The role of equivalent metrics in fixed point theory

Abstract

Metrical fixed point theory is accomplished by a wide class of terms:

$\bullet$ operators (bounded, Lipschitz, contraction, contractive, nonexpansive, noncontractive, expansive, dilatation, isometry, similarity, Picard, weakly Picard, Bessaga, Janos, Caristi, pseudocontractive, accretive, etc.),

$\bullet$ convexity (strict, uniform, hyper, etc.),

$\bullet$ deffect of some properties (measure of noncompactness, measure of nonconvexity, minimal displacement, etc.),

$\bullet$ data dependence (stability, Ulam stability, well-posedness, shadowing property, etc.),

$\bullet$] attractor,

$\bullet$ basin of attraction$\ldots$

The purpose of this paper is to study several properties of these concepts with respect to equivalent metrics.