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

Advances in Difference Equations

Fahd Jarad¹  Thabet Abdeljawad²  Jehad Alzabut² 

[1] Department of Mathematics and Computer Science, ÇDepartment of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia;ankaya University, Ankara, Turkey

关键词: Lyapunov inequality;    conformable derivative;    Green’s function;    boundary value problem;    Sturm-Liouville eigenvalue problem;    34A08;    26D15;   

DOI  :  10.1186/s13662-017-1383-z

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

We prove a generalized Lyapunov-type inequality for a conformable boundary value problem (BVP) of orderα∈(1,2]$\alpha \in (1,2]$. Indeed, it is shown that if the boundary value problem(Tαcx)(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]$[c,d]$, then1∫cd|r(t)|dt>αα(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 form2∫cd|r(t)|dt>3α−1(d−c)2α−1(3α−12α−1)2α−1α,12<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.

【 授权许可】

CC BY