scholarly journals Oscillation of Two-Dimensional Neutral Delay Dynamic Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Xinli Zhang ◽  
Shanliang Zhu
Keyword(s):  
Time Scales ◽  
Dynamic Systems ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Real Sequence ◽  
Two Dimensional ◽  
Positive Real ◽  
Neutral Delay ◽  
All Solutions ◽  
Special Case

We consider a class of nonlinear two-dimensional dynamic systems of the neutral type(x(t)-a(t)x(τ1(t)))Δ=p(t)f1(y(t)),yΔ(t)=-q(t)f2(x(τ2(t))).We obtain sufficient conditions for all solutions of the system to be oscillatory. Our oscillation results whena(t)=0improve the oscillation results for dynamic systems on time scales that have been established by Fu and Lin (2010), since our results do not restrict to the case wheref(u)=u. Also, as a special case when𝕋=ℝ, our results do not requireanto be a positive real sequence. Some examples are given to illustrate the main results.

scholarly journals On the oscillation of a nonlinear two-dimensional difference systems

2001 ◽  
Vol 32 (3) ◽  
pp. 201-209 ◽  
Author(s):  
E. Thandapani ◽  
B. Ponnammal
Keyword(s):  
Asymptotic Behavior ◽  
Sufficient Conditions ◽  
Real Sequence ◽  
Two Dimensional ◽  
Difference System ◽  
Nonoscillatory Solutions ◽  
Difference Systems ◽  
All Solutions ◽  
Necessary And Sufficient ◽  
Strongly Sublinear

The authors consider the two-dimensional difference system$$ \Delta x_n = b_n g (y_n) $$ $$ \Delta y_n = -f(n, x_{n+1}) $$where $ n \in N(n_0) = \{ n_0, n_0+1, \ldots \} $, $ n_0 $ a nonnegative integer; $ \{ b_n \} $ is a real sequence, $ f: N(n_0) \times {\rm R} \to {\rm R} $ is continuous with $ u f(n,u) > 0 $ for all $ u \ne 0 $. Necessary and sufficient conditions for the existence of nonoscillatory solutions with a specified asymptotic behavior are given. Also sufficient conditions for all solutions to be oscillatory are obtained if $ f $ is either strongly sublinear or strongly superlinear. Examples of their results are also inserted.

scholarly journals Oscillation and Asymptotic Behaviour of a Higher-Order Nonlinear Neutral-Type Functional Differential Equation with Oscillating Coefficients

2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Mustafa Kemal Yildiz ◽  
Emrah Karaman ◽  
Hülya Durur
Keyword(s):  
Neutral Type ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Bounded Solutions ◽  
Neutral Delay ◽  
All Solutions ◽  
Neutral Delay Differential Equations ◽  
Oscillating Coefficients ◽  
Functional Differential ◽  
Delay Differential

We will study oscillation of bounded solutions of higher-order nonlinear neutral delay differential equations of the following type:[y(t)+p(t)f(y(τ(t)))](n)+q(t)h(y(σ(t)))=0,t≥t0,t∈R, wherep∈C([t0,∞),R),limt→∞p(t)=0,q∈C([t0,∞),R+),τ(t),σ(t)∈C([t0,∞),R),τ(t),σ(t)<t,lim⁡t→∞τ(t),σ(t)=∞, andf,h∈C(R,R). We obtain sufficient conditions for the oscillation of all solutions of this equation.

scholarly journals Behavior of the Solutions for Predator-Prey Dynamic Systems with Beddington-DeAngelis Type Functional Response on Periodic Time Scales in Shifts

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Neslihan Nesliye Pelen ◽  
Ayşe Feza Güvenilir ◽  
Billur Kaymakçalan
Keyword(s):  
Periodic Solution ◽  
Functional Response ◽  
Time Scales ◽  
Dynamic Systems ◽  
Two Dimensional ◽  
Predator Prey ◽  
Prey System ◽  
Special Case

We consider two-dimensional predator-prey system with Beddington-DeAngelis type functional response on periodic time scales in shifts. For this special case we try to find under which conditions the system hasδ±-periodic solution.

