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

作者：Abdeljawad Thabet; Alzabut Jehad; Jarad Fahd

来源：Advances in Difference Equations, 2017, 2017(1): 321.

DOI：10.1186/s13662-017-1383-z

• Lyapunov inequality
• conformable derivative
• Green's function
• boundary value problem
• Sturm-Liouville eigenvalue problem

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

• 出版日期2017-10-11