» Generalized metric spaces having the fixed point property with respect to contractions

Abstract.

Let $X$ be a nonempty set and $d:X\times X\rightarrow \mathbb{R}_{+}$ a mapping (i.e., a {\em distance}) and $f: X\rightarrow X$ a contraction. In this paper we study the following problem:

Under which conditions on $(X,d)$ do we have that

$F_f:=\{x\in X| f(x)=x\}=\{x^*\} \, ?$

Similar problems for $\mathbb{R}^m_{+}$ -distances, $K$ -distances, $s(\mathbb{R}_{+})$ -distances, and for extending distances are investigated.
Applications to contractions on generalized (dislocated, quasi-, partial, ultra-) metric spaces, are also given.
In order to study these problems we introduce the notion of {\em suitable distance space for contractions}. The paper [Berinde V., P\u acurar M., Rus I.A. \textit{Some classes of distance spaces as generalized metric spaces: terminology, mappings, fixed points and applications in Theoretical Informatics}, Creat. Math. Inform. 34 (2025), no. 2, 155–174] is an heuristic introduction to the present one.

Our results open new perspectives in the fixed point theory and theoretical computer science and have important applicability in denotational semantics, as semantic operators in most programming language paradigms satisfy the requirements of fixed point principles for contractions on generalized metric spaces.