### 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.

\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.

#### Keywords

Fixed point theory; topologically equivalent metric; strongly equivalent metric; Picard operator; weakly Picard operator; contraction; generalized contraction; well-posedness; shadowing property; stability; Ulam stability; invariant subset

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