Subexponential solutions of linear integro-differential equations and transient renewal equations

2002 ◽  
Vol 132 (3) ◽  
pp. 521-543 ◽  
Author(s):  
John A. D. Appleby ◽  
David W. Reynolds
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Rate Of Convergence ◽  
Renewal Equation ◽  
Asymptotically Stable ◽  
Renewal Equations ◽  
Integro Differential Equation ◽  
All Solutions ◽  
Image Position

This paper studies the asymptotic behaviour of the solutions of the scalar integro-differential equation The kernel k is assumed to be positive, continuous and integrable.If it is known that all solutions x are integrable and x(t) → 0 as t → ∞, but also that x = 0 cannot be exponentially asymptotically stable unless there is some γ > 0 such that Here, we restrict the kernel to be in a class of subexponential functions in which k(t) → 0 as t → ∞ so slowly that the above condition is violated. It is proved here that the rate of convergence of x(t) → 0 as t → ∞ is given by The result is proved by determining the asymptotic behaviour of the solution of the transient renewal equation If the kernel h is subexponential, then

On the asymptotic behaviour of nonlinear systems of ordinary differential equations

1985 ◽  
Vol 26 (2) ◽  
pp. 161-170
Author(s):  
Zhivko S. Athanassov
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Ordinary Differential Equations ◽  
Asymptotically Stable ◽  
Future Time ◽  
Zero Function ◽  
Perturbed Systems ◽  
Approach Zero ◽  
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In this paper we study the asymptotic behaviour of the following systems of ordinary differential equations:where the identically zero function is a solution of (N) i.e. f(t, 0)=0 for all time t. Suppose one knows that all the solutions of (N) which start near zero remain near zero for all future time or even that they approach zero as time increases. For the perturbed systems (P) and (P1) the above property concerning the solutions near zero may or may not remain true. A more precise formulation of this problem is as follows: if zero is stable or asymptotically stable for (N), and if the functions g(t, x) and h(t, x) are small in some sense, give conditions on f(t, x) so that zero is (eventually) stable or asymptotically stable for (P) and (P1).

Oscillatory and asymptotic behaviour of all solutions of differential equations with deviating arguments

1978 ◽  
Vol 81 (3-4) ◽  
pp. 195-210 ◽  
Author(s):  
Ch. G. Philos
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Unknown Function ◽  
Continuous Functions ◽  
Linear Differential Equations ◽  
Deviating Arguments ◽  
All Solutions ◽  
Non Linear ◽  
Linear Differential ◽  
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SynopsisThis paper deals with the oscillatory and asymptotic behaviour of all solutions of a class of nth order (n > 1) non-linear differential equations with deviating arguments involving the so called nth order r-derivative of the unknown function x defined bywhere r1, (i = 0,1,…, n – 1) are positive continuous functions on [t0, ∞). The results obtained extend and improve previous ones in [7 and 15] even in the usual case where r0 = r1 = … = rn–1 = 1.

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 ◽  
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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.

Oscillations in Higher-Order Neutral Differential Equations

1993 ◽  
Vol 45 (1) ◽  
pp. 132-158 ◽  
Author(s):  
CH. G. Philos ◽  
I. K. Purnaras ◽  
Y. G. Sficas
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Characteristic Equation ◽  
Higher Order ◽  
Neutral Differential Equation ◽  
Neutral Differential Equations ◽  
Real Roots ◽  
All Solutions ◽  
Image Position

AbstractConsider the n-th order (n ≥ 1 ) neutral differential equation where σ1 < σ 2 < ∞ and μ and η are increasing real-valued functions on [Ƭ1, Ƭ2] and [σ1, σ2] respectively. The function μ is assumed to be not constant on [Ƭ1, Ƭ2] and [Ƭ1, Ƭ2] for every Ƭ ∈ (Ƭ1, Ƭ2) similarly, for each σ ∈ (σ1, σ2), it is supposed that r\ is not constant on [σ1 , σ] and [σ, σ2]. Under some mild restrictions on Ƭ1,- and σ1, (ι = 1,2), it is proved that all solutions of (E) are oscillatory if and only if the characteristic equation of (E) has no real roots.

