Sharp estimates for the unique solution of the Hadamard-type two-point fractional boundary value problems

TL;DR

This paper provides precise estimates for the existence and uniqueness of solutions to Hadamard-type fractional boundary value problems, employing Green's function and Banach contraction, and improves eigenvalue bounds with practical examples.

Contribution

It introduces sharp existence and uniqueness estimates for Hadamard-type fractional differential equations and refines eigenvalue bounds, using Green's function and contraction principles.

Findings

Sharp existence and uniqueness estimates established.

Improved lower bounds for eigenvalues derived.

Examples demonstrate applicability of results.

Abstract

In this short note, we present the sharp estimate for the existence of a unique solution for a Hadamard-type fractional differential equations with two-point boundary value conditions. The method of analysis is obtained by using the integral of the Green's function and the Banach contraction principle. Further, we will also obtain a sharper lower bound of the eigenvalues for an eigenvalue problem. Two examples are presented to clarify the applicability of the essential results.