I - convergence on cone metric spaces

The Journal of Analysis, 2019

Here we have introduced the idea of rough convergence of sequences in a cone metric space. Also it has been investigated how far several basic properties of rough convergence as valid in a normed linear space are affected in a cone metric space. 24: (2003), 285-301.

Thai Journal of Mathematics, 2018

Recently, several articles have been written on the cone metric spaces.Despite the fact that any cone metric space is equivalent to a usual metric space, weaim in this paper to deal with some of the published articles on cone metric spaces byrepairing some gaps, providing new proofs and extending their results to topologicalvector spaces. Several authors have worked with a class of special cones whichknown as strongly minhedral cones where the strongly minihedrality condition (that is, each nonempty bounded above subset has a least upper bound) is veryrestrictive. Another goal of this article is to eliminate or mitigate this condition.Furthermore, we present some examples in order to show that the imagination ofmany authors that the behavior of the ordering induced by a strongly minihedralcone is just as the behavior of the usual ordering on the real line, that has causedan error in their proofs, is not correct. We establish a relationship between strongminihedrality and total order...

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.

2018

The purpose of this new survey paper is, among other things, to collect in one place most of the articles on cone (abstract, K-metric) spaces, published after 2007. This list can be useful to young researchers trying to work in this part of functional and nonlinear analysis. On the other hand, the existing review papers on cone metric spaces are updated. The main contribution is the observation that it is usually redundant to treat the case when the underlying cone is solid and non-normal. Namely, using simple properties of cones and Minkowski functionals, it is shown that the problems can be usually reduced to the case when the cone is normal, even with the respective norm being monotone. Thus, we offer a synthesis of the respective fixed point problems arriving at the conclusion that they can be reduced to their standard metric counterparts. However, this does not mean that the whole theory of cone metric spaces is redundant, since some of the problems remain which cannot be treated in this way, which is also shown in the present article.

Topology and its Applications, 2011

We have shown in this paper that a (complete) cone metric space (X, E, P , d) is indeed (completely) metrizable for a suitable metric D. Moreover, given any finite number of contractions f 1 ,. .. , f n on the cone metric space (X, E, P , d), D can be defined in such a way that these functions become also contractions on (X, D).