θ*-Weak Contractions and Discontinuity at the Fixed Point with Applications to Matrix and Integral Equations

Atiya Perveen,Salvatore Sessa,Waleed M Alfaqih,Mohammad Imdad

doi:10.3390/axioms10030209

Atiya Perveen, Salvatore Sessa + Show 2 more

Open Access

https://doi.org/10.3390/axioms10030209

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Abstract

In this paper, the notion of θ*-weak contraction is introduced, which is utilized to prove some fixed point results. These results are helpful to give a positive response to certain open question raised by Kannan and Rhoades on the existence of contractive definition which does not force the mapping to be continuous at the fixed point. Some illustrative examples are also given to support our results. As applications of our result, we investigate the existence and uniqueness of a solution of non-linear matrix equations and integral equations of Volterra type as well.

Highlights

Introduction and PreliminariesIn order to study the existence of fixed point for discontinuous mappings, Kannan [1]introduced a weaker contraction condition and proved the following theorem: Every self-mapping S defined on a complete metric space ( M, d) satisfying the condition h 1d(Sz, Sw) ≤ β[d(z, Sz) + d(w, Sw)], where β ∈ 0, (1)∀z, w ∈ M, has a unique fixed point
Introduced a weaker contraction condition and proved the following theorem: Every self-mapping S defined on a complete metric space ( M, d) satisfying the condition h 1
In his paper, [3], Rhoades presented 250 contractive definitions (including (1)) and compared them. He found that though most of them do not force the mapping to be continuous in the entire domain but under these definitions, all the mapping are continuous at the fixed point

Summary

Introduction and Preliminaries

In order to study the existence of fixed point for discontinuous mappings, Kannan [1]. Introduced a weaker contraction condition and proved the following theorem: Every self-mapping S defined on a complete metric space ( M, d) satisfying the condition h 1. In his paper, [3], Rhoades presented 250 contractive definitions (including (1)) and compared them He found that though most of them do not force the mapping to be continuous in the entire domain but under these definitions, all the mapping are continuous at the fixed point. Later on, this contraction condition was modified by many authors. This contraction condition was modified by many authors In this direction, Ahmad et al [9] proved the same result by using class of functions Θ1,2,4. Let S be a self-mapping defined on a metric space ( M, d) satisfying condition Θ2. The set of all fixed points of a self-mapping S is denoted by Fix (S)

Main Results

Application to Nonlinear Matrix Equations

Application to Integral Equations