On relaxations of contraction constants and Caristi’s theorem in b-metric spaces

Abstract

In this paper, the following facts are stated in the setting of b-metric spaces. (1) The contraction constant in the Banach contraction principle fully extends to [0, 1), but the contraction constants in Reich’s fixed point theorem and many other fixed point theorems do not fully extend to [0, 1), which answers the early stated question on transforming fixed point theorems in metric spaces to fixed point theorems in b-metric spaces. (2) Caristi’s theorem does not fully extend to b-metric spaces, which is a negative answer to a recent Kirk–Shahzad’s question (Remark 12.6) [Fixed Point Theory in Distance Spaces. Springer, 2014].