scholarly journals Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
G. M. Moremedi ◽  
I. P. Stavroulakis
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Difference Equations ◽  
Delay Difference Equation ◽  
Real Numbers ◽  
First Order ◽  
All Solutions ◽  
Delay Difference ◽  
Constant Argument

Consider the first-order delay difference equation with a constant argument Δxn+pnxn-k=0,  n=0,1,2,…, and the delay difference equation with a variable argument Δxn+pnxτn=0,  n=0,1,2,…, where p(n) is a sequence of nonnegative real numbers, k is a positive integer, Δx(n)=x(n+1)-x(n), and τ(n) is a sequence of integers such that τ(n)≤n-1 for all n≥0 and limn→∞τ(n)=∞. A survey on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given.

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 Sharp conditions for the oscillation of delay difference equations

1989 ◽  
Vol 2 (2) ◽  
pp. 101-111 ◽  
Author(s):  
G. Ladas ◽  
Ch. G. Philos ◽  
Y. G. Sficas
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Sufficient Condition ◽  
Delay Difference Equation ◽  
Real Numbers ◽  
All Solutions ◽  
Existence Of Positive Solutions ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonnegative Sequence

Suppose that {pn} is a nonnegative sequence of real numbers and let k be a positive integer. We prove that limn→∞inf [1k∑i=n−kn−1pi]>kk(k+1)k+1 is a sufficient condition for the oscillation of all solutions of the delay difference equation An+1−An+pnAn−k=0,   n=0,1,2,…. This result is sharp in that the lower bound kk/(k+1)k+1 in the condition cannot be improved. Some results on difference inequalities and the existence of positive solutions are also presented.

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.

scholarly journals ON EXISTENCE OF POSITIVE SOLUTIONS OF NEUTRAL DIFFERENCE EQUATIONS

1994 ◽  
Vol 25 (3) ◽  
pp. 257-265
Author(s):  
J. H. SHEN ◽  
Z. C. WANG ◽  
X. Z. QIAN
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Difference Equations ◽  
Real Numbers ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Existence Of Positive Solutions

Consider the neutral difference equation \[\Delta(x_n- cx_{n-m})+p_nx_{n-k}=0, n\ge N\qquad (*) \] where $c$ and $p_n$ are real numbers, $k$ and $N$ are nonnegative integers, and $m$ is positive integer. We show that if \[\sum_{n=N}^\infty |p_n|<\infty \qquad (**) \] then Eq.(*) has a positive solution when $c \neq 1$. However, an interesting example is also given which shows that (**) does not imply that (*) has a positive solution when $c =1$.

Sharp explicit oscillation conditions for difference equations with several delays

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Kirill M. Chudinov
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Linear Difference Equation ◽  
First Order ◽  
Variable Delays ◽  
All Solutions ◽  
Oscillation Test ◽  
Several Delays ◽  
Lower Limits

Abstract We consider explicit sufficient conditions for all solutions of a first-order linear difference equation with several variable delays and non-negative coefficients to be oscillatory. The conditions have the form of inequalities bounding below the upper and lower limits of the sums of coefficients over a subset of the discrete semiaxis. Our main results are oscillation tests based on a new principle for composing the estimated sums of coefficients. We also give some results in the form of examples, including a counterexample to a wrong oscillation test cited in several recent papers.

scholarly journals Oscillation of nonlinear delay difference equations

2001 ◽  
Vol 28 (5) ◽  
pp. 301-306 ◽  
Author(s):  
Jianchu Jiang
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Delay Difference Equation ◽  
Oscillation Criteria ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonlinear Delay

We obtain some oscillation criteria for solutions of the nonlinear delay difference equation of the formxn+1−xn+pn∏j=1mxn−kjαj=0.

scholarly journals On the oscillation of the solutions to delay and difference equations

2009 ◽  
Vol 43 (1) ◽  
pp. 173-187
Author(s):  
Khadija Niri ◽  
Ioannis P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Discrete Analogue ◽  
Optimal Conditions ◽  
Nondecreasing Sequence ◽  
Real Numbers ◽  
First Order ◽  
All Solutions ◽  
Linear Delay ◽  
Delay Differential

Abstract Consider the first-order linear delay differential equation xʹ(t) + p(t)x(τ(t)) = 0, t≥ t<sub>0</sub>, (1) where p, τ ∈ C ([t<sub>0</sub>,∞, ℝ<sup>+</sup>, τ(t) is nondecreasing, τ(t) < t for t ≥ t<sup>0</sup> and lim<sub>t→∞</sub> τ(t) = ∞, and the (discrete analogue) difference equation Δx(n) + p(n)x(τ(n)) = 0, n= 0, 1, 2,…, (1)ʹ where Δx(n) = x(n + 1) − x(n), p(n) is a sequence of nonnegative real numbers and τ(n) is a nondecreasing sequence of integers such that τ(n) ≤ n − 1 for all n ≥ 0 and lim<sub>n→∞</sub> τ(n) = ∞. Optimal conditions for the oscillation of all solutions to the above equations are presented.

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.

scholarly journals Classification of solutions of delay difference equations

1994 ◽  
Vol 17 (3) ◽  
pp. 619-623
Author(s):  
B. G. Zhang ◽  
Pengxiang Yan
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Nonnegative Integer ◽  
Delay Difference Equation ◽  
Classification Of Solutions ◽  
Delay Difference Equations ◽  
Delay Difference

In this paper we study the classification of solutions of delay difference equation{?2yn=Pnyn-myn=An???for???n=N-(m+1),…,N-1whereAn,n=N-(m+1),…,N-1are given,mis a nonnegative integer.

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