Existence and Nonexistence of Positive Solutions for Fractional Boundary Value Problems with Lidstone-Inspired Fractional Conditions

This paper investigates the existence and nonexistence of positive solutions for a class of nonlinear Riemann-Liouville fractional boundary value problems of order $\alpha + 2n$, where $\alpha \in (m-1, m]$ with $m \geq 3$ and $m, n \in \mathbb{N}$. The conjugate fractional boundary conditions are inspired by Lidstone conditions. The nonlinearity, $(-1)^n\lambda g(t)f(u(t))$, is assumed to be continuous and depends on a positive parameter. We identify parameter constraints that determine the existence or nonexistence of positive solutions. Our method involves constructing a Green's function by convolving the Green's functions of a lower-order fractional boundary value problem and a conjugate boundary value problem and using properties of this Green's function to apply the Krasnosel'skii Fixed Point Theorem. Illustrative examples are provided to demonstrate existence and nonexistence intervals.