Positive Solutions for Fractional Boundary Value Problems with Fractional Conditions Using Induction and Convolution of Lower-Order Problemห

This paper examines the conditions for the existence and nonexistence of positive solutions to a class of nonlinear Riemann-Liouville fractional boundary value problems of order α+2n, where α∈(m−1,m] with m≥3 and m,n∈N. The problem’s nonlinearity is continuous and depends on a positive parameter. We derive constraints on this parameter that dictate whether positive solutions can be found. Our approach involves constructing a Green’s function by combining the Green’s functions of a lower-order fractional boundary value problem and a right-focal boundary value problem. Leveraging the properties of this Green’s function, we apply Krasnosel’skii’s Fixed Point Theorem to establish our results. Several examples are presented to illustrate the existence and nonexistence regions.