Existence of positive solutions to a semipositone singular h-Riemann-Liouville fractional boundary value problem

In this talk we analyze the existence of positive solutions for an h-Riemann–Liouville fractional differential equation (E) that involves a positive parameter and a singular nonlinearity that changes sign, subject to nonlocal boundary conditions (BC) incorporating Riemann–Stieltjes integrals and h-Riemann-Liouville fractional derivatives of various orders. Because the nonlinearity may take negative values, the problem is referred to as a semipositone fractional boundary value problem. The h-Riemann–Liouville fractional derivative extends several well-known fractional derivatives, including the classical Riemann–Liouville derivative when h(t)=t, the Hadamard derivative when h(t)=ln t, as well as other related fractional operators. We establish intervals for the parameter for which problem (E), (BC) admits at least one positive solution. We begin by deriving the Green function corresponding to the problem and examining several of its key properties. Afterwards, through a suitable change of variables, we reformulate the original problem into an equivalent one. An operator is then constructed in a suitable Banach space, and its fixed points correspond precisely to the solutions of the equivalent problem. Our main results are obtained by applying the Guo–Krasnosel’skii fixed point theorem of cone expansion and compression of norm type. In the end, we present examples that highlight the usefulness of the established results.