A $\phi$ - contraction Principle in Partial Metric Spaces with Self-distance Terms

TL;DR

This paper introduces a generalized contraction principle in complete partial metric spaces that incorporates self-distance terms, extending previous fixed point theorems and providing an example demonstrating its broader applicability.

Contribution

It presents a new contraction principle in partial metric spaces with self-distance terms, generalizing prior results and offering a more inclusive fixed point theorem.

Findings

The new contraction principle generalizes existing theorems.

An example illustrates the broader applicability of the result.

The result surpasses previous fixed point theorems in partial metric spaces.

Abstract

We prove a generalized contraction principle with control function in complete partial metric spaces. The contractive type condition used allows the appearance of self distance terms. The obtained result generalizes some previously obtained results such as the very recent " D. Ili\'{c}, V. Pavlovi\'{c} and V. Rako\u{c}evi\'{c}, Some new extensions of Banach's contraction principle to partial metric spaces, Appl. Math. Lett. 24 (2011), 1326--1330". An example is given to illustrate the generalization and its properness. Our presented example does not verify the contractive type conditions of the main results proved recently by S. Romaguera in " Fixed point theorems for generalized contractions on partial metric spaces, Topology Appl. 159 (2012), 194-199" and by I. Altun, F. Sola and H. Simsek in "Generalized contractions on partial metric spaces, Topology and Its Applications 157 (18) (2010), 2778--2785". Therefore, our results have an advantage over the previously obtained.