A unified theory of cone metric spaces and its applications to the fixed point theory

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. © 2013 Proinov; licensee Springer.