead of the norm; this will be enough for our purpose.

The notions of multivalued contraction and -contraction mapping are essential to this work and we include it here for completeness.

*Definition 1. *One says that a set-valued mapping is a multivalued -contraction mapping if there exists a fixed real number , , such that

*Definition 2 (see [7]). *Let be a metric space. A multivalued map is called -contraction if the following conditions hold: (C1)there exists in such that (C2)for every in in and , there exists in such that

Another important concept is the notion of fixed point for a multivalued map.

*Definition 3. *An element is said to be a fixed point of a multivalued map if . One denotes by the set of fixed points of .

Concerning the existence of fixed points for multivalued contractions, a highly relevant result was provided by Nadler (see [2]).

Theorem 4 (Theorem 5 [2]). *Let be a complete metric space and a multivalued contraction mapping. Then has a fixed point.*

Pathak and Shahzad [16] have given a generalization of Theorem 4 under a condition weaker than multivalued contractivity by using the metric , as recalled below.

Theorem 5 (Theorem 3.2 [16]). *Let be a complete metric space. Every -type multivalued contraction mapping with Lipschitz constant has a fixed point.*

In the next section, we obtain some fixed point results for a more general class of multivalued -type contractions. We provide a common structure which allows obtaining as particular cases some well-known fixed point results.

#### 3. Fixed Point Results for Multivalued -Type Weak -Contraction Mappings

In this section, we present the main results, in which the existence of fixed points is deduced for multivalued -type weak -Lipschitz and -contraction mappings, as introduced below.

*Definition 6. *Let be a metric space. A multivalued map is called an -type multivalued weak -Lipschitz mapping if condition (C2) in Definition 2 holds and there exist and such that If, moreover, there exists such that , for every , we say that is an -type multivalued weak -contraction.

*Remark 7. *In Definition 6, if we takewe get the notion of -type multivalued contraction mapping.

On the other hand, if we takethen we obtain the notion of -type multivalued weak contractive mapping (see [23, Definition 3.3]), since inequality (5) is reduced to Finally, if we takethen we obtain the notion of -type multivalued quasi-contraction mapping (compare inequality with [23, equation (2.3)])

We start this study by providing a fixed point result for -type multivalued weak -Lipschitz mappings. This result (Theorem 8) will find an extension below in Theorem 18 and further in Theorem 21, both stated for partial -type multivalued weak -Lipschitz mappings. It is possible to state first these results and, then, obtain Theorem 8 as a corollary. However, by considering Theorem 8 first, we can focus our attention on the properties of function , considering a general expression which immediately connects with some well-known fixed point results. Therefore, we present the main results in several steps:(s.i)First, in Theorem 8, we consider a general expression for function , while for function it is required that , .(s.ii) Then, in Theorem 18, we extend this result to partial -type multivalued weak -Lipschitz mappings.(s.iii) In Theorem 21, the same type of variability is allowed for function and we explore some other possible general expressions for function , also for partial -type multivalued weak -Lipschitz mappings.(s.iv) We include some considerations concerning -type weak -Lipschitz mappings with respect to a binary relation (including the case of partial orderings).(s.v) We complete this study by proving Theorem 32, which considers different restrictions for the selection of functions and .

Theorem 8. *Let be a complete metric space. Let be an -type multivalued weak -Lipschitz mapping such that the functions , satisfy the following: *(i)*There exists such that , for every .*(ii)*For every , the function is monotonically increasing in the variable , provided that the other variables , , remain fixed.*(iii)*For each fixed, is continuous at .*(iv)*There exists a function such that(iv-a) , for every ,(iv-b) is monotonically increasing for each fixed,(iv-c) is monotonically increasing for each fixed,(iv-d)there exists such that , for every ,(iv-e)there exists such that , for every .*(v)

*The constants in (i) and , in (iv) are such that .*

*Then has a fixed point.*

*Proof. *We denote by the mapping defined as By (5) and (i), we get We construct a sequence in in lines similar to [16, Theorem 3.2] or [23, Theorem 3.4] as follows. Let be given and take to be arbitrary. We fix an element in . Now, from property (C2), it is possible to choose such that . At this step, we could choose depending on and . In general, for , if is chosen, then we can select such that At this step, we could choose depending on and . Note that if for some , then and the proof is complete. By (), we can take such that and . In fact, if we set on each step , then, by (13), we obtain, for every , that which implies, for every , thatwhere we have used (ii) and (iv-a). Hence, If for some , then , so that is a fixed point of and the proof is concluded. Suppose, therefore, that , for every .

