Bounded oscillations under the effect of retardations for differential equations of arbitrary order

1977 ◽  
Vol 77 (1-2) ◽  
pp. 129-136 ◽  
Author(s):  
V. A. Staikos ◽  
I. P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Arbitrary Order ◽  
Bounded Solution ◽  
Sufficient Conditions ◽  
Bounded Solutions ◽  
Image Position

SynopsisThis paper is concerned with the effect of the delays on the bounded solutions of the nth order n ≧ 1) differential equationSufficient conditions involving the retardations τJ(j = 1,2, …, m) which insure that every bounded solution of the considered equation is oscillatory are given. The results obtained generalise recent ones in [3 and 4].

On Stability of Solutions of Certain Differential Equations of the Third Order

1967 ◽  
Vol 10 (5) ◽  
pp. 681-688 ◽  
Author(s):  
B.S. Lalli
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Third Order ◽  
The Third ◽  
Image Position

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation1.1using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.

scholarly journals Bifurcation of Bounded Solutions of Impulsive Differential Equations

2016 ◽  
Vol 26 (14) ◽  
pp. 1650242 ◽  
Author(s):  
Kevin E. M. Church ◽  
Xinzhi Liu
Keyword(s):  
Differential Equation ◽  
Integral Operator ◽  
Differential Equations ◽  
Impulsive Differential Equation ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Necessary Condition ◽  
Bounded Solutions ◽  
Pitchfork Bifurcations ◽  
Nonlinear Integral Operator

In this article, we examine nonautonomous bifurcation patterns in nonlinear systems of impulsive differential equations. The approach is based on Lyapunov–Schmidt reduction applied to the linearization of a particular nonlinear integral operator whose zeroes coincide with bounded solutions of the impulsive differential equation in question. This leads to sufficient conditions for the presence of fold, transcritical and pitchfork bifurcations. Additionally, we provide a computable necessary condition for bifurcation in nonlinear scalar impulsive differential equations. Several examples are provided illustrating the results.

scholarly journals On the positive solutions of a higher order functional differential equation with a discontinuity

1982 ◽  
Vol 5 (2) ◽  
pp. 263-273 ◽  
Author(s):  
John R. Graef ◽  
Paul W. Spikes ◽  
Myron K. Grammatikopoulos
Keyword(s):  
Differential Equation ◽  
Functional Differential Equation ◽  
Bounded Solution ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Bounded Solutions ◽  
Nonlinear Functional ◽  
Functional Differential ◽  
Necessary And Sufficient ◽  
Special Case

Then-th order nonlinear functional differential equation[r(t)x(n−υ)(t)](υ)=f(t,x(g(t)))is considered; necessary and sufficient conditions are given for this equation to have: (i) a positive bounded solutionx(t)→B>0ast→∞; and (ii) all positive bounded solutions converging to0ast→∞. Other results on the asymptotic behavior of solutions are also given. The conditions imposed are such that the equation with a discontinuity[r(t)x(n−υ)(t)](υ)=q(t)x−λ,   λ>0is included as a special case.

scholarly journals Oscillation of a functional differential equation arising from an industrial problem

1978 ◽  
Vol 26 (3) ◽  
pp. 323-329 ◽  
Author(s):  
Hiroshi Onose
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
First Order ◽  
Functional Differential ◽  
Image Position ◽  
Industrial Problem

AbstractIn the last few years, the oscillatory behavior of functional differential equations has been investigated by many authors. But much less is known about the first-order functional differential equations. Recently, Tomaras (1975b) considered the functional differential equation and gave very interesting results on this problem, namely the sufficient conditions for its solutions to oscillate. The purpose of this paper is to extend and improve them, by examining the more general functional differential equation

scholarly journals Some further results on oscillation of neutral differential equations

1992 ◽  
Vol 46 (1) ◽  
pp. 149-157 ◽  
Author(s):  
Jianshe Yu ◽  
Zhicheng Wang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Variable Coefficients ◽  
Neutral Differential Equation ◽  
Neutral Differential Equations ◽  
All Solutions ◽  
Image Position

