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

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.

References

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