A Displacement Functional Approach to Fixed Points of Strictly Contractive and Kannan Mappings

Abstract

A minimizing sequence approach to the Banach fixed point theorem was recently introduced, based on minimizing the displacement functional φ(x) = d(x, Tx). In this paper, we clarify how this method relates to approximating fixed point sequences studied in the literature. We first provide an alternative proof of Edelstein’s fixed point theorem for strictly contractive mappings on compact metric spaces by minimizing φ(x). The argument relies entirely on the minimization principle without the use of Picard iteration. We then obtain an alternative proof of Kannan’s fixed point theorem using the same minimizing sequence technique, showing that the method applies in settings where continuity is not assumed. Finally, we establish a new fixed point theorem for strictly contractive mappings on proper metric spaces under a coercivity condition on the displacement functional. This result identifies attainment of the minimum of A = {φ(x): x∈X} as the fundamental mechanism of the method, rather than global compactness of the underlying space.