Journal of Applied Mathematics

Volume 2013 (2013), Article ID 621614, 6 pages

http://dx.doi.org/10.1155/2013/621614

Research Article

Fixed Points of Multivalued Nonself Almost Contractions

Maryam A. Alghamdi,¹ Vasile Berinde,² and Naseer Shahzad³

¹Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, P.O. Box 4087, Jeddah 21491, Saudi Arabia

²Department of Mathematics and Computer Science, North University of Baia Mare, Baia Mare, Romania

³Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21859, Saudi Arabia

Received 29 March 2013; Accepted 20 May 2013

Academic Editor: Wei-Shih Du

Copyright © 2013 Maryam A. Alghamdi 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.

We consider multivalued nonself-weak contractions on convex metric spaces and establish the existence of a fixed point of such mappings. Presented theorem generalizes results of M. Berinde and V. Berinde (2007), Assad and Kirk (1972), and many others existing in the literature.

1. Introduction

The study of fixed points of single-valued self-mappings or multivalued self-mappings satisfying certain contraction conditions has a great majority of results in metric fixed point theory. All these results are mainly generalizations of Banach contraction principle.

The Banach contraction principle guarantees the existence and uniqueness of fixed points of certain self-maps in complete metric spaces. This result has various applications to operator theory and variational analysis. So, it has been extended in many ways until now. One of these is related to multivalued mappings. Its starting point is due to Nadler Jr. [1].

The fixed point theory for multivalued nonself-mappings developed rapidly after the publication of Assad and Kirk’s paper [2] in which they proved a non-self-multivalued version of Banach’s contraction principle. Further results for multivalued non-self-mappings were proved in, for example, [3–7]. For other related results, see also [8–38].

On the other hand, Berinde [11–13] introduced a new class of self-mappings (usually called weak contractions or almost contractions) that satisfy a simple but general contraction condition that includes most of the conditions in Rhoades’ classification [39]. He obtained a fixed point theorem for such mappings which generalized the results of Kannan [40], Chatterjea [41], and Zamfirescu [42]. As shown in [43], the weakly contractive metric-type fixed point result in [12] is “almost” covered by the related altering metric one due to Khan et al. [21].

In [9], M. Berinde and V. Berinde extended Theorem 8 to the case of multivalued weak contractions.

Definition 1. Let be a metric space and a nonempty subset of . A map is called a multivalued almost contraction if there exist a constant and some such that

Theorem 2 (see [9]). Let be a complete metric space and a multivalued almost contraction. Then has a fixed point.

The aim of this paper is to prove a fixed point theorem for multivalued nonself almost contractions on convex metric spaces. This theorem extends several important results (including the above) in the fixed point theory of self-mappings to the case on nonself-mappings and generalizes several fixed point theorems for nonself-mappings.

2. Preliminaries

We recall some basic definitions and preliminaries that will be needed in this paper.

Let be a metric space and the set of all nonempty bounded and closed subsets of . For , define

It is known that is a metric on and is called the Hausdorff metric or Pompeiu-Hausdorff metric induced by . It is also known that is a complete metric space whenever is a complete metric space.

Definition 3. Let be a multivalued map. An element is said to be a fixed point of if .

In this paper we assume that is a convex metric space which is defined as follows.

Definition 4. A metric space is convex if for each with there exists , , such that

This notion is similar to the definition of metric space of hyperbolic type. The class of metric spaces of hyperbolic type includes all normed linear spaces and all spaces with hyperbolic metric.

It is known that in a convex metric space each two points are the endpoints of at least one metric segment (see [2]).

Proposition 5 (see [2]). Let be a closed subset of a complete and convex metric space . If and , then there exists a point (the boundary of ) such that

The following lemma will be required in the sequel.

Lemma 6 (see [1, 2]). Let be a metric space and . If , then, for each positive number , there exists such that

The definition of an almost contraction given by Berinde [12] is as follows.

