Compactness result and its applications in integral equations

TL;DR

This paper proves a version of the Arzelà-Ascoli theorem for σ-locally compact Hausdorff spaces and applies it to establish the compactness of certain integral operators, leading to new fixed point theorems.

Contribution

It introduces a generalized Arzelà-Ascoli theorem for σ-locally compact spaces and applies it to derive fixed point theorems for integral operators.

Findings

Proved a version of Arzelà-Ascoli theorem for σ-locally compact spaces

Established compactness of Fredholm, Hammerstein, and Urysohn operators

Derived fixed point theorems for Hammerstein and Urysohn operators

Abstract

A version of Arzel\`a-Ascoli theorem for $X$ being $\sigma$-locally compact Hausdorff space is proved. The result is used in proving compactness of Fredholm, Hammerstein and Urysohn operators. Two fixed point theorems, for Hammerstein and Urysohn operator, are derived on the basis of Schauder theorem.