A Lyapunov-Type Inequality for Partial Differential Equation Involving the Mixed Caputo Derivative

Jie Wang,Shuqin Zhang

doi:10.3390/math8010047

Abstract

In this work, we derive a Lyapunov-type inequality for a partial differential equation on a rectangular domain with the mixed Caputo derivative subject to Dirichlet-type boundary conditions. The obtained inequality provides a necessary condition for the existence of nontrivial solutions to the considered problem and an example is given to illustrate it. Moreover, we present some applications to demonstrate the effectiveness of the new results.

Highlights

IntroductionWe focus on the representation of the Lyapunov-type inequality for the following boundary value problem:
In this paper, we focus on the representation of the Lyapunov-type inequality for the following boundary value problem: C rD0 u( x, y) + q( x, y)u( x, y) = 0, ( x, y) ∈ (0, a) × (0, b), u(0, y) = 0, u( a, y) = 0, y ∈ [0, b], u( x, 0) = 0, u( x, b) = 0, x ∈ [0, a], (1)where a, b > 0, r = (r1, r2 ), 1 < r1, r2 < 2, CD0r is the mixed Caputo derivative of order r and q : [0, a] × [0, b] → R is a given Lebesgue integrable function
With the help of the properties of its Green function, we establish a new Lyapunov-type inequality, which provides a necessary condition for the existence of nontrivial solutions to problem (1)

Summary

Introduction

We focus on the representation of the Lyapunov-type inequality for the following boundary value problem:. We apply the obtained inequality to prove the uniqueness of solutions for the nonhomogenous boundary value problem and derive an estimation related to the eigenvalue of the corresponding equation. Hyperbolic partial differential equations and inclusions of fractional order have been intensely studied by many researchers, see for instance [24,25,26,27,28,29,30] and the references therein These papers mainly studied the existence of solutions for initial value problems of partial differential equations with the mixed fractional derivatives, and few scholars studied boundary value problem for the corresponding equations. The obtained inequality provides a necessary condition for the existence of nontrivial solutions to the considered problem and an example is given to illustrate it.

Preliminaries

Applications

Conclusions