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

The aim of this paper is to generalize a famous result for Banach-type contractive mapping from ρ(k) ∈ [0, 1/s) to ρ(k) ∈ [0, 1) in cone b-metric space over Banach algebra with coefficient s ≥ 1, where ρ(k) is the spectral radius of the generalized Lipschitz constant k. Moreover, some similar generalizations for the contractive constant k from k ∈ [0, 1/s) to k ∈ [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.