Third-order Neutral Dynamic Equations Research Articles

Existence of nonoscillatory solutions tending to zero of third-order neutral dynamic equations on time scales

We study the existence of nonoscillatory solutions tending to zero of a class of third-order nonlinear neutral dynamic equations on time scales by employing Krasnoselskii’s fixed point theorem. Two examples are given to illustrate the significance of the conclusions.

Existence of nonoscillatory solutions to nonlinear third-order neutral dynamic equations on time scales

Existence of nonoscillatory solutions to nonlinear third-order neutral dynamic equations on time scales

On the asymptotic and oscillatory behavior of solutions of third-order neutral dynamic equations on time scales

The oscillatory and asymptotic behavior results for a class of third-order nonlinear neutral dynamic equations on time scales are presented. The results obtained can be extended to more general third-order neutral dynamic equations of the type considered here. Examples are provided to illustrate the applicability of the results.

Oscillation Criteria for Nonlinear Third-Order Neutral Dynamic Equations with Damping on Time Scales

We establish several oscillation criteria for a class of third-order nonlinear dynamic equations with a damping term and a nonpositive neutral coefficient by using the Riccati transformation. Two illustrative examples are presented to show the significance of the results obtained.

Oscillation criteria for third-order neutral dynamic equations with continuously distributed delay

It is the purpose of this paper to give oscillation criteria for the third-order neutral dynamic equations with continuously distributed delay, [ r ( t ) ( [ x ( t ) + ∫ a b p ( t , η ) x [ τ ( t , η ) ] Δ η ] Δ Δ ) γ ] Δ + ∫ c d q(t,ξ)f ( x [ ϕ ( t , ξ ) ] ) Δξ=0, on a time scale T, where γ is the quotient of odd positive integers. By using a generalized Riccati transformation and an integral averaging technique, we establish some new sufficient conditions which ensure that every solution of this equation oscillates or converges to zero.

Asymptotic properties of solutions of third-order nonlinear neutral dynamic equations

The purpose of this article is to give oscillation criteria for the third-order neutral dynamic equation ( r 2 ( t ) [ ( r 1 ( t ) [ y ( t ) + p ( t ) y ( τ ( t ) ) ] △ ) △ ] γ ) △ +f(t,y(δ(t)))=0, where γ⩾1 is a ratio of odd positive integers with r 1 (t), r 2 (t), and p(t) are positive real-valued rd-continuous functions defined on T. We give new results for the third-order neutral dynamic equations and an example to illustrate the importance of our results.