Definition 7. Let be a metric space. A map is called almost contraction if there exist a constant and some such that

Theorem 8 (see [12]). Let be a complete metric space and an almost contraction. Then

(1);(2)for any , the Picard iteration converges to some ;(3)the following estimate holds

Let us recall (see [30]) that a mapping possessing properties (1) and (2) is called a weakly Picard operator.

In fact, Theorem 8 generalizes some important fixed point theorems in the literature such as Banach contraction principle, Kannan fixed point theorem [40], Chatterjea fixed point theorem [41], and Zamfirescu fixed point theorem [42].

3. Main Results

Theorem 9. Let be a complete convex metric space and a nonempty closed subset of . Suppose that is a multivalued almost contraction, that is,

with and some such that . If satisfies Rothe’s type condition, that is, , then there exists such that ; that is, has a fixed point in .

Proof. We construct two sequences and in the following way. Let and . If , let . If , then there exists such that

Thus , and, by Lemma 6 and , we can choose such that

If , let . If , then there exists such that

Thus , and, by Lemma 6 and , we can choose such that

Continuing the arguments we construct two sequences and such that

(i); (ii),   where(iii); (iv) whenever , and then is such that

Now we claim that is a Cauchy sequence. Suppose that

Obviously, if , then and belong to . Now, we conclude that there are three possibilities.

Case  1. If , then . Thus

since .

Case  2. If , , then . We have

Case  3. If , , then , , , and . We have

Since , then

Since

then

Thus, combining Cases 1, 2, and 3, it follows that

where

Following [2], by induction it follows that for

where

Now, for , we have

This implies that the sequence is a Cauchy sequence. Since is complete and is closed, it follows that there exists such that

By construction of , there is a subsequence such that

We will prove that . In fact, by (i), . Since as , we have

as . Note that

which on letting implies that ; it, then, follows that .

By Theorem 9 we obtain as a particular case, a fixed point theorem for multivalued nonself-contractions due to Assad and Kirk [2] that appears to be the first fixed point result for nonself-mappings in the literature.

Corollary 10 (see [2]). Let be a complete convex metric space and a nonempty closed subset of . Suppose that is a multivalued contraction; that is,

with . If satisfies Rothe’s type condition, that is, , then there exists such that ; that is, has a fixed point in .

Example 11. Let be the set of real numbers with the usual norm, the unit interval, and be given by , for , , and , for .

In order to show that is a multivalued almost contraction, we have to discuss 8 possible cases.

Case 1. Consider . Then condition (8) reduces to

Since, for , one has and , in order to have the previous inequality satisfied, it suffices to take and arbitrarily.

Case 2. Consider . Then condition (8) reduces to

Since, for , one has and , in order to have the previous inequality satisfied, it suffices to take and arbitrarily.

Case 3. Take . In this case we have

and so condition (8) is satisfied with and arbitrarily.

Case 4. Consider . In this case we have

and so condition (8) is satisfied with and arbitrarily.

Case 5. Take . Then condition (8) reduces to

Since for , one has and , in order to have the previous inequality satisfied, it suffices to take and arbitrarily.

Case 6. Consider . Then condition (8) reduces to

Since, for , one has and , in order to have the previous inequality satisfied, it suffices to take and arbitrarily.

Case 7. Take . Then condition (8) reduces to

Since, for , one has and , in order to have the previous inequality satisfied, it suffices to take and .

Case 8. Consider . Then condition (8) reduces to

Since, for , one has and , in order to have the previous inequality satisfied, it suffices to take and arbitrarily.

Now, by summarizing all cases, we conclude that condition (8) is satisfied with and . Note that the additional condition is also satisfied.

Hence, is a multivalued almost contraction that satisfies all assumptions in Theorem 9, and has two fixed points; that is, .

Note that Corollary 10 cannot be applied to in Example 11. Indeed, if we take and in (32), then one obtains

That is, , which leads to the contradiction .

Acknowledgments

The authors thank the referees for the very useful suggestions and remarks that contributed to the improvement of the paper and especially for drawing attention to [21]. The second author’s research was partially supported by the Grant PN-II-RU-TE-2011-3-239 of the Romanian Ministry of Education and Research. The research of the first and third authors was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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