Banach Contraction Principle for Cyclical Mappings on Partial Metric Spaces

TL;DR

This paper extends the Banach contraction principle to cyclical mappings in 0-complete partial metric spaces, providing new fixed point results and counterexamples, and generalizing existing theorems.

Contribution

It introduces an extension of the Banach contraction principle to cyclical mappings in 0-complete partial metric spaces, which was not previously established.

Findings

Extended Banach contraction principle to cyclical mappings in 0-complete partial metric spaces.

Provided examples illustrating the effectiveness of the new results.

Generalized fixed point theorems for cyclic decompositions with 0-compact sets.

Abstract

In this paper, we prove that the Banach contraction principle proved by S. G. Matthews in 1994 on 0--complete partial metric spaces can be extended to cyclical mappings. However, the generalized contraction principle proved by D. Ili\'{c}, V. Pavlovi\'{c} and V. Rako\u{c}evi\'{c} in "Some new extensions of Banach's contraction principle to partial metric spaces, Appl. Math. Lett. 24 (2011), 1326--1330" on complete partial metric spaces can not be extended to cyclical mappings. Some examples are given to illustrate the effectiveness of our results. Moreover, we generalize some of the results obtained by W. A. Kirk, P. S. Srinivasan and P. Veeramani in "Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory 4 (1) (2003),79--89". Finally, an Edelstein's type theorem is also extended in case one of the sets in the cyclic decomposition is 0-compact.