Now, assuming that , for some , by (16) and (iv-b), we get and, by (iv-d), which is a contradiction since and (due to inequality ). Hence, we have proved that , for every , so that, by (16) and (iv-c), we have By (iv-d), we obtain Repeating the same procedure -times, we get , for all , where by hypothesis . Hence, is a Cauchy sequence since, for with ,Therefore, by the complete character of , there exists such that . Suppose that . ThenWe note that and , so that , which implies that as .

Hence, from , , and , as , and using hypothesis (iii), we get from (22) Moreover, by (iv-a) and (iv-e), we obtain Using that we deduce that Sinceand , it follows thatHence, if , then . On the other hand, if , the assumption leads to a contradiction by virtue of hypothesis (v) (), which implies that . Finally, since is closed, it is proved that ; that is, is a fixed point of .

*Remark 9. *Note that if , condition (v) in Theorem 8 implies that , so that in this case the -type multivalued weak -Lipschitz mapping in Theorem 8 is in fact an -type multivalued weak -contraction. In this sense, both functions and may contribute to the contractivity of the multivalued mapping through condition (v) which establishes a relation among the constants involved.

Corollary 10. *If one takes , with , and , then is monotonically increasing in all the variables and continuous. Besides, for , one has **In this setting of -type multivalued contraction mappings, we have, as a corollary of Theorem 8, Theorem 3.2 [16] (see Theorem 5).*

In the context of -type multivalued weak contractive mappings, we obtain the following corollary which corrects Theorem 3.4 [23] [see the proof of Theorem 3.4 [23], where it was assumed that ].

Corollary 11. *Let be a complete metric space and an -type multivalued weak contractive mapping with . Then has a fixed point.*

*Proof. *If we take , with , and is monotonically increasing in all the variables and continuous. Choosing , we have is monotonically increasing in each variable, , and thus (iv-d) holds for , , for every , so that (iv-e) holds for and , so that () is fulfilled for the choice . Hence Theorem 8 applies.

In the context of -type multivalued quasi-contraction mappings, we obtain the following corollary which coincides with Theorem 3.6 [23].

Corollary 12. *Let be a complete metric space and an -type multivalued quasi-contraction mapping with . Then has a fixed point.*

*Proof. *Taking , with andwe have that is monotonically increasing in all the variables and also continuous. Taking , we get is monotonically increasing in each variable for the other fixed, , and thus (iv-d) holds for , , for every , so that (iv-e) holds for and , so that (v) is fulfilled for .

On the other hand, if a result similar to Theorem 8 was established for -multivalued contractions, we would have the following theorem.

Theorem 13. *Let be a complete metric space. Let be an -type multivalued weak -Lipschitz mapping, that is, such that there exist and with **Suppose that the functions , satisfy conditions (i)–(iv) in Theorem 8 and*()*the constants in (i) and , in (iv) are such that .**Then has a fixed point.*

*Proof. *Identical to the proof of Theorem 8, except the last part, we have Sinceand , thenIf , then . If , leads to a contradiction in the previous inequality due to (), which implies that and the proof is complete.

*Remark 14. *As a consequence of Theorem 13, using the Hausdorff metric , the restriction required on is (r.i)for multivalued contractions ,(r.ii)for multivalued weak contractive mappings ,(r.iii)for multivalued quasi-contraction mappings .

*Remark 15. *In Theorems 8 and 13, conditions (iv-b), (iv-c), and (iv-d) can be removed, adding the property that()there exists such that , for every ,while condition (v) (resp., ) has to be replaced by the fact that ()the constants in (i) and , in (iv) are such that (for ) or (for ).

