Positive solutions to a system of h-Riemann–Liouville fractional differential equations with coupled boundary conditions

Fractional-order differential and integral operators, as well as fractional differential equations, play a significant role in modeling real-world phenomena across a wide range of scientific and engineering fields. Compared to classical integer-order differential equations, fractional differential equations provide a more effective framework for capturing hereditary and memory effects inherent in many materials and processes. In this talk, we investigate the existence of positive solutions for a system of two h-Riemann–Liouville fractional differential equations (S) in the unknown functions u and v, involving singular sign-changing nonlinearities and two positive parameters. This system is supplemented with general coupled boundary conditions (BCs) that incorporate various h-Riemann–Liouville fractional derivatives together with Riemann–Stieltjes integrals. The h-Riemann–Liouville fractional derivative extends both the classical Riemann–Liouville derivative (when h(t)=t) and the Hadamard derivative (when h(t)=ln t). We first derive the Green functions associated with problem (S)–(BC) and establish several upper and lower bounds for them. Then we make a change to the unknown functions (u,v) and (x,y), and the new problem (P) is written equivalently as a system of two integral equations (I). We then construct a suitable Banach space X and define an operator A in this space corresponding to system (I), noting that (x,y) is a solution of (I) (or problem (P)) if and only if (x,y) is a fixed point of operator A. Under appropriate assumptions of the nonlinearities of the system (S), we identify intervals of the parameters for which the operator A admits at least one fixed point or at least two fixed points, which correspond to positive solutions of the problem (P). The proof of our main results rely on the Leray–Schauder nonlinear alternative and the Guo–Krasnosel’skii fixed point theorem. Finally, we provide two examples that illustrate the applicability of the obtained results.