scholarly journals Oscillatory and asymptotic behaviour of second order nonlinear difference equations

1996 ◽  
Vol 39 (3) ◽  
pp. 525-533 ◽  
Author(s):  
Bing Liu ◽  
Jurang Yan
Keyword(s):  
Asymptotic Behaviour ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Nonlinear Difference Equations ◽  
Nonoscillatory Solutions ◽  
Asymptotic Behaviour Of Solutions ◽  
All Solutions ◽  
Image Position

In this paper we are dealing with oscillatory and asymptotic behaviour of solutions of second order nonlinear difference equations of the formSome sufficient conditions for all solutions of (E) to be oscillatory are obtained. Asymptotic behaviour of nonoscillatory solutions of (E) is considered also.

On the asymptotic behaviour of solutions of x″ + a(t)f(x) = 0

1971 ◽  
Vol 70 (1) ◽  
pp. 77-88 ◽  
Author(s):  
T. Burton ◽  
R. Grimmer
Keyword(s):  
Asymptotic Behaviour ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Asymptotic Behaviour Of Solutions ◽  
All Solutions ◽  
Image Position ◽  
Necessary And Sufficient

We consider the equation:where a: [0, ∞) → R1, a(t) > 0, a'(t) is continuous,f:( −∞, +∞) → R1, f is continuous, and xf(x) > 0 for x ≠ 0. The problem is to give conditions on a(t) and f(x) to ensure that all solutions of (1) tend to zero as t → ∞. First, however, we give some sufficient conditions and some necessary and sufficient conditions to ensure that all solutions of (1) are oscillatory or bounded.

Oscillatory and asymptotic behaviour of solutions of a class of linear ordinary differential equations

1981 ◽  
Vol 90 (1-2) ◽  
pp. 25-40 ◽  
Author(s):  
Takaŝi Kusano ◽  
Manabu Naito ◽  
Kyoko Tanaka
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Ordinary Differential Equations ◽  
Sufficient Conditions ◽  
Solution Space ◽  
Extreme Situation ◽  
Nonoscillatory Solutions ◽  
Linear Ordinary Differential Equations ◽  
Image Position ◽  
Monotone Solutions

SynopsisThe equation to be considered iswhere pi(t), 0≦i≦n, and q(t) are continuous and positive on some half-line [a, ∞). It is known that (*) always has “strictly monotone” nonoscillatory solutions defined on [a, ∞), so that of particular interest is the extreme situation in which such strictly monotone solutions are the only possible nonoscillatory solutions of (*). In this paper sufficient conditions are given for this situation to hold for (*). The structure of the solution space of (*) is also studied.

scholarly journals Neutral Difference System and its Nonoscillatory Solutions

2018 ◽  
Vol 71 (1) ◽  
pp. 139-148
Author(s):  
Jana Pasáčková
Keyword(s):  
Difference Equations ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Difference System ◽  
Nonlinear Difference Equations ◽  
Nonoscillatory Solutions ◽  
Weak Property ◽  
Property B

Abstract The paper deals with a system of four nonlinear difference equations where the first equation is of a neutral type. We study nonoscillatory solutions of the system and we present sufficient conditions for the system to have weak property B.

scholarly journals Remarks on Stability Conditions for the Differential Equation x″ + a(t)ƒ(x) = 0

1969 ◽  
Vol 9 (3-4) ◽  
pp. 496-502 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Differential Equation ◽  
Continuous Function ◽  
Nonlinear Differential Equation ◽  
Real Axis ◽  
Sufficient Conditions ◽  
Second Order ◽  
Stability Conditions ◽  
Negative Real Axis ◽  
All Solutions ◽  
Image Position

Consider the following second order nonlinear differential equation: where a(t) ∈ C3[0, ∞) and f(x) is a continuous function of x. We are here concerned with establishing sufficient conditions such that all solutions of (1) satisfy (2) Since a(t) is differentiable and f(x) is continuous, it is easy to see that all solutions of (1) are continuable throughout the entire non-negative real axis. It will be assumed throughout that the following conditions hold: Our main results are the following two theorems: Theorem 1. Let 0 < α < 1. If a(t) satisfieswhere a(t) > 0, t ≧ t0 and = max (−a′(t), 0), andthen every solution of (1) satisfies (2).

scholarly journals Oscillation criteria for certain second order nonlinear difference equations

1999 ◽  
Vol 60 (1) ◽  
pp. 95-108 ◽  
Author(s):  
S.R. Grace ◽  
H.A. El-Morshedy
Keyword(s):  
Difference Equations ◽  
Difference Operator ◽  
Nonnegative Function ◽  
Second Order ◽  
Real Sequence ◽  
Nonlinear Difference Equations ◽  
Oscillation Criteria ◽  
Forward Difference ◽  
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This paper is concerned with nonlinear difference equations of the formwhere δ is the forward difference operator defined by δun−1 = un − un −1 δ2un −1= δ(δun-1) and {an} is a real sequence which is not assumed to be nonnegative. The function f is such that uf(u) < 0 for all u ≠ 0 and f(u) − f(v) = g(u, v)(u − v), for all u, v ≠ 0, and for some nonnegative function g. Our results are not only new but also improve and generalise some recent oscillation criteria. Examples illustrating the importance of our main results are also given.

scholarly journals Asymptotic behavior of nonoscillatory solutions of second order functional differential equations

1975 ◽  
Vol 13 (2) ◽  
pp. 291-299 ◽  
Author(s):  
Takaŝi Kusano ◽  
Hiroshi Onose
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Nonoscillatory Solutions ◽  
Oscillatory Solutions ◽  
Functional Differential ◽  
Approach Zero ◽  
Image Position

The asymptotic behavior of nonoscillatory solutions of the second order functional differential equationis studied. First, in the case when a(t)is oscillatory, sufficient conditions are given in order that all bounded non-oscillatory solutions of (*) approach zero as t → ∞. Secondly, in the case when a(t) is nonnegative, conditions are provided under which all nonoscillatory solutions of (*) tend to zero as t → ∞.

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