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The Scientific World Journal

Volume 2013 (2013), Article ID 481601, 3 pages

http://dx.doi.org/10.1155/2013/481601

## On Multivalued Contractions in Cone Metric Spaces without Normality

^{1}Department of Mathematics, International Islamic University, H-10, Islamabad 44000, Pakistan^{2}Department 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.

#### Abstract

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 [1] 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. [9], Kadelburg and Radenovic [16], Klim and Wardowski [17], Latif and Shaddad [18], Radenovic and Kadelburg [19], Rezapour and Haghi [20], and Wardowski [14, 21] obtained fixed points of set-valued mappings in normal cone metric spaces. On the other hand, it is shown in [11] 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 [14] 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 .

#### References

- L.-G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 332, no. 2, pp. 1468–1476, 2007. View at Publisher · View at Google Scholar - T. Abdeljawad, P. Murthy, and K. Tas, “A Gregus type common fixed point theorem of set-valued mappings in cone metric spaces,”
*Journal of Computational Analysis and Applications*, vol. 13, no. 4, pp. 622–628, 2011. View at Google Scholar - J. Ahmad, M. Arshad, and C. Vetro, “On a theorem of khan in a generalized metric space,”
*International Journal of Analysis*, vol. 2013, Article ID 852727, 6 pages, 2013. View at Publisher · View at Google Scholar - M. Arshad, J. Ahmad, and E. Karapınar, “Some common fixed point results in rectangular metric spaces,”
*Journal of Analysis*, vol. 2013, Article ID 307234, 7 pages, 2013. View at Publisher · View at Google Scholar - A. G. Ahmad, Z. M. Fadail, M. Abbas, Z. Kadelburg, and S. Radenović, “Some fixed and periodic points in abstract metric spaces,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 908423, 15 pages, 2012. View at Publisher · View at Google Scholar - M. Arshad, A. Azam, and P. Vetro, “Some common fixed point results in cone metric spaces,”
*Fixed Point Theory and Applications*, vol. 2009, Article ID 493965, 2009. View at Publisher · View at Google Scholar - S. H. Cho, J. S. Bae, and K. S. Na, “Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings in cone metric
spaces,”
*Fixed Point Theory and Applications*, vol. 2012, article 133, 10 pages, 2012. View at Publisher · View at Google Scholar - S. H. Cho and J. S. Bae, “Fixed point theorems for multi-valued maps in cone metric spaces,”
*Fixed Point Theory and Applications*, vol. 2011, article 87, 2011. View at Google Scholar - Y. J. Choa, S. Hirunworakit, and N. Petrot, “Set-valued fixed-point theorems for generalized contractive mappings without the Hausdorff metric,”
*Applied Mathematics Letters*, vol. 24, no. 11, pp. 1959–1967, 2011. View at Publisher · View at Google Scholar - L. Gajić and V. Rakočević, “Qusai-contractions on a nonnormal cone metric space,”
*Fund Analyzer Application*, vol. 46, pp. 75–79, 2012. View at Google Scholar - S. Janković, Z. Kadelburg, and S. Radenović, “On cone metric spaces: a survey survey,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 74, pp. 2591–2601, 2011. View at Google Scholar - G. Jungck, S. Radenović, S. Radojević, and V. Rakočević, “Common fixed point theorems for weakly compatible pairs on cone metric spaces,”
*Fixed Point Theory and Applications*, vol. 2009, Article ID 643840, 2009. View at Publisher · View at Google Scholar - S. R. Kumar, “Common fixed point theorems for sub compatible and sub sequentially continuous maps in 2 metric spaces,”
*International MatheMatical Forum*, vol. 7, no. 24, pp. 1187–1200, 2012. View at Google Scholar - D. Wardowski, “On set-valued contractions of Nadler type in cone metric spaces,”
*Applied Mathematics Letters*, vol. 24, no. 3, pp. 275–278, 2011. View at Publisher · View at Google Scholar · View at Scopus - Z. Kadelburg, S. Radenović, and V. Rakočević, “A note on the equivalence of some metric and cone metric fixed point results,”
*Applied Mathematics Letters*, vol. 24, pp. 370–374, 2011. View at Google Scholar - Z. Kadelburg and S. Radenović, “Some results on set-valued contractions in abstract metric spaces,”
*Computers and Mathematics with Applications*, vol. 62, no. 1, pp. 342–350, 2011. View at Publisher · View at Google Scholar · View at Scopus - D. Klim and D. Wardowski, “Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 71, no. 11, pp. 5170–5175, 2009. View at Publisher · View at Google Scholar · View at Scopus - A. Latif and F. Y. Shaddad, “Fixed point results for multivalued maps in cone metric spaces,”
*Fixed Point Theory and Applications*, vol. 2010, article 87, Article ID 941371, 2010. View at Publisher · View at Google Scholar · View at Scopus - S. Radenović and Z. Kadelburg, “Some results on fixed points of multifunctions on abstract metric spaces,”
*Mathematical and Computer Modelling*, vol. 53, no. 5-6, pp. 746–754, 2011. View at Publisher · View at Google Scholar · View at Scopus - Sh. Rezapour and R. H. Haghi, “Fixed point of multifunctions on cone metric spaces,”
*Numerical Functional Analysis and Optimization*, vol. 30, no. 7-8, pp. 1–8, 2009. View at Publisher · View at Google Scholar · View at Scopus - D. Wardowski, “Endpoints and fixed points of set-valued contractions in cone metric spaces,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 71, no. 1-2, pp. 512–516, 2009. View at Publisher · View at Google Scholar · View at Scopus