Existence of Multiple Positive Solutions for Nonlinear Fractional Boundary Value Problems on the Half-Line

In this paper, we deal with the following nonlinear fractional differential problem in the half-line R+=(0,+∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^{+}=(0,+ \infty)}$$\end{document}Dαu(x)+f(x,u(x),Dpu(x))=0,x∈R+,u(0)=u′0=⋯=um-2(0)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{\begin{array}{l}D^{\alpha }u(x)+f(x,u(x),D^{p}u(x))=0,\quad x \in \mathbb{R}^{+},\\ u(0)=u^{\prime } \left( 0\right) = \cdots =u^{\left( m-2\right) }(0)=0,\end{array}\right.$$\end{document}where m∈N,m≥2,m-1<α≤m,0<p≤α-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m\in \mathbb{N}, m \geq 2, m-1 < \alpha \leq m, 0 < p \leq \alpha -1}$$\end{document}, the differential operator is taken in the Riemann–Liouville sense and f is a Borel measurable function in R+×R+×R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^{+} \times \mathbb{R}^{+} \times \mathbb{R} ^{+}}$$\end{document} satisfying certain conditions. More precisely, we show the existence of multiple unbounded positive solutions, by means of Schäuder fixed point theorem. Some examples illustrating our main result are also given.