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

摘要

We state and prove new generalized Lyapunov-type and Hartman-type inequalities for a conformable boundary value problem of order α ∈ (1, 2] with mixed non-linearities of the form (Tαax)(t)+r1(t)|x(t)|η−1x(t)+r2(t)|x(t)|δ−1x(t)=g(t),t∈(a,b), satisfying the Dirichlet boundary conditions x(a) = x(b) = 0, where r1, r2, and g are real-valued integrable functions, and the non-linearities satisfy the conditions 0  η  1  δ  2. Moreover, Lyapunov-type and Hartman-type inequalities are obtained when the conformable derivative Tαa is replaced by a sequential conformable derivative Tαa∘Tαa, α ∈ (1/2, 1]. The potential functions r1, r2 as well as the forcing term g require no sign restrictions. The obtained inequalities generalize some existing results in the literature.