Ch. G. Philos

Stable steady state of some population models

Applying an analytical method and several limiting equations arguments, some sufficient conditions are provided for the existence of a unique positive equilibriumK for the delay differential equationx=−γx+D(xt), which is the general form of many population models. The results are concerned with the global attractivity, uniform stability, and uniform asymptotic stability ofK. Application of the results to some known population models, which shows the effectiveness of the methods applied here, is also presented.

The dynamics of some discrete population models

IN THIS paper we are interested in providing sufficient conditions guaranteeing the fact that all positive solutions of the discrete delay difference equation A II+1 = AA, + F(A,_,) (E) converge as n + 00. In (E), A is a constant real number with 0 0 sufficiently small. Thus, formally, equation (E) comes from x(t + h) x(t) h = -yx(t) + D(x(t r)) for small h. If we set ~,z(f) := x(ht), then the preceding equation for t = n and r/h = m becomes yh@ + 1) = (1 ?‘h)Y, (n) + hD(Y,(n m)), which is of the form (E), where A := 1 yh and F(u) := hD(u), u 2 0. Clearly h > 0 is small if and only if A E (0, 1). 1069 @h)

Global attractivity in a nonlinear difference equation

Abstract We consider the (nonlinear) difference equation x n = a + ∑ k=1 m b k x n −k , n = 0,1,2… where a and bk (k = 1, 2, …, m) are nonnegative numbers with B ≡ ∑mk = 1bk > 0, and we are interested in whether all positive solutions are attracted by the positive equilibrium L = ( a 2 ) + ( a 2 ) 2 + B .

An asymptotic result for some delay difference equations with continuous variable

We consider a nonhomogeneous linear delay difference equation with continuous variable and establish an asymptotic result for the solutions. Our result is obtained by the use of a positive root with an appropriate property of the so called characteristic equation of the corresponding homogeneous linear (autonomous) delay difference equation. More precisely, we show that, for any solution, the limit of a specific integral transformation of it, which depends on a suitable positive root of the characteristic equation, exists as a real number and it is given explicitly in terms of the positive root of the characteristic equation and the initial function.

The behavior of solutions of linear integro-differential equations with unbounded delay

Abstract A basic theorem on the behavior of solutions of scalar linear integro-differential equations with unbounded delay is established. As a consequence of this theorem, a stability criterion is obtained.

A result on the behavior of the solutions for scalar first order linear autonomous neutral delay differential equations

Abstract. Scalar first order linear autonomous neutral delay differential equations are considered. A basic asymptotic criterion is obtained. Also, a nonoscillation result is derived. Moreover, a useful exponential estimate for the solutions is established and a stability criterion is given.

Oscillations in superlinear differential equations of second order

Abstract A new oscillation criterion is given for general superlinear ordinary differential equations of second order of the form x″(t) + a(t)f[x(t)] = 0, where a ϵ C([t 0 , ∞)), f ϵ C(R) with yf(y) > 0 for y ≠ 0 and ∝ ±1 ±∞ [ 1 f(y) ] dy and f is continuously differentiable on R − {0} with f ′( y ) ⩾ 0 for all y ≠ 0. The coefficient a is not assumed to be eventually nonnegative and the oscillation cirterion obtained involves the average behavior of the integral of a . In the special case of the differential equation x″(t) + a(t) ¦x(t)¦ λ sgn x(t) = 0 (λ > 1) this criterion improves a recent oscillation result due to Wong [Oscillation theorems for second-order nonlinear differential equations, Proc. Amer. Math. Soc. 106 (1989), 1069–1077].

On the behavior of the solutions for linear autonomous neutral delay difference equations

A class of linear autonomous neutral delay difference equations is considered, and some new results on the asymptotic behavior and the stability are given, via a positive root of the correspondng characteristic equation.

An oscillation criterion for first order linear delay differential equations

A new oscillation criterion is given for the delay differential equation x0(t) + p(t)x (t ú(t)) = 0, where p, ú 2 C ([0Ò1)Ò [0Ò1)) and the function T defined by T(t) = t ú(t), t 1⁄2 0 is increasing and such that limt!1 T(t) = 1. This criterion concerns the case where lim inf t!1 R t T(t) p(s) ds e . Received by the editors November 27, 1996. AMS subject classification: 34K15.

On second order sublinear oscillation

AbstractThe basic purpose of this paper is to present a new oscillation criterion for second order sublinear ordinary differential equations of the formx″(t) +a(t)f[x(t)] = 0,t ≧t0>0, wherea is a continuous function on [t0, ∞) without any restriction on its sign andf is a continuous function on the real line, which is continuously differentiable, except possibly at 0, and satisfiesyf(y)>0 andf′(y)>0 fory ≠ 0, and