An investigation of a new Lyapunov-type inequality for Katugampola–Hilfer fractional BVP with nonlocal and integral boundary conditions

Abstract

Abstract In this manuscript, we focus our attention on investigating new Lyapunov-type inequalities (LTIs) for two classes of boundary value problems (BVPs) in the framework of Katugampola–Hilfer fractional derivatives, supplemented by nonlocal, integral, and mixed boundary conditions. The equivalent integral equations of the proposed Katugampola–Hilfer fractional BVPs are established in the context of Green functions. Also, the properties of these Green functions are proved. The LTIs are investigated as sufficient criteria for the existence and nonexistence of nontrivial solutions for the subjected problems. Our systems are more general than in the literature, as a consequence there are many new and known specific cases included. Finally, our results are applied for estimating eigenvalues of two given BVPs.

Keywords

• Fractional boundary value problem
• Katugampola–Hilfer fractional derivative
• Existence and nonexistence solutions
• Lyapunov-type inequality