A fixed point theorem in the space of integrable functions and applications

We give sufficient conditions to ensure when a mapping T:E→E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T:\varvec{E} \rightarrow \varvec{E}$$\end{document} has a unique fixed point, E is a set of measurable functions that is uniformly continuous, closed, and convex. The proof of the existence of the fixed point depends on a certain type of sequential compactness for uniformly integrable functions that is also studied. The fixed point theorem is applied in the study of the uniqueness and existence of some Fredholm and Caputo equations.