scholarly journals Oscillation and nonoscillation of nonlinear neutral delay difference equations

2007 ◽  
Vol 38 (4) ◽  
pp. 323-333 ◽  
Author(s):  
E. Thandapani ◽  
P. Mohan Kumar
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Oscillation And Nonoscillation

In this paper, the authors establish some sufficient conditions for oscillation and nonoscillation of the second order nonlinear neutral delay difference equation$$ \Delta^2 (x_n-p_nx_{n-k}) + q_nf(x_{n-\ell}) = 0, ~~n \ge n_0 $$where $ \{p_n\} $ and $ \{q_n\} $ are non-negative sequences with $ 0$

scholarly journals Some Oscillation Results for Even Order Delay Difference Equations with a Sublinear Neutral Term

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Govindasamy Ayyappan ◽  
Gunasekaran Nithyakala
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
Neutral Term ◽  
All Solutions ◽  
Even Order ◽  
Delay Difference Equations ◽  
Delay Difference

In this paper, some new results are obtained for the even order neutral delay difference equationΔanΔm-1xn+pnxn-kα+qnxn-lβ=0, wherem≥2is an even integer, which ensure that all solutions of the studied equation are oscillatory. Our results extend, include, and correct some of the existing results. Examples are provided to illustrate the importance of the main results.

scholarly journals Necessary and sufficient conditions for oscillation of first order neutral delay difference equations

2015 ◽  
Vol 3 (2) ◽  
pp. 61
Author(s):  
A. Murgesan ◽  
P. Sowmiya
Keyword(s):  
Difference Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
First Order ◽  
Constant Coefficients ◽  
Necessary And Sufficient ◽  
Delay Difference Equations ◽  
Delay Difference

<p>In this paper, we obtained some necessary and sufficient conditions for oscillation of all the solutions of the first order neutral delay difference equation with constant coefficients of the form <br />\begin{equation*} \quad \quad \quad \quad \Delta[x(n)-px(n-\tau)]+qx(n-\sigma)=0, \quad \quad n\geq n_0 \quad \quad \quad \quad \quad \quad {(*)} \end{equation*}<br />by constructing several suitable auxiliary functions. Some examples are also given to illustrate our results.</p>

Oscillation of neutral delay difference equations of second order with positive and negative coefficients

2009 ◽  
Vol 59 (4) ◽  
Author(s):  
Seshadev Padhi ◽  
Chuanxi Qian
Keyword(s):  
Difference Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Neutral Delay ◽  
Neutral Difference Equations ◽  
Positive And Negative Coefficients ◽  
Delay Difference Equations ◽  
Delay Difference

AbstractThis paper is concerned with a class of neutral difference equations of second order with positive and negative coefficients of the forms $$ \Delta ^2 (x_n \pm c_n x_{n - \tau } ) + p_n x_{n - \delta } - q_n x_{n - \sigma } = 0 $$ where τ, δ and σ are nonnegative integers and {p n}, {q n} and {c n} are nonnegative real sequences. Sufficient conditions for oscillation of the equations are obtained.

scholarly journals Positive Solutions of a Second-Order Nonlinear Neutral Delay Difference Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-30 ◽  
Author(s):  
Zeqing Liu ◽  
Wei Sun ◽  
Jeong Sheok Ume ◽  
Shin Min Kang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Difference Equation ◽  
Positive Solutions ◽  
Point Theorem ◽  
Sufficient Conditions ◽  
Second Order ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
Delay Difference

The purpose of this paper is to study solvability of the second-order nonlinear neutral delay difference equationΔ(a(n,ya1n,…,yarn)Δ(yn+bnyn-τ))+f(n,yf1n,…,yfkn)=cn,  ∀n≥n0. By making use of the Rothe fixed point theorem, Leray-Schauder nonlinear alternative theorem, Krasnoselskill fixed point theorem, and some new techniques, we obtain some sufficient conditions which ensure the existence of uncountably many bounded positive solutions for the above equation. Five nontrivial examples are given to illustrate that the results presented in this paper are more effective than the existing ones in the literature.

Improved oscillation criteria for second order nonlinear delay difference equations with non-positive neutral term

2016 ◽  
Vol 56 (1) ◽  
pp. 155-165 ◽  
Author(s):  
E. Thandapani ◽  
S. Selvarangam ◽  
R. Rama ◽  
M. Madhan
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Second Order ◽  
Delay Difference Equation ◽  
Oscillation Criteria ◽  
Neutral Term ◽  
Positive Integers ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonlinear Delay

Abstract In this paper, we present some oscillation criteria for second order nonlinear delay difference equation with non-positive neutral term of the form $$\Delta (a_n (\Delta z_n )^\alpha ) + q_n f(x_{n - \sigma } ) = 0,\;\;\;n \ge n_0 > 0,$$ where zn = xn − pnxn−τ, and α is a ratio of odd positive integers. Examples are provided to illustrate the results. The results obtained in this paper improve and complement to some of the existing results.

scholarly journals Necessary Conditions for the Solutions of Second Order Non-linear Neutral Delay Difference Equations to Be Oscillatory or Tend to Zero

2007 ◽  
Vol 2007 ◽  
pp. 1-16 ◽  
Author(s):  
R. N. Rath ◽  
J. G. Dix ◽  
B. L. S. Barik ◽  
B. Dihudi
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Difference Operator ◽  
Necessary Conditions ◽  
Delay Difference Equation ◽  
Real Numbers ◽  
Neutral Delay ◽  
Forward Difference ◽  
Delay Difference Equations ◽  
Delay Difference

We find necessary conditions for every solution of the neutral delay difference equationΔ(rnΔ(yn−pnyn−m))+qnG(yn−k)=fnto oscillate or to tend to zero asn→∞, whereΔis the forward difference operatorΔxn=xn+1−xn, andpn, qn, rnare sequences of real numbers withqn≥0, rn>0. Different ranges of{pn}, includingpn=±1, are considered in this paper. We do not assume thatGis Lipschitzian nor nondecreasing withxG(x)>0forx≠0. In this way, the results of this paper improve, generalize, and extend recent results. Also, we provide illustrative examples for our results.

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