Second-order Nonlinear Neutral Dynamic Equations Research Articles

New oscillation criteria for second-order neutral delay dynamic equations with a non-positive neutral term on time scales

In this paper, we investigate the oscillatory behavior of solutions of a class of secondorder nonlinear neutral delay dynamic equations with non-positive coefficient where α ≥ β are ratios of odd positive integers and z(t) := x(t) - p(t)x(τ (t)). The obtained results guarantee that all solutions are oscillatory. Illustrative examples are provided.

Existence of positive solutions for second-order nonlinear neutral dynamic equations on time scales

AbstractIn this article we study the existence of positive solutions for second-order nonlinear neutral dynamic equations on time scales. The main tool employed here is Schauder’s fixed point theorem. The results obtained here extend the work of Culakova, Hanustiakova and Olach [12]. Two examples are also given to illustrate this work.

Oscillation of second-order nonlinear neutral dynamic equations with “Maxima” on time scales

This paper is concerned with the oscillation of solutions of the second order nonlinear neutral delay dynamic equation with “Maxima” of the formon a time scale 𝕋. The obtained results improve and generalize some recent published results. Examples are given to illustrate the main results.

Uncountably many nonoscillatory bounded solutions to second-order nonlinear neutral dynamic equations

This work is devoted to the study of the existence of uncountably many nonoscillatory bounded solutions to second-order nonlinear neutral dynamic equations by means of the Darbo fixed point theorem. We construct assumptions without sign conditions on the nonlinear part of the equation. Moreover, we prove the necessary condition for the existence of an asymptotically zero solution to the problem under consideration.

Oscillation criteria for second-order nonlinear delay dynamic equations of neutral type

We investigate oscillatory behavior of solutions to a class of second-order nonlinear neutral delay dynamic equations with nonpositive neutral coefficients. In particular, we study the corresponding noncanonical neutral differential equations. New oscillation criteria are established that complement and improve related contributions to the subject. An example is given to illustrate the main results.

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

In this paper, we discuss oscillation criteria for second-order nonlinear neutral delay dynamic equations with damping on time scales by using the generalized Riccati transformation and the inequality technique. Under certain conditions, we establish four new oscillation criteria. Our results in this paper are new even for the cases of T = R and T = Z.

New Oscillation Results of Second-Order Mixed Nonlinear Neutral Dynamic Equations with Damping on Time Scales

New Oscillation Results of Second-Order Mixed Nonlinear Neutral Dynamic Equations with Damping on Time Scales

On the Oscillation of Second-Order Nonlinear Neutral Delay Dynamic Equations on Time Scales

Based on Riccati transformation and the inequality technique, we establish some new sufficient conditions for oscillation of the second-order neutral delay dynamic equations on time scales. Our results not only extend and improve some known theorems, but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation on time scales. At the end of this paper, we give an example to illustrate the main results.

A note on "oscillation criteria for second-order nonlinear neutral dynamic equations on time scales

In this work, we give a counter example for the main result of the paper \small[Tamkang J. of Math., 43 (1)(2012), 109--122.] and we give the correct formula for some theorems given in their work.

Oscillation of second-order nonlinear neutral dynamic equations with distributed deviating arguments on time scales

This paper concerns second-order nonlinear neutral dynamic equations with distributed deviating arguments on time scales of the form $$\bigl(r(t) \bigl(\bigl(y(t)+p(t)y\bigl(\tau(t)\bigr)\bigr)^{\Delta}\bigr)^{\gamma}\bigr)^{\Delta}+\int_{a}^{b}f \bigl(t,y\bigl(\delta (t,\xi)\bigr)\bigr)\Delta\xi=0, $$ where $\gamma>0$ is a quotient of odd positive integers. By using the generalized Riccati technique and integral averaging techniques, we derive new oscillation criteria for the above equations, which generalize and improve some existing results in the literature.

Oscillation of Second-Order Nonlinear Neutral Dynamic Equations with Noncanonical Operators

The study of half-linear differential equations has become an important area of research due to the fact that such equations occur in a variety of real-world problems such as in the study of \(p\)-Laplace equations, non-Newtonian fluid theory, and the turbulent flow of a polytrophic gas in a porous medium. On the basis of these background details, we study oscillatory behavior of a class of second-order neutral functional dynamic equations on a time scale. New criteria improve and complement related results reported in the literature. Some examples are included to illustrate the results obtained. In particular, an example regarding the second-order neutral differential equation is also provided to show that these theorems improve those in the continuous case.

