A generalized Lyapunov-type inequality in the frame of conformable derivatives

Abstract

We prove a generalized Lyapunov-type inequality for a conformable boundary value problem (BVP) of order $\alpha \in (1,2]$ . Indeed, it is shown that if the boundary value problem $$ \bigl(\textbf{T}_{\alpha }^{c} x\bigr) (t)+r(t)x(t)=0,\quad t \in (c,d), x(c)=x(d)=0 $$ has a nontrivial solution, where r is a real-valued continuous function on $[c,d]$ , then 1 $$ \int_{c}^{d} \bigl\vert r(t) \bigr\vert \,dt> \frac{\alpha^{\alpha }}{(\alpha -1)^{\alpha -1}(d-c)^{ \alpha -1}}. $$ Moreover, a Lyapunov type inequality of the form 2 $$ \int_{c}^{d} \bigl\vert r(t) \bigr\vert \,dt> \frac{3\alpha -1}{(d-c)^{2\alpha -1}} \biggl( \frac{3 \alpha -1}{2\alpha -1} \biggr) ^{\frac{2\alpha -1}{\alpha }},\quad \frac{1}{2}< \alpha \leq 1, $$ is obtained for a sequential conformable BVP. Some examples are given and an application to conformable Sturm-Liouville eigenvalue problem is analyzed.