Positive Mild Solutions of A Fractional Boundary Value Problem on the Half-Line

Abstract

In this paper, we investigate the existence and uniqueness of a mild solution to a fractional boundary value problem involving a Riemann-Liouville type fractional derivative. Our approach is based on the application of a relatively recent fixed point theorem for F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}$$\end{document}-contractions in complete metric spaces, which allows us to work under more general conditions than classical contraction principles. This framework provides a powerful and flexible tool for dealing with nonlocal problems arising in various applied fields. In addition to establishing existence and uniqueness, we demonstrate that, under certain additional assumptions, the mild solution is positive. This qualitative property is of particular interest in real-world applications where negative solutions may lack physical meaning. Finally, to illustrate the theoretical results, we present a concrete example that satisfies all the hypotheses and confirms the main conclusions.