On a Riesz Fractional Boundary Value Problem in a Weighted Lebesgue Space

In this paper, we derive a necessary condition for the existence of solutions to a boundary value problem involving the Riesz fractional derivative, a two-sided fractional operator. The problem is transformed into a singular integral equation within a weighted Lebesgue space. By applying the Kolmogorov compactness criterion and using the boundedness properties of the Hilbert transform, we establish the compactness and boundedness conditions necessary for the operator to satisfy the requirements of Krasnoselskii’s fixed point theorem. This allows us to prove the existence of solutions in the Lebesgue space $$L^{p}$$ . Additionally, we derive Lyapunov-type inequalities in a weighted Lebesgue space.