Topological structures and the coincidence point of two mappings in cone bmetric spaces

Let (X, d,K) be a cone b-metric space over a ordered Banach space (E,ς) with respect to cone P. In this paper, we study two problems: (1) We introduce a b-metric pc and we prove that the b-metric space induced by b-metric pc has the same topological structures with the cone b-metric space. (2) We prove the existence of the coincidence point of two mappings T , f : X → X satisfying a new quasi-contraction of the type d(Tx, Ty) fd(fx, fy), d(fx, Ty), d(fx, Tx), d(fy, Ty), d(fy, Tx)g, where Λ : E → E is a linear positive operator and the spectral radius of KΛ is less than 1. Our results are new and extend the recent results of [N. Hussain, M. H. Shah, Comput. Math. Appl., 62 (2011), 1677–1684], [M. Cvetkovi´c, V. Rakoˇcevi´c, Appl. Math. Comput., 237 (2014), 712–722], [Z. Kadelburg, S. Radenovi´c, J. Nonlinear Sci. Appl., 3 (2010), 193–202]. c 2017 All rights reserved.