New Criteria on Oscillatory and Asymptotic Behavior of Third-Order Nonlinear Dynamic Equations with Nonlinear Neutral Terms.

In the paper, we provide sufficient conditions for the oscillatory and asymptotic behavior of a new type of third-order nonlinear dynamic equations with mixed nonlinear neutral terms. Our theorems not only improve and extend existing theorems in the literature but also provide a new approach as far as the nonlinear neutral terms are concerned. The main results are illustrated by some particular examples.

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

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.