On the extended version of Krasnosel'skii's fixed point theorem for Kannan type equicontraction mappings

TL;DR

This paper extends Krasnosel'skii's fixed point theorem to Kannan type equicontraction mappings using Sadovskii's theorem and measure of noncompactness, with applications to initial value problems.

Contribution

It introduces a new sufficient condition for solutions involving Kannan type mappings and measure of noncompactness, extending classical fixed point results.

Findings

Established existence of solutions under new conditions

Applied results to initial value problems

Utilized Sadovskii's theorem and measure of noncompactness

Abstract

A sufficient condition is established for the existence of a solution to the equation $\mathcal{T}(u,\mathcal{C}(u))=u$, by considering a class of Kannan type equicontraction mappings $\mathcal{T}:\mathcal{A}\times \overline{\mathcal{C}(\mathcal{A})}\to \Xi$, where $\mathcal{A}$ is a convex, closed and bounded subset of a Banach space $\Xi$ and $\mathcal{C}$ is a compact mapping. To fulfil the desired purpose, we engage the Sadovskii's theorem, involving the measure of noncompactness. The relevance of the acquired results has been illustrated by considering a certain class of initial value problems.