This comes from the following ideas: from (), following the proof of Theorem 8, we can take such that and (for the Hausdorff metric , we choose them in such a way that ). Supposing that , for every , if we do not have monotonicity of , from (16) and (), we get so that, using , Repeating the same procedure -times, we get , for all , where by hypothesis (). The rest of the proof is valid.

Now, following the lines in [21] for -set-valued quasi-contractions, we give the following definition.

*Definition 16. *Let be a metric space and let be given. A multivalued map is called a partial -type multivalued weak -Lipschitz mapping if condition (C2) in Definition 2 holds and there exist and such thatIf, moreover, there exists such that , for every , we say that is a partial -type multivalued weak -contraction.

*Remark 17. *Taking , with , and then we obtain the notion of partial -type -set-valued quasi-contraction (compare with [21, Definition 3.1]).

In relation with Definitions 2.21, 3.1 and Theorem 3.2 [21], we can establish the following result.

Theorem 18. *Let be a complete metric space, let be given, and let be a partial -type multivalued weak -Lipschitz mapping such that *(h1)* is -admissible; that is, for each and with , one has , for every ,*(h2)*there exist and such that ,*(h3)*if is a sequence in such that , for all , and there exists such that as , then , for every ,*(h4)*the functions , are such that conditions (i), (ii), (iii), (iv) [(a)–(e)], and (v) in Theorem 8 hold.**
Then has a fixed point in .*

*Proof. *We proceed similarly to the proof of Theorem 8 and also Theorem 3.2 [21]. By (40) and (i), we get We take given. We start the sequence with the terms and and then, using (C2), there exists such that Note that since is -admissible, we have . In general, for , if is chosen such that , then we can select such that and we also have . Again, could have been chosen at each step depending on and . For this, we remark that if for some , the proof is concluded, so that we can assume that , for every and take on each step , where are fixed, by (v), in such a way that and . Then, by (43), we get, for every , that which implies, for every , thatusing (ii) and (iv-a). As in the proof of Theorem 8, we can suppose that , for every . Similarly to the proof of Theorem 8, we deduce that , for every and so that is a Cauchy sequence and, by the completeness of , there exists such that

Finally, if , inequality (22) holds; since , for all , and as , then, by hypotheses, , for every , so that Similarly to the proof of Theorem 8, we get that henceso that if , then and if , the assumption leads to a contradiction again and the proof is complete.

*The previous result allows formulating a fixed point result for -type partial quasi-contraction mappings while, in [21], the Pompeiu-Hausdorff metric is used in -metric spaces. The procedure in metric spaces could be adapted to -metric spaces in the lines of [21].*

*Remark 19. *In Theorem 18, if we consider the Pompeiu-Hausdorff metric , then condition (v) can be relaxed to hypothesis () in the statement of Theorem 13.

In particular, for partial -set-valued quasi-contractions, , , and , we obtain, as a corollary, the assertion of Theorem 3.2 [21] for metric spaces.

*Remark 20. *Inequality (40) is trivially valid if one of the following conditions hold (see Corollaries 3.4–3.6 [21]): (r.i) , for every ;(r.ii) , for every , where ;(r.iii) , for every , where .

*Theorem 21. Let be a complete metric space, let be given, and let be a partial -type multivalued weak -Lipschitz mapping such that conditions (h1), (h2), and (h3) hold. Suppose also that the functions , satisfy the following:(A1) is monotonically increasing and .(A2)If as , then .(A3)Conditions (ii), (iii), and (iv) [(a)–(e)] in Theorem 8 hold.(A4)For given in (iv-e), one has that , for every .(A5)There exists such that the operator (where is the identity mapping on ) satisfies the following properties:(A5.i) , for every ;(A5.ii) , for every ;(A5.iii) , for every , where .*

Then has a fixed point in .

*Proof. *We proceed similarly to the proof of Theorem 18. We take and given by (h2), with ; then a sequence is obtained by using (C2) and -admissibility. The process is summarized as follows: for , if is chosen such that , then, by (C2), we can select such thatand we also have .

We remark that, due to the monotonically increasing character of , the mapping is always strictly increasing. Note that if for some , the proof is concluded, so that we can assume that , for every , so that we can take, on each step, . Then, for every , by (ii) and (iv-a), If , for some , then is a fixed point of , so we assume that , for every .

