The role of equivalent metrics in fixed point theory

Abstract

Metrical fixed point theory is accomplished by a wide class of terms: \roster \item"$\bullet$" operators (bounded, Lipschitz, contraction, contractive, nonexpansive, noncontractive, expansive, dilatation, isometry, similarity, Picard, weakly Picard, Bessaga, Janos, Caristi, pseudocontractive, accretive, etc.), \item"$\bullet$" convexity (strict, uniform, hyper, etc.), \item"$\bullet$" deffect of some properties (measure of noncompactness, measure of nonconvexity, minimal displacement, etc.), \item"$\bullet$" data dependence (stability, Ulam stability, well-posedness, shadowing property, etc.), \item"$\bullet$" attractor, \item"$\bullet$" basin of attraction$\ldots$ \endroster The purpose of this paper is to study several properties of these concepts with respect to equivalent metrics.