scholarly journals A Final Result on the Oscillation of Solutions of the Linear Discrete Delayed EquationΔx(n)=−p(n)x(n−k)with a Positive Coefficient

2011 ◽  
Vol 2011 ◽  
pp. 1-28 ◽  
Author(s):  
J. Baštinec ◽  
L. Berezansky ◽  
J. Diblík ◽  
Z. Šmarda
Keyword(s):  
Positive Solution ◽  
Positive Sequence ◽  
Positive Coefficient ◽  
Opposite Inequality ◽  
All Solutions ◽  
Delayed Equation ◽  
Oscillation Of Solutions

A linear(k+1)th-order discrete delayed equationΔx(n)=−p(n)x(n−k)wherep(n)a positive sequence is considered forn→∞. This equation is known to have a positive solution if the sequencep(n)satisfies an inequality. Our aim is to show that, in the case of the opposite inequality forp(n), all solutions of the equation considered are oscillating forn→∞.

A Note on an Oscillation Criterion for an Equation with a Functional Argument

1968 ◽  
Vol 11 (4) ◽  
pp. 593-595 ◽  
Author(s):  
Paul Waltman
Keyword(s):  
Oscillatory Solution ◽  
Oscillation Criterion ◽  
All Solutions ◽  
Image Position ◽  
Functional Argument ◽  
Oscillation Of Solutions

It might be thought that, as far as the oscillation of solutions is concerned, the behaviour ofandwould be the same as long as t - α(t) → ∞ as t→∞. To motivate the theorem presented in this note, we show first that this is not the case. Consider the above equation with α(t) = 3t/4, a(t) = l/2t2 i.e.This equation has the non-oscillatory solution y(t) = t1/2 although all solutions ofare oscillatory [1, p. 121].

scholarly journals Oscillation of Solutions of a Linear Second-Order Discrete-Delayed Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
J. Baštinec ◽  
J. Diblík ◽  
Z. Šmarda
Keyword(s):  
Second Order ◽  
Delayed Equation ◽  
Oscillation Of Solutions

scholarly journals Oscillation in second order functional equations with deviating arguments

1981 ◽  
Vol 4 (1) ◽  
pp. 137-146
Author(s):  
Bhagat Singh
Keyword(s):  
Positive Solution ◽  
Functional Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Deviating Arguments ◽  
All Solutions

For the pair of functional equations(A)(r(t)y′(t))+p(t)h(h(g(t)))=f(t)and(B)(r(t)y′(t))−p(t)h(y(g(t)))=0sufficient conditions have been found to cause all solutions of equation (A) to be oscillatory. These conditions depend upon a positive solution of equation (B).

On the Oscillation of Solutions of First Order Differential Equations with Retarded Arguments

2003 ◽  
Vol 10 (1) ◽  
pp. 63-76 ◽  
Author(s):  
M. K. Grammatikopoulos ◽  
R. Koplatadze ◽  
I. P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Integral Conditions ◽  
First Order ◽  
All Solutions ◽  
Oscillation Of Solutions ◽  
First Order Differential Equations

Abstract For the differential equation where 𝑝𝑖 ∈ 𝐿 loc (𝑅+; 𝑅+), τ 𝑖 ∈ 𝐶(𝑅+; 𝑅+), τ 𝑖(𝑡) ≤ 𝑡 for 𝑡 ∈ 𝑅+, , optimal integral conditions for the oscillation of all solutions are established.

scholarly journals Nonlinear Neutral Delay Differential Equations of Fourth-Order: Oscillation of Solutions

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 129 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Omar Bazighifan ◽  
Maria Alessandra Ragusa
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Neutral Delay ◽  
Neutral Differential Equations ◽  
All Solutions ◽  
Neutral Delay Differential Equations ◽  
Riccati Substitution ◽  
Delay Differential ◽  
Oscillation Of Solutions ◽  
Comparison Technique

The objective of this paper is to study oscillation of fourth-order neutral differential equation. By using Riccati substitution and comparison technique, new oscillation conditions are obtained which insure that all solutions of the studied equation are oscillatory. Our results complement some known results for neutral differential equations. An illustrative example is included.

