ON OSCILLATION OF A FOOD-LIMITED POPULATION MODEL WITH TIME DELAY

is known as Hutchinson’s equation, if r and K are positive constants and h(t)= t− τ for a positive constant τ. Hutchinson’s equation has been investigated by several authors (see, e.g., [13, 14, 18, 23]). Delay logistic equation (1.1) was studied by Gopalsamy and Zhang [7, 25] who gave sufficient conditions for the oscillation and the nonoscillation of (1.1). Publications [1, 2, 3, 4, 5, 6, 10, 12, 15, 16, 17, 19, 22, 24] are devoted to various generalizations of logistic equation (1.1). In 1963, Smith [20] proposed an alternative to the logistic equation for a food-limited population

In 1963, Smith [20] proposed an alternative to the logistic equation for a food-limited population Here N, r, and K are the mass of the population, the rate of increase with unlimited food, and value of N at saturation, respectively.The constant 1/c is the rate of replacement of mass in the population at saturation (this includes both the replacement of metabolic loss and of dead organisms).
In [8,9,11], Gopalsamy, Kulenovic, Ladas, Grove, and Qian considered the autonomous delay food-limited equation So and Yu [21] investigated stability properties of the following nonlinear differential equation with a constant delay: which is a generalization of food-limited equations (1.2) and (1.3).
In this paper, we consider oscillation properties of a nonautonomous foodlimited equation with a nonconstant delay which also generalizes (1.3).We compare oscillation properties of (1.5) and some linear delay differential equations.As a corollary, we obtain explicit oscillation and nonoscillation conditions for (1.5).For the autonomous equation (1.3), our conditions and the known ones in [8] coincide.
We also consider two generalizations of (1.5), the first one is (1.4) with a nonconstant delay and the second one is (1.5) with several delays.
Our proof of the main result is based on some application of Schauder's fixedpoint theorem which was employed for a generalized logistic equation in [4].According to this method, the differential equation is transformed into an operator equation where operator A is a monotone increasing operator and B is a monotone decreasing one.We prove that there exist two functions v, w, Then operator Tu = AuBu acts in the interval v(t) ≤ u(t) ≤ w(t) and therefore, we can use Schauder's fixedpoint theorem.Functions w and v are the limits of two sequences {w n } and {v n }, respectively, and for the construction of the first approximation w 1 , we apply a positive solution of some linear delay differential equation.The paper is organized as follows.In Sections 2 and 3, we consider an equation which is obtained from (1.5) by the following substitution: On the base of these results, in Section 4, we investigate generalized delay logistic equation (1.5).

Preliminaries
Consider a scalar delay differential equation under the following assumptions: (A1) r(t) and s(t) are Lebesgue measurable locally essentially bounded functions, r(t) ≥ 0 and s(t Together with (2.1), we consider for each t 0 ≥ 0 an initial value problem , t ≥ t 0 , (2.2) We also assume that the following hypothesis holds: Equation (2.1) has a nonoscillatory solution if it has an eventually positive or an eventually negative solution.Otherwise, all solutions of (2.1) are oscillatory.
We present here Lemma 2.2 which will be used in the proof of the main results.

Oscillation conditions
In this section and Section 4, we assume that (A1), (A2), and (A3) hold and consider only such solutions of (2.1) for which the following condition holds: We begin with the following lemma.
Proof.First suppose x(t) is an eventually positive solution of (2.1).Lemma 3.1 implies that there exists t 1 ≥ 0 such that 0 < x(t) < for t ≥ t 1 .We suppose (3.3) holds for t ≥ t 2 ≥ t 1 .For t ≥ t 2 , we have Lemma 2.2 yields that (3.9) has a nonoscillatory solution.We have a contradiction.
60 Oscillation of a food-limited model Now suppose − < x(t) < 0 for t ≥ t 1 and (3.3) holds for t ≥ t 2 ≥ t 1 .Then for Hence, (3.9) has a nonoscillatory solution and we again obtain a contradiction which completes the proof.
Theorem 3.4.Suppose for some > 0 there exists a nonoscillatory solution of the linear delay differential equation Then there exists a nonoscillatory solution of (2.1).

L. Berezansky and E. Braverman 61
Hence by induction, There exist pointwise limits of nonincreasing nonnegative sequence w n (t) and of nondecreasing sequence v n (t).If we denote w(t) = lim n→∞ w n (t) and v(t) = lim n→∞ v n (t), then by the Lebesgue Convergence theorem, we conclude that We fix b ≥ t 0 and define operator T : where L ∞ [t 0 ,b] is the space of all essentially bonded on [t 0 ,b] functions with the usual norm.
For every function u from the interval v ≤ u ≤ w, we have v ≤ Tu ≤ w.The result of [4, Lemma 3] implies that operator T is a compact operator on the space L ∞ [t 0 ,b].Then by Schauder's fixed-point theorem there exists a nonnegative solution of equation u = Tu.Denote where the parameters of this equation satisfy conditions (A1) and (A2), K > 0, and the initial function ψ satisfies (A3).Similar to the case m = 1, the following generalizations of Theorems 4.1 and 4.2 can be obtained.Theorem 4.7.Suppose (3.2) holds and for some > 0 all solutions of the linear equationẋ(t) + (1 − )Then there exists a nonoscillatory about K solution of (4.12).
then(4.12)has a nonoscillatory about K solution.