Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 914158, 12 pages
http://dx.doi.org/10.1155/2015/914158

Research Article

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

Hemant Kumar Pathak¹ and Rosana Rodríguez-López²

¹School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, Chhattisgarh 492010, India
²Departamento 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 take

we get the notion of -type multivalued contraction mapping.
On the other hand, if we take

then we obtain the notion of -type multivalued weak contractive mapping (see [23, Definition 3.3]), since inequality (5) is reduced to

Finally, if we take

then 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 , that

where 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 . Then

We 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

Since

and , it follows that

Hence, 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 and

we 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

Since

and , then

If , 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 that

If, 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 , that

using (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

hence

so 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 that

and 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 get

This implies, from (A5) and the property that for every , that

To 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 that

Therefore, similarly to inequality (22), we have, by the monotonicity of ,

Analogously to the proof of Theorem 8 and using (iii),

By (A2), we have

Hence, from this inequality and (57), by (iv-a) and (iv-e), we get

Similarly 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 get

which 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 ; then

In 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 have

a 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 is called an -type multivalued weak -Lipschitz mapping with respect to if condition (C2) in Definition 2 holds and there exist and such that

If, moreover, there exists such that , for every , one says that is an -type multivalued weak -contraction with respect to .

Theorem 28. Let be a complete metric space, a binary relation on , and an -type multivalued weak -Lipschitz mapping with respect to such that

(H1) is weakly preserving, 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)one of the following conditions holds:

(H4-i)the functions , satisfy conditions (i), (ii), (iii), (iv) [(a)–(e)], and (v) in Theorem 8 or(H4-ii)the functions , satisfy (A1)–(A5).

Then has a fixed point in .

Remark 29. The condition of weakly preservance of , (H1), is fulfilled if the following condition holds:

where, given , we define

Remark 30. In the proof of Theorem 28, we can relax slightly the definition of -type multivalued weak -Lipschitz mappings with respect to , by replacing hypotheses (C2) and weakly preservance of , (H1), by the following combination of both:

()for every , with and , there exists such that and .

For self-mappings , we have the following corollary of Theorem 24.

Corollary 31. Let be a complete metric space, a binary relation on , and such that

(c.i)for each with , one has ,(c.ii)there exists such that ,

and there exist and such that

Suppose also that (H3) holds and that the functions , satisfy conditions (I), (II), and (III) in Theorem 24.
Then has a fixed point in .

In the previous result, if the metric space admits a partial ordering , then the relation can be chosen as , so that the conditions in Corollary 31 reduce to those in Corollary 2.4 [19].

Also in the lines of [19], we consider the family of functions defined as follows: if and only if is nondecreasing and such that

It is clear that if , then , for every (see [19] and the references therein).

We have the following result, where condition (C2) is removed.

Theorem 32. Let be a complete metric space, let be given, and let be a multifunction such that conditions (h1), (h2), and (h3) hold. Suppose, further, that there exist and such that

where the functions , satisfy that

(a1) is strictly increasing and ,(a2)there exists such that , for every , where ,(a3)if , then .

Then has a fixed point in .

Proof. We proceed similarly to the proof of Theorem 2.1 [19]. We include it here for completeness. We take and in with given by hypothesis. The proof is finished if or if . Then, we assume that and take arbitrarily fixed. Then, by the definition of and inequality (74),

so that we can choose such that ( due to ). By -admissibility, we have that . Moreover, since , then and . Since is strictly increasing, we have . Next, we take . Now, if , the proof is finished, so we assume that and, similarly, we get

then there exists () such that

Besides, .
Using that is strictly increasing, we have , and we can take . Again, if , the proof is finished, so we assume that and, similarly, we get

and then there exists () such that

Besides, .
In general, for , if () is chosen such that and , we can take . If , the proof is finished, so we assume that and, similarly, we get

and then there exists () such that

and .
Hence, we can take a sequence with , , , and for every .
This allows proving that is a Cauchy sequence since, for with , we get

which tends to as since .
Now, by completeness of , there exists such that By hypothesis, , for every . Suppose that . Then

for every . Since , for all , then, for every ,

so that the hypothesis implies that . This, joint to and the convergence of , implies that .

Remark 33. Note that condition (74) coincides with (40). In Theorem 32, if we consider the Hausdorff distance in (74) and , the conclusion holds.

Remark 34. If we take , then and if , then . Hence, for the Hausdorff distance , the existence of fixed point follows, such as in Theorem 2.1 [19], since . On the other hand, for the metric , if we consider , then the condition is valid for and Theorem 32 applies.

Theorem 35. Let be a complete metric space, a binary relation on , and such that conditions (H1), (H2), and (H3) hold. Suppose that there exist and such that

where the functions , satisfy conditions (a1), (a2), and (a3) in Theorem 32. Then has a fixed point in .

Remark 36. Conditions (69) and (85) are the same.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the editor and the anonymous referees for their interesting and helpful comments and suggestions. The research of Hemant Kumar Pathak was supported by University Grants Commission, New Delhi (MRP-MAJOR-MATH-2013-18394). The research of Rosana Rodríguez-López was partially supported by Ministerio de Economía y Competitividad, project MTM2013-43014-P, and cofinanced by the European Community fund FEDER.

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