Multivalued fixed point theorems in tvs-cone metric spaces
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RESEARCH
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Multivalued fixed point theorems in tvs-cone metric spaces Akbar Azam* and Nayyar Mehmood *
Correspondence: [email protected] Department of Mathematics, COMSATS Institute of Information Technology, Chak Shahzad, Islamabad 44000, Pakistan
Abstract In this paper we extend the Kannan, Chatterjea and Zamfirescu theorems for multivalued mappings in a tvs-cone metric space without the assumption of normality on cones and generalize many results in literature. MSC: 47H10; 54H25 Keywords: tvs-cone metric space; non-normal cones; multivalued contraction; fixed points
1 Introduction The notion of cone metric space was introduced by Huang and Zhang in []. They replaced the set of real numbers by an ordered Banach space and defined a cone metric space. They extended Banach fixed point theorems for contractive type mappings. Many authors [– ] studied the properties of cone metric spaces and generalized important fixed point results of complete metric spaces. The concept of cone metric space in the sense of HuangZhang was characterized by Al-Rawashdeh et al. in []. Indeed, (X, d) is a cone metric space if and only if (X, dE ) is an E-metric space, where E is a normed ordered space, with int(E+ ) = φ ([], Theorem .). Recently Beg et al. [] introduced and studied topological vector space-valued cone metric spaces (tvs-cone metric spaces), which generalized the cone metric spaces []. Let (X, d) be a metric space. A mapping T : X → X is called a contraction if there exists λ ∈ [, ) such that d(Tx, Ty) ≤ λd(x, y) for all x, y ∈ X. A mapping T is called Kannan if there exists α ∈ [, ) such that d(Tx, Ty) ≤ α d(x, Tx) + d(y, Ty) . The main difference between contraction and Kannan mappings is that contractions are always continuous, whereas Kannan mappings are not necessarily continuous. Another type of contractive condition, due to Chatterjea [], is based on an assumption analogous to Kannan mappings as follows: there exists α ∈ [, ) such that d(Tx, Ty) ≤ α d(x, Ty) + d(y, Tx) . © 2013 Azam and Mehmood; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Azam and Mehmood Fixed Point Theory and Applications 2013, 2013:184 http://www.fixedpointtheoryandapplications.com/content/2013/1/184
It is well known that the Banach contractions, Kannan mappings and Chatterjea mappings are independent in general. Zamfirescu [] proved a remarkable fixed point theorem by combining the results of Banach, Kannan and Chatterjea. Afterwards, some authors investigated these results in many directions [–]. In the papers [–], the authors studied fixed point theorems for multivalued mappings in cone metric spaces. Seong and Jong [] invented the generalized Hausdorff distance in a cone metric space and proved multivalued results in cone metric spaces. Shatanawi e
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