Oscillatory behavior of higher order nonlinear homogeneous neutral delay dynamic equations with positive and negative coefficients

2018 ◽  
Vol 24 (2) ◽  
pp. 139-154
Author(s):  
Saroj Panigrahi ◽  
P. Rami Reddy
Keyword(s):  
Asymptotic Behavior ◽  
Fourth Order ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Oscillatory Behavior ◽  
Dynamic Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Neutral Delay ◽  
Positive And Negative Coefficients ◽  
Delay Dynamic Equations

Abstract In this paper, we derive some sufficient conditions for the oscillatory and asymptotic behavior of solutions of the higher order nonlinear neutral delay dynamic equation with positive and negative coefficients. The results of this paper extend and generalize the results of [S. Panigrahi and P. Rami Reddy, Oscillatory and asymptotic behavior of fourth order non-linear neutral delay dynamic equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 20 2013, 143–163] and [S. Panigrahi, J. R. Graef and P. Rami Reddy, Oscillation results for fourth order nonlinear neutral dynamic equations, Commun. Math. Anal. 15 2013, 11–28]. Examples are included to illustrate the validation of the results.

On oscillatory and asymptotic behavior of fourth order nonlinear neutral delay dynamic equations with positive and negative coefficients

2014 ◽  
Vol 64 (2) ◽  
Author(s):  
John Graef ◽  
Saroj Panigrahi ◽  
P. Reddy
Keyword(s):  
Fixed Point ◽  
Fourth Order ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Dynamic Equations ◽  
Neutral Delay ◽  
Positive And Negative Coefficients ◽  
Delay Dynamic Equations ◽  
Krasnosel’Skii’S Fixed Point Theorem ◽  
Neutral Dynamic Equations

AbstractIn this paper, oscillatory and asymptotic properties of solutions of nonlinear fourth order neutral dynamic equations of the form $(r(t)(y(t) + p(t)y(\alpha _1 (t)))^{\Delta ^2 } )^{\Delta ^2 } + q(t)G(y(\alpha _2 (t))) - h(t)H(y(\alpha _3 (t))) = 0(H)$ and $(r(t)(y(t) + p(t)y(\alpha _1 (t)))^{\Delta ^2 } )^{\Delta ^2 } + q(t)G(y(\alpha _2 (t))) - h(t)H(y(\alpha _3 (t))) = f(t),(NH)$ are studied on a time scale $\mathbb{T}$ under the assumption that $\int\limits_{t_0 }^\infty {\tfrac{t} {{r(t)}}\Delta t = \infty } $ and for various ranges of p(t). In addition, sufficient conditions are obtained for the existence of bounded positive solutions of the equation (NH) by using Krasnosel’skii’s fixed point theorem.

scholarly journals Oscillatory behaviour of higher-order nonlinear neutral delay dynamic equations on time scales

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2635-2649
Author(s):  
M.M.A. El-Sheikh ◽  
M.H. Abdalla ◽  
A.M. Hassan
Keyword(s):  
Sufficient Conditions ◽  
Higher Order ◽  
Comparison Theorems ◽  
Dynamic Equations ◽  
Oscillatory Behaviour ◽  
Neutral Delay ◽  
Riccati Technique ◽  
Delay Dynamic Equations ◽  
Positive Integers ◽  
Oscillation Of Solutions

In this paper, new sufficient conditions are established for the oscillation of solutions of the higher order dynamic equations [r(t)(z?n-1(t))?]? + q(t) f(x(?(t)))=0, for t ?[t0,?)T, where z(t):= x(t)+ p(t)x(?(t)), n ? 2 is an even integer and ? ? 1 is a quotient of odd positive integers. Under less restrictive assumptions for the neutral coefficient, we employ new comparison theorems and Generalized Riccati technique.

