Existence of positive solutions for nonlinear boundary value problems involving fractional boundary conditions

This paper focuses on investigating the existence of positive solutions to boundary value problems (BVPs) of higher-order nonlinear fractional differential equations (FDES) involving fractional boundary conditions. We examine the characteristics of Green's functions. By utilizing the powerful methodologies of Schauder's fixed-point theorem, upper and lower solutions, and cone theory techniques, we obtain significant existence results for nonsingular boundary value problems. For singular problems, we employ cone theory techniques in conjunction with the Leray-Schauder nonlinear alternative to obtain positive solutions. To further clarify and validate the primary findings, several illustrative examples are provided.