Fixed Point Theorems on Spaces Endowed with Vector-Valued Metrics
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RESEARCH
Open Access
A unified theory of cone metric spaces and its applications to the fixed point theory Petko D Proinov* *
Correspondence: [email protected] Faculty of Mathematics and Informatics, Plovdiv University, Plovdiv, 4000, Bulgaria
Abstract In this paper, we develop a unified theory for cone metric spaces over a solid vector space. As an application of the new theory, we present full statements of the iterated contraction principle and the Banach contraction principle in cone metric spaces over a solid vector space. We propose a new approach to such cone metric spaces. We introduce a new notion of strict vector ordering, which is quite natural and it is easy to use in the cone metric theory and its applications to the fixed point theory. This notion plays the main role in the new theory. Among the other results in this paper, the following is perhaps of most interest. Every ordered vector space with convergence can be equipped with a strict vector ordering if and only if it is a solid vector space. Moreover, if the positive cone of an ordered vector space with convergence is solid, then there exists only one strict vector ordering on this space. Also, in this paper we present some useful properties of cone metric spaces, which allow us to establish convergence results for Picard iteration with a priori and a posteriori error estimates. MSC: 54H25; 47H10; 46A19; 65J15; 06F30 Keywords: cone metric space; solid vector space; fixed points; Picard iteration; iterated contraction principle; Banach contraction principle
1 Introduction In , the famous French mathematician Maurice Fréchet [, ] introduced the concept of metric spaces. In , his PhD student, the Serbian mathematician, Ðuro Kurepa [], introduced more abstract metric spaces, in which the metric takes values in an ordered vector space. In the literature, the metric spaces with vector valued metric are known under various names: pseudometric spaces [, ], K -metric spaces [–], generalized metric spaces [, ], vector-valued metric spaces [], cone-valued metric spaces [, ], cone metric spaces [, ]. It is well known that cone metric spaces and cone normed spaces have deep applications in the numerical analysis and the fixed point theory. Some applications of cone metric spaces can be seen in Collatz [], Zabrejko [], Rus, Petruşel, Petruşel [] and in references therein. Schröder [, ] was the first who pointed out the important role of cone metric spaces in the numerical analysis. The famous Russian mathematician Kantorovich [] was the first who showed the importance of cone normed spaces for the numerical analysis. Starting from , many authors have studied cone metric spaces over solid Banach spaces and fixed point theorems in such spaces (Huang and Zhang [], Rezapour and © 2013 Proinov; 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
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