Asymptotic behavior of second-order nonlinear neutral dynamic equations

Abstract This paper is concerned with oscillation and asymptotic behavior of a second-order neutral delay dynamic equation on an arbitrary time scale. We obtain two theorems which guarantee that every solution of the studied equation oscillates or converges to zero. These results improve and complement some known results given in the literature. MSC:34K11, 34N05, 39A10, 39A12, 39A13, 39A21.

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

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

Oscillation criteria for second-order nonlinear neutral dynamic equations with distributed deviating arguments on time scales

In this article, we establish some new oscillation criteria and give sufficient conditions to ensure that all solutions of nonlinear neutral dynamic equation of the form

Oscillatory properties of second-order nonlinear neutral dynamic equations with positive and negative coefficients

This paper is to investigate the oscillatory properties of second-order nonlinear neutral dynamic equation with positive and negative coefficients of the form on an arbitrary time scale $\mathbb{T}$ . We establish several oscillation criteria for the equation by using a generalized Riccati technique. The obtained results extend and improve certain existing results in the literature. Examples are shown to illustrate our main results.

Oscillation of second-order nonlinear neutral dynamic equations on time scales with distributed deviating arguments

This article is concerned with oscillation of second-order neutral dynamic equations with distributed deviating arguments of the form ( r ( t ) ( ( y ( t ) + p ( t ) y ( τ ( t ) ) ) Δ ) γ ) Δ + ∫ c d f ( t , y ( θ ( t , ξ ) ) ) Δ ξ = 0 , where γ > 0 is a ratio of odd positive integers with r ( t ) and p ( t ) real-valued rd-continuous positive functions defined on T . We establish some new oscillation criteria and give sufficient conditions to insure that all solutions of nonlinear neutral dynamic equation are oscillatory on a time scale T .

Oscillation of second-order quasi-linear neutral functional dynamic equations with distributed deviating arguments

In this paper, some sufficient conditions for the oscillation of second-order nonlinear neutral functional dynamic equation \[( r(t) ( [x(t) + p(t)x[\tau (t)]]^\Delta)^\gamma )^\Delta +\int^b_a q(t; \xi)x^\gamma [g(t; \xi)]\Delta\xi= 0; t \in \mathbb{T}\] are established. An example is given to illustrate an application of our results.

On The Bounded Oscillation of Certain Second-order Nonlinear Neutral Delay Dynamic Equations with Oscillating Coefficients

We investigate the bounded oscillation of the second-order nonlinear neutral delay dynamic equation with oscillating coefficients \[\Big(r(t)\Big|\big[x(t)+p(t)x(\tau(t))\big]^\Delta\Big|^{\alpha-1}\big[x(t)+p(t)x(\tau(t))\big]^\Delta\Big)^\Delta+q(t)|x(t)|^{\beta-1}x(t)=0 \] on an arbitrary time scale $\mathbb{T}$, where $p$ is an oscillating function defined on $\mathbb{T}$ and $\alpha,\beta>0$ are constants, and obtain several sufficient conditions for the oscillation of all bounded solutions of the equation when $\beta>\alpha, \beta=\alpha$ and $\beta<\alpha$, respectively. Our results extend and complement some known results where $p(t)\equiv 0$ and $\alpha,\beta$ are quotients of odd positive integers.

Nonlinear Oscillation and Asymptotic Behavior of Second-order Neutral Dynamic Equations on Time Scales

The paper is to study the oscillation and asymptotic behavior of certain second-order nonlinear neutral dynamic equations on time scales. We obtain some sufficient conditions which ensure that every solution of the equations oscillates or converges to zero. Our results improve and extend some existing results.

Oscillation of second-order nonlinear neutral dynamic equations on time scales

This paper is concerned with the oscillation of second-order nonlinear neutral dynamic equations of the form r ( t ) ( y ( t ) + p ( t ) y ( τ ( t ) ) ) Δ γ Δ + f ( t , y ( δ ( t ) ) ) = 0 , on a time scale T . By using a generalized Riccati technique and integral averaging techniques, we establish new oscillation results which handle some cases not covered by known criteria.