Topological Vector Space-Valued Cone Metric Spaces and Fixed Point Theorems
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Research Article Topological Vector Space-Valued Cone Metric Spaces and Fixed Point Theorems ´ 2 and Vladimir Rakocevi ˇ Zoran Kadelburg,1 Stojan Radenovic, c´ 3 1
Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Serbia Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia 3 Department of Mathematics, Faculty of Sciences and Mathematics, University of Niˇs, Viˇsegradska 33, 18000 Niˇs, Serbia 2
Correspondence should be addressed to Stojan Radenovi´c, [email protected] Received 18 December 2009; Revised 14 July 2010; Accepted 19 July 2010 Academic Editor: Hichem Ben-El-Mechaiekh Copyright q 2010 Zoran Kadelburg 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. We develop the theory of topological vector space valued cone metric spaces with nonnormal cones. We prove three general fixed point results in these spaces and deduce as corollaries several extensions of theorems about fixed points and common fixed points, known from the theory of normed-valued cone metric spaces. Examples are given to distinguish our results from the known ones.
1. Introduction Ordered normed spaces and cones have applications in applied mathematics, for instance, in using Newton’s approximation method 1–4 and in optimization theory 5. K-metric and K-normed spaces were introduced in the mid-20th century 2, see also 3, 4, 6 by using an ordered Banach space instead of the set of real numbers, as the codomain for a metric. Huang and Zhang 7 reintroduced such spaces under the name of cone metric spaces but went further, defining convergent and Cauchy sequences in the terms of interior points of the underlying cone. These and other authors see, e.g., 8–22 proved some fixed point and common fixed point theorems for contractive-type mappings in cone metric spaces and cone uniform spaces. In some of the mentioned papers, results were obtained under additional assumptions about the underlying cone, such as normality or even regularity. In the papers 23, 24, the authors tried to generalize this approach by using cones in topological vector spaces tvs instead of Banach spaces. However, it should be noted that an old result see, e.g., 3 shows
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Fixed Point Theory and Applications
that if the underlying cone of an ordered tvs is solid and normal, then such tvs must be an ordered normed space. So, proper generalizations when passing from norm-valued cone metric spaces of 7 to tvs-valued cone metric spaces can be obtained only in the case of nonnormal cones. In the present paper we develop further the theory of topological vector space valued cone metric spaces with nonnormal cones. We prove three general fixed point results in these spaces and deduce as corollaries several extensions of theorems about fixed points and common fixed points, known from the theory of n
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