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Lyapunov-type inequality for fractional BVPs involving two Hadamard fractional derivatives of different orders

Zaid Laadjal

Abstract

This paper establishes a Lyapunov-type inequality for a class of fractional boundary value problems (BVPs) involving two Hadamard fractional derivatives of different orders with Dirichlet boundary conditions. The method is based on the construction of the corresponding Green's function and establishing its maximum value through rigorous analytical techniques. The obtained inequality provides the necessary conditions for the existence of nontrivial solutions to the proposed problem. Finally, we illustrate the applicability of our results by establishing nonexistence criteria for nontrivial solutions to certain problems, providing examples.

Lyapunov-type inequality for fractional BVPs involving two Hadamard fractional derivatives of different orders

Abstract

This paper establishes a Lyapunov-type inequality for a class of fractional boundary value problems (BVPs) involving two Hadamard fractional derivatives of different orders with Dirichlet boundary conditions. The method is based on the construction of the corresponding Green's function and establishing its maximum value through rigorous analytical techniques. The obtained inequality provides the necessary conditions for the existence of nontrivial solutions to the proposed problem. Finally, we illustrate the applicability of our results by establishing nonexistence criteria for nontrivial solutions to certain problems, providing examples.
Paper Structure (5 sections, 8 theorems, 81 equations)

This paper contains 5 sections, 8 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Green's Function Construction and Maximum Analysis
  4. Lyapunov-type inequality for the Problem at Hand
  5. Applications

Key Result

Lemma 4

Let $\sigma ,\kappa >0$, where $n-1<\sigma \leq n$ with $n\in \mathbb{N}$. Then, the following properties hold:

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Remark 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Theorem 7
  • Theorem 8
  • Corollary 9
  • Theorem 10
  • ...and 3 more