scholarly journals On the global behavior of the nonlinear difference equationxn+1=f(pn,xn−m,xn−t(k+1)+1)

2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
Taixiang Sun ◽  
Hongjian Xi
Keyword(s):  
Difference Equation ◽  
Positive Solution ◽  
Sufficient Conditions ◽  
Positive Sequence ◽  
Nonlinear Difference Equation ◽  
Global Behavior ◽  
Initial Values

We consider the following nonlinear difference equation:xn+1=f(pn,xn−m,xn−t(k+1)+1),n=0,1,2,…, wherem∈{0,1,2,…}andk,t∈{1,2,…}with0≤m<t(k+1)−1, the initial valuesx−t(k+1)+1,x−t(k+1)+2,…,x0∈(0,+∞), and{pn}n=0∞is a positive sequence of the periodk+1. We give sufficient conditions under which every positive solution of this equation tends to the periodk+1solution.

scholarly journals On a Class of Nonautonomous Max-Type Difference Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Wanping Liu ◽  
Xiaofan Yang ◽  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Difference Equations ◽  
Positive Sequence ◽  
Finite Limit

This paper addresses the max-type difference equationxn=max{fn/xn−kα,B/xn−mβ},n∈ℕ0, wherek,m∈ℕ,B>0, and(fn)n∈ℕ0is a positive sequence with a finite limit. We prove that every positive solution to the equation converges tomax{(limn→∞fn)1/(α+1),B1/(β+1)}under some conditions. Explicit positive solutions to two particular cases are also presented.

scholarly journals ON OSCILLATION OF SOLUTIONS TO QUASI-LINEAR EMDEN – FOWLER TYPE HIGHER-ORDER DIFFERENTIAL EQUATIONS

2017 ◽  
Vol 21 (6) ◽  
pp. 12-22
Author(s):  
I.V. Astashova
Keyword(s):  
Differential Equation ◽  
Periodic Solutions ◽  
Nonlinear Equations ◽  
Order Differential Equation ◽  
Oscillatory Solutions ◽  
All Solutions ◽  
Even Order ◽  
And Behavior ◽  
Oscillation Of Solutions ◽  
Higher Order Differential Equations

Existence and behavior of oscillatory solutions to nonlinear equations with regular and singular power nonlinearity are investigated. In particular, the existence of oscillatory solutions is proved for the equation y(n) + P(x; y; y ′ ; : : : ; y(n−1))|y|k sign y = 0; n 2; k ∈ R; k 1; P ̸= 0; P ∈ C(Rn+1): A criterion is formulated for oscillation of all solutions to the quasilinear even-order differential equation y(n) + nΣ−1 i=0 aj(x) y(i) + p(x) |y|ksigny = 0; p ∈ C(R); aj ∈ C(R); j = 0; : : : ; n − 1; k 1; n = 2m; m ∈ N; which generalizes the well-known Atkinson’s and Kiguradze’s criteria. The existence of quasi-periodic solutions is proved both for regular (k 1) and singular (0 k 1) nonlinear equations y(n) + p0 |y|ksigny = 0; n 2; k ∈ R; k 0; k ̸= 1; p0 ∈ R; with (−1)np0 0: A result on the existence of periodic oscillatory solutions is formulated for this equation with n = 4; k 0; k ̸= 1; p0 0:

scholarly journals Subdominant positive solutions of the discrete equationΔu(k+n)=−p(k)u(k)

2004 ◽  
Vol 2004 (6) ◽  
pp. 461-470 ◽  
Author(s):  
Jaromír Baštinec ◽  
Josef Diblík
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Discrete Equation ◽  
Positive Coefficient ◽  
Open Questions ◽  
Existence Of Positive Solutions

A delayed discrete equationΔu(k+n)=−p(k)u(k)with positive coefficientpis considered. Sufficient conditions with respect topare formulated in order to guarantee the existence of positive solutions ifk→∞. As a tool of the proof of corresponding result, the method described in the author's previous papers is used. Except for the fact of the existence of positive solutions, their upper estimation is given. The analysis shows that every positive solution of the indicated family of positive solutions tends to zero (ifk→∞) with the speednot smaller than the speed characterized by the functionk·(n/(n+1))k. A comparison with the known results is given and some open questions are discussed.

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