On The Existence of \(P\)-Contractions that are not Enriched Contractions

Abstract

We construct two examples clarifying the position of enriched $P$-contractions among related classes of mappings on normed spaces. Our first example is a $P$-contraction operator on a finite subset of a normed linear space that is not an enriched contraction. In the second example we have a continuous operator on the whole of a normed linear space that is a $P$-contraction but not an enriched contraction. These examples answer in negative a question raised in \cite{AltunHancerAtes2024} asking whether every enriched $P$-contraction is an enriched contraction.