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

We prove a generalized Lyapunov-type inequality for a conformable boundary value problem (BVP) of order α ∈ (1,2] . Indeed, it is shown that if the boundary value problem (Tαᶜ x) (t)+r(t)x(t)=0, t ∈ (c,d), x(c)=x(d)=0 has a nontrivial solution, where r is a real-valued continuous function on $[c,d]$ , then 1 ∫cᵈ r(t) \,dt> αα(α -1)α -1(d-c)α -1. Moreover, a Lyapunov type inequality of the form 2 ∫cᵈ r(t) \,dt> 3α -1(d-c)2α -1 ( 3 α -1/2α -1 ) 2α -1/α, 1/2< α ≤ 1, is obtained for a sequential conformable BVP. Some examples are given and an application to conformable Sturm-Liouville eigenvalue problem is analyzed.