A Short Note on Lyapunov Type Inequalities for Hilfer Fractional Boundary Value Problems

TL;DR

This paper establishes Lyapunov-type inequalities for Hilfer fractional boundary value problems, providing new bounds and properties for solutions involving fractional derivatives of order between 1 and 2.

Contribution

It introduces Lyapunov inequalities specifically for Hilfer fractional boundary value problems with separated and anti-periodic conditions, using Green's functions.

Findings

Derived Lyapunov inequalities for Hilfer fractional BVPs

Constructed and analyzed Green's functions for these problems

Provided bounds that can be used to study solution properties

Abstract

This paper deals with fractional boundary value problems involving the Hilfer fractional differential operator of order $1 < \alpha \leq 2$ and type $0 \leq \beta \leq 1$. We derive the corresponding Lyapunov-type inequalities for two prominent classes of Hilfer fractional boundary value problems (HFBVPs) involving separated and anti-periodic boundary conditions. For this purpose, we construct the associated Green's functions and deduce their important properties.