A sharp generalization on cone b-metric space over Banach algebra


Authors

Huaping Huang - School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, China. Stojan Radenovic - Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120, Beograd, Serbia. Guantie Deng - School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, China.


Abstract

The aim of this paper is to generalize a famous result for Banach-type contractive mapping from \(\rho(k)\in[0,\frac{1}{s})\) to \(\rho(k)\in[0,1)\) in cone b-metric space over Banach algebra with coefficient \(s\geq 1\), where \(\rho(k)\) is the spectral radius of the generalized Lipschitz constant \(k\). Moreover, some similar generalizations for the contractive constant \(k\) from \(k\in[0,\frac{1}{s})\) to \(k \in [0, 1)\) in cone b-metric space and in b-metric space are also obtained. In addition, two examples are given to illustrate that our generalizations are in fact real generalizations.


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