We obtain new sufficient conditions for the oscillation of all solutions of the neutral differential equation with variable coefficientswhere P, Q, R ∈ C([t0, ∞), R+), r ∈ (0, ∞) and τ, σ ∈ [0, ∞). Our results improve several known results in papers by: Chuanxi and Ladas; Lalli and Zhang; Wei; Ruan.

scholarly journals Asymptotic behavior of nonoscillatory solutions of a higher order functional differential equation

1981 ◽  
Vol 24 (1) ◽  
pp. 85-92 ◽  
Author(s):  
Hiroshi Onose
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Nonoscillatory Solutions ◽  
Nonlinear Functional ◽  
Functional Differential ◽  
Approach Zero ◽  
Image Position

The asymptotic behavior of nonoscillatory solutions of nth order nonlinear functional differential equationsis investigated. Sufficient conditions are provided which ensure that all nonoscillatory solutions approach zero as t → ∞.

scholarly journals Nonoscillation theorems for functional differential equations of arbitrary order

1984 ◽  
Vol 7 (2) ◽  
pp. 249-256 ◽  
Author(s):  
John R. Graef ◽  
Myron K. Grammatikopoulos ◽  
Yuichi Kitamura ◽  
Takasi Kusano ◽  
Hiroshi Onose ◽  
...  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Arbitrary Order ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Oscillatory Solutions ◽  
Nonlinear Functional ◽  
Functional Differential ◽  
Nonoscillation Theorem

The authors give sufficient conditions for all oscillatory solutions of a sublinear forced higher order nonlinear functional differential equation to converge to zero. They then prove a nonoscillation theorem for such equations. A few intermediate results are also obtained.

A further instability theorem for a certain fifth-order differential equation

1979 ◽  
Vol 86 (3) ◽  
pp. 491-493 ◽  
Author(s):  
J. O. C. Ezeilo
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Trivial Solution ◽  
Constant Coefficient ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Non Linear ◽  
Image Position ◽  
Fifth Order

In our previous consideration in (1) of the constant-coefficient fifth-order differential equation:an attempt was made to identify (though not exhaustively) different sufficient conditions on a1,…,a5 for the instability of the trivial solution x = 0 of (1·1). It was our expectation that the conditions so identified could be generalized in some form or other to equations (1·1) in which a1,…,a5 were not necessarily constants, thereby giving rise to instability theorems for some non-linear fifth-order differential equations; and this turned out in fact to be so except only for the case:with R0 = R0(a1, a2, a3, a4) > 0 sufficiently large, about which we were unable at the time to derive any worthwhile generalization to any equation (1·1) in which a1, …,a5 are not all constants.

scholarly journals Oscillations of higher order neutral differential equations

1992 ◽  
Vol 52 (2) ◽  
pp. 261-284 ◽  
Author(s):  
S. J. Bilchev ◽  
M. K. Grammatikopoulos ◽  
I. P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Neutral Differential Equations ◽  
Real Roots ◽  
All Solutions ◽  
Image Position ◽  
Necessary And Sufficient

AbstractConsider the nth-order neutral differential equation where n ≥ 1, δ = ±1, I, K are initial segments of natural numbers, pi, τi, σk ∈ R and qk ≥ 0 for i ∈ I and k ∈ K. Then a necessary and sufficient condition for the oscillation of all solutions of (E) is that its characteristic equation has no real roots. The method of proof has the advantage that it results in easily verifiable sufficient conditions (in terms of the coefficients and the arguments only) for the oscillation of all solutionso of Equation (E).

scholarly journals Oscillations for first order neutral differential equations with variable coefficients

1991 ◽  
Vol 43 (1) ◽  
pp. 147-152 ◽  
Author(s):  
Shigui Ruan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Variable Coefficients ◽  
Neutral Differential Equation ◽  
Neutral Differential Equations ◽  
First Order ◽  
Image Position

In this paper, sufficient conditions for oscillations of the first order neutral differential equation with variable coefficientsare obtained, where c, τ, σ and µ are positive constants, p, q ∈ C ([t0, ∞), R+).

Sign in / Sign up

Export Citation Format

Share Document