Non-normal cone metric and cone b-metric spaces and fixed point results

Abstract

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.