ON THE BANACH PRINCIPLE IN b-METRIC SPACES.

In 1989, Bakhtin proved a version of the Banach contraction principle in the context of b-metric spaces. More precisely, let (X, d; s) be a complete b-metric space with parameter s ⩾ 1 and let T a contractive map on X, that is a self-map T of X satisfying d(Tx, Ty) ⩽ λd(x, y), ∀x, y ∈ X, (Bλ) with some λ ∈ [0, 1). Bakhtin proved that if λ ∈ [0, 1 s), then T has a unique fixed point. In the two last decades many research papers were produced by many authors in the setting of b-metric spaces. In 2021, the author published a note on the Banach contraction principle in b-metric spaces in Divilgaciones Mat'ematicas dealing with some complements to Bakhtin results. The aim of this paper is to provide other complements. In particular, we discuss the remaining case where λ ∈ [1 s, 1). We give an evaluation of the order of convergence for the iterative Picard process and a posteriori error estimate for this process and we estimate the diameter of the T-orbits. We investigate well-posedness of the fixed problem of a map T satisfying Bλ when X is Torbitally complete and we establish two shadowing properties for these maps without requiring completeness of X. [ABSTRACT FROM AUTHOR]