Oscillation results for higher order nonlinear neutral differential equations with positive and negative coefficients

2015 ◽  
Vol 21 (2) ◽  
Author(s):  
Saroj Panigrahi ◽  
Rakhee Basu
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Asymptotic Behavior Of Solutions ◽  
Banach Fixed Point Theorem ◽  
Neutral Differential Equations ◽  
Positive And Negative Coefficients

AbstractIn this paper, the authors investigated oscillatory and asymptotic behavior of solutions of a class of nonlinear higher order neutral differential equations with positive and negative coefficients. The results in this paper generalize the results of Tripathy, Panigrahi and Basu [Fasc. Math. 52 (2014), 155–174]. We establish new conditions which guarantees that every solution either oscillatory or converges to zero. Moreover, using the Banach Fixed Point Theorem sufficient conditions are obtained for the existence of bounded positive solutions. Examples are considered to illustrate the main results.

scholarly journals New Oscillatory Behavior of Third-Order Nonlinear Delay Dynamic Equations on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Li Gao ◽  
Quanxin Zhang ◽  
Shouhua Liu
Keyword(s):  
Time Scales ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Dynamic Equations ◽  
Riccati Transformation ◽  
Third Order ◽  
Generalized Riccati Transformation ◽  
Delay Dynamic Equations ◽  
Inequality Technique ◽  
Nonlinear Delay

A class of third-order nonlinear delay dynamic equations on time scales is studied. By using the generalized Riccati transformation and the inequality technique, four new sufficient conditions which ensure that every solution is oscillatory or converges to zero are established. The results obtained essentially improve earlier ones. Some examples are considered to illustrate the main results.

scholarly journals Some Oscillation Criteria for Higher-Order Neutral Emden-Fowler Delay Dynamic Equations on Time Scales

2018 ◽  
Vol 228 ◽  
pp. 01003
Author(s):  
Ying Sui ◽  
Yulong Shi ◽  
Yibin Sun ◽  
Shurong Sun
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Higher Order ◽  
Delay Equation ◽  
Dynamic Equations ◽  
Neutral Delay ◽  
Oscillation Criteria ◽  
Delay Dynamic Equations

New oscillation criteria are established for higher-order Emdn-Fowler dynamic equation $ q(v)x^{\beta } (\delta (v)) + (r(v)(z^{{\Delta ^{{n - 1}} }} (v))^{\alpha } )^{\Delta } = 0 $ on time scales, $ z(v): = p(v)x(\tau (v)) + x(v) $ Our results extend and supplement those reported in literatures in the sense that we study a more generalized neutral delay equation and do not require $ r^{\Delta } (v) \ge 0 $ and the commutativity of the jump and delay operators.

scholarly journals Oscillatory Behavior of Quasilinear Neutral Delay Dynamic Equations on Time Scales

2010 ◽  
Vol 2010 ◽  
pp. 1-25 ◽  
Author(s):  
Zhenlai Han ◽  
Shurong Sun ◽  
Tongxing Li ◽  
Chenghui Zhang
Keyword(s):  
Time Scales ◽  
Oscillatory Behavior ◽  
Dynamic Equations ◽  
Neutral Delay ◽  
Delay Dynamic Equations

scholarly journals Oscillatory and asymptotic behavior of solutions for second-order mixed nonlinear integro-dynamic equations with maxima on time scales

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 2907-2929
Author(s):  
Hassan Agwa ◽  
Mokhtar Naby ◽  
Heba Arafa
Keyword(s):  
Asymptotic Behavior ◽  
Time Scales ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Dynamic Equations ◽  
Asymptotic Behavior Of Solutions

This paper is concerned with the oscillatory and asymptotic behavior for solutions of the following second-order mixed nonlinear integro-dynamic equations with maxima on time scales (r(t)(z?(t))?)? + ?t0 a(t,s) f(s, x(s))?s + ?n,i=1 qi(t) max s?[?i(t),?i(t)] x?(s) = 0, where z(t) = x(t) + p1(t)x(?1(t)) + p2(t)x(?2(t)), t ? [0,+?)T. The oscillatory behavior of this equation hasn?t been discussed before, also our results improve and extend some results established by Grace et al. [2] and [8].

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.

Sign in / Sign up

Export Citation Format

Share Document