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