# Topological structures and the coincidence point of two mappings in cone b-metric spaces

Volume 10, Issue 4, pp 1334--1344 Publication Date: April 20, 2017
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### Authors

Congjun Zhang - School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, Jiangsu, 210023, China.
Sai Li - School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, Jiangsu, 210023, China.
Baoqing Liu - School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, Jiangsu, 210023, China.

### Abstract

Let (X, d,K) be a cone b-metric space over a ordered Banach space ($E,\preceq$) with respect to cone P. In this paper, we study two problems: (1) We introduce a b-metric $\rho_c$ and we prove that the b-metric space induced by b-metric $\rho_c$ has the same topological structures with the cone b-metric space. (2) We prove the existence of the coincidence point of two mappings $T , f : X \rightarrow X$ satisfying a new quasi-contraction of the type $d(Tx, Ty) \preceq \Lambda\{d(fx, fy), d(fx, Ty), d(fx, Tx), d(fy, Ty), d(fy, Tx)\}$, where $\Lambda : E \rightarrow E$ is a linear positive operator and the spectral radius of $K\Lambda$ 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. Cvetković, V. Rakočević, Appl. Math. Comput., 237 (2014), 712–722], [Z. Kadelburg, S. Radenović, J. Nonlinear Sci. Appl., 3 (2010), 193–202].

### Keywords

• Topological structures
• cone b-metric spaces
• quasi-contraction
• points of coincidence
• common fixed points.

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