On oscillatory fourth order nonlinear neutral differential equations – III

2018 ◽  
Vol 68 (6) ◽  
pp. 1385-1396 ◽  
Author(s):  
Arun Kumar Tripathy ◽  
Rashmi Rekha Mohanta
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Functional Differential Equations ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Neutral Differential Equations ◽  
All Solutions ◽  
Functional Differential ◽  
T Tau

Abstract In this paper, several sufficient conditions for oscillation of all solutions of fourth order functional differential equations of neutral type of the form $$\begin{array}{} \displaystyle \bigl(r(t)(y(t)+p(t)y(t-\tau))''\bigr)''+q(t)G\bigl(y(t-\sigma)\bigr)=0 \end{array}$$ are studied under the assumption $$\begin{array}{} \displaystyle \int\limits^{\infty}_{0}\frac{t}{r(t)}{\rm d} t =\infty \end{array}$$

scholarly journals The Existence and Global Exponential Stability of Almost Periodic Solutions for Neutral-Type CNNs on Time Scales

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 321 ◽  
Author(s):  
Bing Li ◽  
Yongkun Li ◽  
Xiaofang Meng
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Periodic Solutions ◽  
Time Scales ◽  
Global Exponential Stability ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Leakage Delays

In this paper, neutral-type competitive neural networks with mixed time-varying delays and leakage delays on time scales are proposed. Based on the contraction fixed-point theorem, some sufficient conditions that are independent of the backwards graininess function of the time scale are obtained for the existence and global exponential stability of almost periodic solutions of neural networks under consideration. The results obtained are brand new, indicating that the continuous time and discrete-time conditions of the network share the same dynamic behavior. Finally, two examples are given to illustrate the validity of the results obtained.

scholarly journals Oscillation for a Nonlinear Dynamic System with a Forced Term on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Xinli Zhang ◽  
Shanliang Zhu
Keyword(s):  
Dynamic System ◽  
Time Scale ◽  
Nonlinear Dynamic ◽  
Sufficient Conditions ◽  
Nonlinear Dynamic System ◽  
Two Dimensional ◽  
Differential Systems ◽  
Difference Systems ◽  
All Solutions ◽  
Forced Term

We consider a class of two-dimensional nonlinear dynamic system with a forced term on a time scale𝕋and obtain sufficient conditions for all solutions of the system to be oscillatory. Our results not only unify the oscillation of two-dimensional differential systems and difference systems but also improve the oscillation results that have been established by Saker, 2005, since our results are not restricted to the case whereb(t)≠0for allt∈𝕋andg(u)=u. Some examples are given to illustrate the results.

scholarly journals On Inequalities of Lyapunov for Two-Dimensional Nonlinear Dynamic Systems on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Qiao-Luan Li ◽  
Wing-Sum Cheung ◽  
Xu-Yang Fu
Keyword(s):  
Time Scales ◽  
Dynamic Systems ◽  
Nonlinear Dynamic ◽  
Type Equation ◽  
Nonlinear Dynamic Systems ◽  
Two Dimensional

We establish some new Lyapunov-type inequalities for two-dimensional nonlinear dynamic systems on time scales. As for application, boundedness of the Emden-Fowler-type equation is proved.

Oscillation and asymptotic behavior of a higher-order neutral delay difference equation with variable delays under Δ m

2021 ◽  
Vol 71 (1) ◽  
pp. 129-146
Author(s):  
Chittaranjan Behera ◽  
Radhanath Rath ◽  
Prayag Prasad Mishra
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
Variable Delays ◽  
All Solutions ◽  
Delay Difference

Abstract In this article we obtain sufficient conditions for the oscillation of all solutions of the higher-order delay difference equation Δ m ( y n − ∑ j = 1 k p n j y n − m j ) + v n G ( y σ ( n ) ) − u n H ( y α ( n ) ) = f n , $$\begin{array}{} \displaystyle \Delta^{m}\big(y_n-\sum_{j=1}^k p_n^j y_{n-m_j}\big) + v_nG(y_{\sigma(n)})-u_nH(y_{\alpha(n)})=f_n\,, \end{array}$$ where m is a positive integer and Δ xn = x n+1 − xn . Also we obtain necessary conditions for a particular case of the above equation. We illustrate our results with examples for which it seems no result in the literature can be applied.

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