Fundamental questions and new counterexamples for b-metric spaces and Fatou Property

Abstract

In this paper, we give new examples to show that the continuity actually strictly stronger than the Fatou property in b-metric spaces. We establish a new fixed point theorem for new essential and fundamental sufficient conditions such that a Ciric type contraction with contraction constant λ ∈ [1/s, 1) in a complete b-metric space with s > 1 have a unique fixed point. Many new examples illustrating our results are also given. Our new results extend and improve many recent results and they are completely original and quite different from the well known results on the topic in the literature.