Lyapunov-type inequalities for an anti-periodic fractional boundary value problem involving ψ-Caputo fractional derivative

Abstract

A Lyapunov-type inequality is established for the anti-periodic fractional boundary value problem $\begin{array}{cc}& \left({}^C{D}_a^{\alpha ,\psi }u\right)\left(x\right)+f(x,u(x\left)\right)=0,a<x<b,\\ & u\left(a\right)+u\left(b\right)=0,u^{\text{'}}\left(a\right)+u^{\text{'}}\left(b\right)=0,\end{array}$ where $(a,b)\in R^2$ , $a<b$ , $1<\alpha <2$ , $\psi \in C^2\left(\right[a,b\left]\right)$ , ${\psi }^{\text{'}}\left(x\right)>0$ , $x\in [a,b]$ , ${}^{C}D_a^{\alpha ,\psi }$ is the ψ-Caputo fractional derivative of order α, and $f:[a,b]\times R\to R$ is a given function. Next, we give an application of the obtained inequality to the corresponding eigenvalue problem.

Keywords: Lyapunov-type inequalities; anti-periodic fractional boundary value problem; eigenvalues; ψ-Caputo fractional derivative.