A new survey: Cone metric spaces

Nonlinear Analysis: Theory, Methods & Applications, 2011

Using an old M. Krein's result and a result concerning symmetric spaces from [S. Radenović, Z. Kadelburg, Quasi-contractions on symmetric and cone symmetric spaces, Banach J. Math. Anal. 5 (1) (2011), 38-50], we show in a very short way that all fixed point results in cone metric spaces obtained recently, in which the assumption that the underlying cone is normal and solid is present, can be reduced to the corresponding results in metric spaces. On the other hand, when we deal with non-normal solid cones, this is not possible. In the recent paper [M.A. Khamsi, Remarks on cone metric spaces and fixed point theorems of contractive mappings, Fixed Point Theory Appl. 2010, 7 pages, Article ID 315398, doi:10.1115/2010/315398] the author claims that most of the cone fixed point results are merely copies of the classical ones and that any extension of known fixed point results to cone metric spaces is redundant; also that underlying Banach space and the associated cone subset are not necessary. In fact, Khamsi's approach includes a small class of results and is very limited since it requires only normal cones, so that all results with non-normal cones (which are proper extensions of the corresponding results for metric spaces) cannot be dealt with by his approach.

Various types of cones in topological vector spaces are discussed. In particular, the usage of (non)-solid and (non)-normal cones in fixed point results is presented. A recent result about normable cones is shown to be wrong. Finally, a Geraghty-type fixed point result in spaces with cones which are either solid or normal is obtained.

2013

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

Applied Mathematics Letters, 2011

In the present work, using Minkowski functionals in topological vector spaces, we establish the equivalence between some fixed point results in metric and in (topological vector space) cone metric spaces. Thus, a lot of results in the cone metric setting can be directly obtained from their metric counterparts. In particular, a common fixed point theorem for f -quasicontractions is obtained. Our approach is even easier than that of Du [Wei-Shih Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010) 2259-2261] where similar conclusions were obtained using scalarization functions.

In this paper we introduce cone D-metric spaces, we prove some fixed point theorems on the D-cone metric spaces.

Scientific Publications of the State University of Novi Pazar Series A: Applied Mathematics, Informatics and mechanics, 2016

We show that most fixed point results obtained so far in cone metric spaces over solid non-normal cones can be easily reduced to the case of solid normal cones and, hence, their proofs can be made much simpler. Also, cone tvs-valued spaces over solid cones are not an essential generalization of cone metric spaces. These results are consequences of the simple fact that each solid cone in a topological vector space is in fact normal under a suitably defined norm. The proof follows by using the technique of Minkowski functional. As an application of these results, we prove an extension of the classical Nemytzki-Edelstein fixed point result to (tvs)-(b)-cone metric spaces over solid cones.

2010

A notion of generalized cone metric space is introduced, and some convergence properties of sequences are proved. Also some fixed point results for mappings satisfying certain contractive conditions are obtained. Our results complement, extend and unify several well known results in the literature. , G(a, b, b) Note that X is nonsymmetric G−cone metric space as G .

Journal of Ultra Scientist of Physical Sciences Section A

The main purpose of this paper is to prove some fixed point theorems and its applications in partial and generalized partial cone metric spaces. Our results are satisfying various contractive conditions on cone spaces. We also prove the uniqueness of such fixed points theorems.

In the present paper, we have proved some convergence properties of a sequence of elements in a partial cone metric space and thereby we have established some fixed point theorems on it.

2011

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.