Lyapunov-Type and Hartman–Wintner-Type Inequalities for a Class of Composite Fractional Integral Operators

In this paper, we establish Lyapunov-type and Hartman–Wintner-type integral inequalities for boundary value problems involving a class of composite fractional integral operators. By employing an explicit Green function representation and sharp uniform bounds for the associated kernel, we derive necessary conditions for the existence of nontrivial solutions. A distinctive feature of the obtained results is the higher-order scaling behavior induced by the interaction of left- and right-sided components of the operator, which cannot be observed in classical one-sided fractional models. As applications, we obtain quantitative nonexistence criteria and explicit lower bounds for the principal eigenvalue of the corresponding eigenvalue problem. These estimates extend classical results to a higher-order fractional framework and highlight the influence of the domain geometry on the stability of the solutions.