Fixed Point in Topological Vector Space-Valued Cone Metric Spaces
© Akbar Azam et al. 2010
Received: 16 December 2009
Accepted: 2 June 2010
Published: 14 June 2010
We obtain common fixed points of a pair of mappings satisfying a generalized contractive type condition in TVS-valued cone metric spaces. Our results generalize some well-known recent results in the literature.
1. Introduction and Preliminaries
Many authors [1–16] studied fixed points results of mappings satisfying contractive type condition in Banach space-valued cone metric spaces. In a recent paper  the authors obtained common fixed points of a pair of mapping satisfying generalized contractive type conditions without the assumption of normality in a class of topological vector space-valued cone metric spaces which is bigger than that of studied in [1–16]. In this paper we continue to study fixed point results in topological vector space valued cone metric spaces.
If is a real Banach space then is called (Banach space-valued) cone metric space .
2. Main Results
Huang and Zhang  proved Theorem 2.1 by using the following additional assumptions.
(d)One of the following is satisfied:
(i) with [5, Theorem 1],
(ii) with [5, Theorem 3],
(iii) with [5, Theorem 4].
Note that the assumptions (d) of results [9, Theorems 1, 3, and 4] and [15, Theorems 2.3, 2.6, 2.7, and 2.8] are not satisfied to find a fixed point of In order to apply inequality (2.1) consider mapping for each then for , and satisfy all the conditions of Corollary 2.4 and we obtain .
The authors are thankful to referee for precise remarks to improve the presentation of the paper.
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