On iterated function systems and algebraic properties of Lipschitz maps in partial metric spaces

TL;DR

This paper explores algebraic, analytic, and topological properties of partial iterated function systems in partial metric spaces, proving key theorems and establishing foundational results on fixed points and semigroup structures.

Contribution

It introduces new results on partial IFSs, including the Collage theorem, fixed point continuity, and semigroup structures of Lipschitz maps in partial metric spaces.

Findings

Proves the Collage theorem for partial IFSs.

Establishes the completeness of the space of contractions.

Shows the semigroup structure of Lipschitz transformations.

Abstract

This paper discusses, certain algebraic, analytic, and topological results on partial iterated function systems($IFS_p$'s). Also, the article proves the Collage theorem for partial iterated function systems. Further, it provides a method to address the points in the attractor of a partial iterated function system and obtain results related to the address of points in the attractor. The completeness of the partial metric space of contractions with a fixed contractivity factor is proved, under suitable conditions. Also, it demonstrates the continuity of the map that associates each contraction in a complete partial metric space to its corresponding unique fixed point. Further, it defines the $IFS_p$ semigroup and shows that under function composition, the set of Lipschitz transformations and the set of contractions are semigroups.