scholarly journals Linear differential equations with unbounded delays and a forcing term

2004 ◽  
Vol 2004 (4) ◽  
pp. 337-345 ◽  
Author(s):  
Jan Čermák ◽  
Petr Kundrát
Keyword(s):  
Differential Equation ◽  
Functional Equation ◽  
Continuous Function ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Functional Equations ◽  
Continuous Functions ◽  
Positive Continuous Function ◽  
All Solutions ◽  
Linear Differential

The paper discusses the asymptotic behaviour of all solutions of the differential equationy˙(t)=−a(t)y(t)+∑i=1nbi(t)y(τi(t))+f(t),t∈I=[t0,∞), with a positive continuous functiona, continuous functionsbi,f, andncontinuously differentiable unbounded lags. We establish conditions under which any solutionyof this equation can be estimated by means of a solution of an auxiliary functional equation with one unbounded lag. Moreover, some related questions concerning functional equations are discussed as well.

Oscillations of retarded differential equations

1974 ◽  
Vol 75 (1) ◽  
pp. 95-101 ◽  
Author(s):  
Y. G. Sficas ◽  
V. A. Staikos
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Half Line ◽  
Retarded Differential Equation ◽  
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Retarded Differential Equations

In this paper we are dealing with the oscillatory and asymptotic behaviour of the nth order, (n > 1), retarded differential equationwhere g is differentiable on the half-line (t0,∞) with (I) g(t) ≤ t for every t ≥ t0 (II) g'(t) ≥ 0 for every t ≥ t0 (III) = ∞.

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).

Asymptotic Behaviour of Disconjugate nth Order Differential Equations

1971 ◽  
Vol 23 (2) ◽  
pp. 293-314 ◽  
Author(s):  
D. Willett
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Linear Differential Equation ◽  
Ordered Set ◽  
Minimal Solution ◽  
Linear Differential ◽  
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An ordered set (u1, …, un) of positive Cn(a, b)-solutions of the linear differential equation1.1will be called fundamental principal system on [a, b) provided that1.2and1.3A system (u1, …, un) satisfying just (1.2) will be called a principal system on [a, b). In any principal system (u1, …, un) the solution u1 will be called a minimal solution.

Oscillation in Differential Equations with Positive and Negative Coefficients

1990 ◽  
Vol 33 (4) ◽  
pp. 442-451 ◽  
Author(s):  
G. Ladas ◽  
C. Qian
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Neutral Differential Equations ◽  
Positive And Negative Coefficients ◽  
All Solutions ◽  
Linear Delay ◽  
Image Position ◽  
Delay Differential

AbstractWe obtain sufficient conditions for the oscillation of all solutions of the linear delay differential equation with positive and negative coefficientswhereExtensions to neutral differential equations and some applications to the global asymptotic stability of the trivial solution are also given.

The asymptotic behaviour of the solutions of the Kassoy problem with a modified source term

1989 ◽  
Vol 113 (3-4) ◽  
pp. 347-356 ◽  
Author(s):  
C. Budd ◽  
Y. Qi
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Asymptotic Behaviour ◽  
Blow Up ◽  
Similarity Solutions ◽  
Parabolic Partial Differential Equation ◽  
Equation Problem ◽  
All Solutions ◽  
Image Position ◽  
Semilinear Parabolic

SynopsisWe study the asymptotic behaviour as x →∞ of the solutions of the ordinary differential equation problemThis equation generalises the ordinary differential equation obtained by studying the blow-up of the similarity solutions of the semilinear parabolic partial differential equation vt=vxx = ev. We show that if λ≦1, all solutions of (*) tend to —∞ as rapidly as the function —exp (x2/4) (E- solutions). However, if λ>1, then there also exists a solution which tends to –∞, like 2λlog(x) (L-solutions). Thus, the case λ = 1, for which (*) reduces tothe Kassoy equation, is the borderline between two quite different forms of asymptotic behaviour of the function u(x).

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