scholarly journals Oscillation of Solutions of Nonlinear Difference Equation With a Super-linear Neutral Term

2018 ◽  
Vol 5 (1) ◽  
pp. 52-58 ◽  
Author(s):  
C. Dharuman ◽  
E. Thandapani
Keyword(s):  
Difference Equation ◽  
Sufficient Conditions ◽  
Nonlinear Difference Equation ◽  
Neutral Difference Equation ◽  
Neutral Term ◽  
Positive Integers ◽  
Oscillation Of Solutions

Abstract This paper deals with the oscillation of solutions of certain class of neutral difference equation ∆(an∆(χn + pnχαn−k)) + qnχβn+1−l = 0, where α and β are ratio of odd positive integers. New sufficient conditions are obtained for the oscillation of studied equation and examples illustrating the main results are provided.

scholarly journals Oscillation theorems for second order difference equations woth negative neutral term

2015 ◽  
Vol 46 (4) ◽  
pp. 441-451 ◽  
Author(s):  
Ethiraju Thandapani ◽  
Devarajulu Seghar ◽  
Sandra Pinelas
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Second Order ◽  
Neutral Difference Equation ◽  
Order Difference ◽  
Oscillation Criteria ◽  
Neutral Term ◽  
Oscillation Theorems ◽  
Positive Integers

In this paper we obtain some new oscillation criteria for the neutral difference equation \begin{equation*} \Delta \Big(a_n (\Delta (x_n-p_n x_{n-k}))\Big)+q_n f(x_{n-l})=0 \end{equation*} where $0\leq p_n\leq p0$ and $l$ and $k$ are positive integers. Examples are presented to illustrate the main results. The results obtained in this paper improve and complement to the existing results.

scholarly journals Oscillation of solutions of impulsive neutral difference equations with continuous variable

2006 ◽  
Vol 2006 ◽  
pp. 1-7 ◽  
Author(s):  
Gengping Wei ◽  
Jianhua Shen
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Continuous Variable ◽  
Neutral Difference Equations ◽  
Forward Difference ◽  
All Solutions ◽  
Positive Integers ◽  
Oscillation Of Solutions

We obtain sufficient conditions for oscillation of all solutions of the neutral impulsive difference equation with continuous variableΔτ(y(t)+p(t)y(t−mτ))+Q(t)y(t−lτ)=0,t≥t0−τ,t≠tk,y(tk+τ)−y(tk)=bky(tk),k∈ℕ(1), whereΔτdenotes the forward difference operator, that is,Δτz(t)=z(t+τ)−z(t),p(t)∈C([t0−τ,∞),ℝ),Q(t)∈C([t0−τ,∞),(0,∞)),m,lare positive integers,τ>0andbkare constants,0≤t0<t1<t2<⋯<tk<⋯withlimk→∞tk=∞.

scholarly journals Oscillation and non-oscillation of some neutral differential equations of odd order

1992 ◽  
Vol 15 (3) ◽  
pp. 509-515 ◽  
Author(s):  
B. S. Lalli ◽  
B. G. Zhang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Nonoscillatory Solution ◽  
Neutral Differential Equation ◽  
Neutral Differential Equations ◽  
Neutral Term ◽  
Odd Order ◽  
Existence Criterion ◽  
Oscillation Of Solutions

An existence criterion for nonoscillatory solution for an odd order neutral differential equation is provided. Some sufficient conditions are also given for the oscillation of solutions of somenth order equations with nonlinearity in the neutral term.

scholarly journals Oscillations in a nonautonomous delay logistic difference equation

1992 ◽  
Vol 35 (1) ◽  
pp. 121-131 ◽  
Author(s):  
Ch. G. Philos
Keyword(s):  
Difference Equation ◽  
Equilibrium Point ◽  
Positive Constant ◽  
Sufficient Conditions ◽  
Existence Of A Solution ◽  
Sharp Condition ◽  
All Solutions ◽  
Image Position ◽  
Positive Integers ◽  
Logistic Difference Equation

Consider the nonautonomous delay logistic difference equationwhere (pn)n≧0 is a sequence of nonnegative numbers, (ln)n≧0 is a sequence of positive integers with limn→∞(n−ln) = ∞ and K is a positive constant. Only solutions which are positive for n≧0 are considered. We established a sharp condition under which all solutions of (E0) are oscillatory about the equilibrium point K. Also we obtained sufficient conditions for the existence of a solution of (E0) which is nonoscillatory about K.

scholarly journals On the recursive sequencexn+1=−1/xn+A/xn−1

2001 ◽  
Vol 27 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Positive Solution ◽  
Sufficient Conditions ◽  
Nonlinear Difference Equation

We investigate the periodic character of solutions of the nonlinear difference equationxn+1=−1/xn+A/xn−1. We give sufficient conditions under which every positive solution of this equation converges to a period two solution. This confirms a conjecture in the work of DeVault et al. (2000).

scholarly journals Qualitative Inspection Fourth order Non Linear Difference Equations

2019 ◽  
Vol 8 (11) ◽  
pp. 3467-3469
Keyword(s):  
Difference Equation ◽  
Fourth Order ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Nonlinear Difference Equation ◽  
Linear Difference Equations ◽  
All Solutions ◽  
Non Linear

In this paper, the authors obtained some new sufficient conditions for the oscillation of all solutions of the fourth order nonlinear difference equation of the form ( ) ( 1 ) 0 3  anxn  pnxn  qn f xn  n = 0,1,2, … ., where an, pn, qn positive sequences. The established results extend, unify and improve some of the results reported in the literature. Examples are provided to illustrate the main result.

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.

scholarly journals Monotonicity of Eventually Positive Solutions for a Second Order Nonlinear Difference Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Huiqin Chen ◽  
Zhen Jin ◽  
Shugui Kang
Keyword(s):  
Difference Equation ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Second Order ◽  
Nonlinear Difference Equation ◽  
Monotone Solutions

We derive several sufficient conditions for monotonicity of eventually positive solutions on a class of second order perturbed nonlinear difference equation. Furthermore, we obtain a few nonexistence criteria for eventually positive monotone solutions of this equation. Examples are provided to illustrate our main results.

scholarly journals Boundedness and asymptotic behavior of solutions of a forced difference equation

1994 ◽  
Vol 17 (2) ◽  
pp. 397-400 ◽  
Author(s):  
John R. Graef ◽  
Paul W. Spikes
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Nonlinear Difference Equation

The authors consider the nonlinear difference equation?[yn+pnyn-h]+qnf(yn-k)=rnwhere?yn=yn+1-yn,{pn},{qn}, and{rn}are real sequences, anduf(u)>0foru?0. Sufficient conditions for boundedness and convergence to zero of certain types of solutions axe given. Examples illustrating the results are also included.

scholarly journals Existence of Monotone Solutions of a Difference Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
Taixiang Sun ◽  
Hongjian Xi ◽  
Weizhen Quan
Keyword(s):  
Difference Equation ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Nonlinear Difference Equation ◽  
Initial Values ◽  
Monotone Positive Solutions ◽  
Monotone Solutions

We consider the nonlinear difference equationxn+1=f(xn−k,xn−k+1,…,xn),n=0,1,…,wherek∈{1,2,…}and the initial valuesx−k,x−k+1,…,x0∈(0,+∞). We give sufficient conditions under which this equation has monotone positive solutions which converge to the equilibrium, extending and including in this way some results of the literature.

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