Assuming that , for some , by (iv-b) and (iv-d), we obtain which is a contradiction due to (A5) since . This proves that , for every . Therefore, by (iv-c), we getThis implies, from (A5) and the property that for every , thatTo check that is a Cauchy sequence, we observe that, for with , we get Since is complete, there exists such that If , using that , for all , and as , then, by (h3), we have , for every ; hence so thatTherefore, similarly to inequality (22), we have, by the monotonicity of ,Analogously to the proof of Theorem 8 and using (iii),By (A2), we haveHence, from this inequality and (57), by (iv-a) and (iv-e), we getSimilarly to the proof of Theorem 8, we have Therefore, if , then and the proof is concluded. If , we get to a contradiction due to and (A4). Hence, and the proof is complete, since is closed, so that .

*Remark 22. *Theorem 8 is a particular case of Theorem 21. Indeed, if we take , where , and assuming that condition (v) holds, then we have the following: (r.i) is monotonically increasing and ; hence (A1) holds.(r.ii) is continuous (so that (A2) holds).(r.iii)Condition (A4) is valid since , for every (this is true since it is equivalent to ).(r.iv)Concerning (A5), since we assume that , then we can take such that and the operator is monotonically increasing and(r.a), for every ;(r.b), for every ;(r.c), for every .

*
It is also obvious that if is an -type multivalued weak -Lipschitz mapping and there exists such that , for , then is also an -type multivalued weak -Lipschitz mapping for .*

*Remark 23. *In the proof of Theorems 18 and 21, it is easy to observe that a proper combination of condition (C2) with the -admissible character of the mapping, (h1), would allow relaxing slightly the definition of partial -type multivalued weak -Lipschitz mappings. Indeed, in these theorems, it is possible to replace these hypotheses ((C2) and -admissibility of ) by the following: ()for every , with and , there exists such that and .

*If we consider self-mappings , a procedure similar to Theorem 2.3 [19] gives the following result.*

*Theorem 24. Let be a complete metric space, let be given, and let be such that (t.i)for each with , one has ,(t.ii)there exists such that ,*

and there exist and with Suppose also that (h3) holds and that the functions , satisfy the following conditions: (I) is monotonically increasing, is increasing in the fourth variable, and conditions (iv) [(a)–(d)] in Theorem 8 hold, where .(II)Consider , for every .(III)One of the following conditions holds:(III.i)For fixed, if are small enough and is close enough to , then .(III.ii)(A2) is satisfied and conditions (iii) and (iv-e) in Theorem 8 hold, where .

Then has a fixed point in .

*Proof. *We take with and define the sequence such that , . If , for some , then we have a fixed point of ; thus we assume that , for every (i.e., , for every ). Using the hypotheses, it is easy to prove that , for every . Then, for each , by the nondecreasing character of and (in the fourth variable) and (iv-a),If , for some , by the monotonicity of , (iv-b), (iv-d), , and (II), we getwhich is a contradiction. Then, , for every ; hence, by the monotonicity of , (iv-c), (iv-d), and ,Using (II), the sequence is a Cauchy sequence. Then there exists such that as . By (h3), , for every . Suppose that ; thenIn case (III.i), for being large enough, we have so that , which is a contradiction; hence and the proof is complete. On the other hand, in case (III.ii), by the monotonicity of and (II), we havea contradiction again and the proof is finished.

*Remark 25. *If in Theorem 24, then we have Theorem 2.3 [19]. Indeed, all the conditions are satisfied (see the proof of Corollary 11) and, for fixed, if are small enough and is close enough to , then ; hence (III.i) is also fullfilled.

*Remark 26. *Certain conditions in Theorem 24 also extend some hypotheses in [20].

*As indicated in [21], a function can be defined in connection with a binary relation in (which could be, e.g., a partial ordering in ). Thus, Theorem 4.7 [21] can be extended to -type multivalued weak -Lipschitz mappings with respect to , which are defined as follows. Here, we only consider the case of metric spaces, but -metric spaces could be considered accordingly.*

*Definition 27. *Let be a metric space and a binary relation on . A multivalued map