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The Scientific World Journal
Volume 2013 (2013), Article ID 481601, 3 pages
On Multivalued Contractions in Cone Metric Spaces without Normality
1Department of Mathematics, International Islamic University, H-10, Islamabad 44000, Pakistan
2Department of Mathematics, COMSATS Institute of Information Technology, Chak Shahzad, Islamabad 44000, Pakistan
Received 4 April 2013; Accepted 15 May 2013
Academic Editors: A. Agouzal, A. Ibeas, F. Khani, and F. Kittaneh
Copyright © 2013 Muhammad Arshad and Jamshaid Ahmad. 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.
Wardowski (2011) in this paper for a normal cone metric space and for the family of subsets of established a new cone metric and obtained fixed point of set-valued contraction of Nadler type. Further, it is noticed in the work of Jankovic et al., 2011 that the ?fixed-point problem in the setting of cone metric spaces is appropriate only in the case when the underlying cone is nonnormal. In the present paper we improve Wardowski's result by proving the same without the assumption of normality on cones.
1. Introduction and Preliminaries
Huang and Zhang  generalized the notion of metric space by replacing the set of real numbers by ordered Banach space and defined cone metric space and extended Banach type fixed-point theorems for contractive type mappings. Subsequently, some other authors (e.g., see [2–15] and references therein) studied properties of cone metric spaces and fixed points results of mappings satisfying contractive type condition in cone metric spaces. Recently, Choa et al. , Kadelburg and Radenovic , Klim and Wardowski , Latif and Shaddad , Radenovic and Kadelburg , Rezapour and Haghi , and Wardowski [14, 21] obtained fixed points of set-valued mappings in normal cone metric spaces. On the other hand, it is shown in  that most of the fixed points results of mappings satisfying contractive type condition in cone metric spaces with a normal cone can be reduced to the corresponding results from metric space theory. The fixed-point problem in the setting of cone metric spaces is appropriate only in the case when the underlying cone is nonnormal, because the results concerning fixed points and other results in the case of cone metric spaces with nonnormal solid cones cannot be proved by reducing to metric spaces. In this paper, we prove the result of Wardowski  without the assumption of normality of cones. We need the following definitions and results, consistent with [1, 11, 14].
Let be a Banach Space and a subset of . Then, is called a cone whenever(i)is closed, nonempty, and ,(ii) for all and nonnegative real numbers ,(iii).
Each cone induces a partial ordering on by if and only if . So will stand for and , while will stand for , where denotes the interior of . The cone is called normal if there is a number such that, for all , The least positive number satisfying (1) is called the normal constant of .
Definition 1. Let be a nonempty set. Suppose the mapping satisfies for all and if and only if , for all , for all . Then, is called a cone metric on , and is called a cone metric space.
Let be a cone metric space, and a sequence in . Then, converges to whenever for every with there is a natural number such that for all . We denote this by or . is a Cauchy sequence whenever for every with there is a natural number such that for all . is called a complete cone metric space if every Cauchy sequence in is convergent.
A set is called closed if, for any sequence convergent to , we have . Denote by the collection of all nonempty subsets of and by a collection of all nonempty closed subsets of . Denote by a set of all fixed points of a mapping . In the present paper, we assume that is a real Banach space, is a cone in with nonempty interior (such cones are called solid), and is a partial ordering with respect to . In accordance with [14, Definition 3.1 and Lemma 3.1], we minutely modify the idea of-cone metric to make it more comparable with a standard metric.
Definition 2. Let be a cone metric space and be a collection of nonempty subsets of . A map is called an -cone metric on induced by if the following conditions hold: for all andif and only if ,for all, for all, If with, then for each there exists such that.
Examples can be seen in [14, examples 3.1 and 3.2].
2. Main Result
Theorem 3. Let be a complete cone metric space. Let be a nonempty collection of nonempty closed subsets of , and let be an -cone metric induced by . If for a map there exists such that for all then .
Proof. Let be an arbitrary but fixed element of and . If , then , and if , using the fact that we may choose such that and Similarly, in case , we may choose such that and We can continue this process to find a sequence of points of such that Now for any , Let be given. Choose a symmetric neighborhood of such that . Also, choose a natural number such that , for all . Then, , for all . Thus, for all . Therefore, is a Cauchy sequence. Since is complete, there exists such that . Since for each , we have such that Now, choose a natural number such that Then for all , It follows that , and it implies that .
Example 4. Suppose with the norm , , , and . Then,,??, and . For all , since . Therefore, is non-normal. Define as follows: Let be a family of subsets of of the form , and define as follows: It is easy to observe that satisfies ()–() of Definition 2. Define as Note that satisfies the conditions of Theorem 3 with and .
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