Lyapunov-type inequalities for mixed non-linear forced differential equations within conformable derivatives

作者：Abdeljawad Thabet; Agarwal Ravi P; Alzabut Jehad^*; Jarad Fand; Ozbekler Abdullah

来源：JOURNAL OF INEQUALITIES AND APPLICATIONS, 2018, 2018(1): 143.

DOI：10.1186/s13660-018-1731-x

• Lyapunov inequality
• Hartman inequality
• Conformable derivative
• Green's function
• Boundary value problem
• Mixed non-linearities

We state and prove new generalized Lyapunov-type and Hartman-type inequalities fora conformable boundary value problem of order alpha is an element of (1,2] with mixed non-linearities of the form
((T alpha X)-X-a)(t) + r(1)(t)vertical bar X(t)vertical bar(eta-1) X(t) + r(2)(t)vertical bar x(t)vertical bar(delta-1) X(t) = g(t), t is an element of (a, b),
satisfying the Dirichlet boundary conditions x(a) = x(b) = 0, where r(1), r(2), and g are real-valued integrable functions, and the non-linearities satisfy the conditions 0 < eta < 1 < delta < 2. Moreover, Lyapunov-type and Hartman-type inequalities are obtained when the conformable derivative T-alpha(a) is replaced by a sequential conformable derivative T-alpha(a) circle T-alpha(a), alpha is an element of (1/2,1]. The potential functions r(1), r(2) as well as the forcing term g require no sign restrictions. The obtained inequalities generalize some existing results in the literature.

• 出版日期2018-6-20