Existence and Uniqueness of Positive Solutions for a Class of Nonlinear Fractional Differential Equations with Singular Boundary Value Conditions

Abstract

This paper focuses on a singular boundary value (SBV) problem of nonlinear fractional differential (NFD) equation defined as follows: $D_{0^{+}}^{\beta }\upsilon \left(\tau \right)+f\left(\tau ,\upsilon \left(\tau \right)\right)=0,\tau \in \left(\mathrm{0,1}\right)$ , $\upsilon \left(0\right)={\upsilon }^{\prime }\left(0\right)={\upsilon }^{″}\left(0\right)={\upsilon }^{″}\left(1\right)=0$ , where $3<\beta \le 4,D_{0^{+}}^{\beta }$ is the standard Riemann–Liouville fractional (RLF) derivative. The nonlinear function $f\left(\tau ,\upsilon \left(\tau \right)\right)$ might be singular on the spatial and temporal variables. This paper proves that a positive solution to the SBV problem exists and is unique, taking advantage of Green’s function through a fixed-point (FP) theory on cones and mixed monotone operators.