On Oscillation of Solutions of Differential Equations with Distributed Delay

We obtain sufficient conditions for oscillation of solutions to a linear differential equation with distributed delay. We construct examples showing that constants in the conditions are unimprovable.


Introduction
The problem of definite-sign solutions and the opposite problem of oscillating solutions (having an unbounded sequence of zeros, from the right) for ordinary differential equations are well known and significant.These problems for functional differential equations are nontrivial even for first-order equations, whose solutions, as is known, can have zeros and oscillate.
In particular, the problem of conditions for the existence of oscillating solutions for the equation ẋ(t) + a(t)x(h(t)) = 0, t 0, ( has been studied in detail.We cite the two most known and well-supplementing each other conditions for the oscillation of solutions to equation (1.1).
The first condition goes back to paper [17].Later it was generalized in [12,14,21], and took the following complete form in [11].
The constant 1/e is unimprovable.If a(t) ≡ a = const, h(t) = t − r, where r = const, then the condition ar > 1/e is necessary and sufficient for the oscillation of every solution of equation (1.1).
The first variants of the other condition were obtained in papers [13,22].Its most general form, obtained in [7], is the following.
Since all solutions of equation (2.1) are continuous, it follows from definition 2.1 that a solution which is not oscillatory has definite sign everywhere to the right from some point.Such solutions are said to be definite-sign.Using the linearity of equation (2.1) we can say, without loss of generality, that a solution is definite-sign if it is positive starting from some point.
In order to obtain conditions for oscillation, we use a proposition known as the lemma on differential inequality.The lemma occurs in papers [2, p. 57, Lemma 2.4.3],[3,6,9] in different equivalent reformulations.Here we formulate it in the form suitable for us in connection with equation (2.1).

Lemma 2.3. If there exists an absolutely continuous function v and a number T
0 such that v(t) > 0 and (Lv)(t) 0 for all t T, then equation (2.1) has a definite-sign solution.

Autonomous equations
We begin obtaining oscillation conditions with autonomous equations.Suppose in equation (2.1) where k is a locally summable function.We get the equation The function F(λ) = −λ + r p k(t)e λt dt, F : C → C, is said to be the characteristic function of equation (3.1).It was shown in [19,Lemma 3.1] that F is analytic everywhere in C and has a countable set of roots to the left from every vertical line Re λ = const, the set of roots of F in every vertical band being finite.Note that roots of F depend on the parameters p and r continuously.Proof.If the function F has a real root λ = λ 0 , then the function x(t) = e λ 0 t is a positive solution of equation (3.1).Hence, equation (3.1) is not oscillatory.Conversely, suppose that the function F has no real roots.Then it has a finite number n of roots whose real part is maximal.Denote them by λ j , j = 1, . . ., n. Denote α max = Re λ j , β j = Im λ j .
Consider an arbitrary solution of (3.1).It is known (see [23]) that it has the form where A j (t) and B j (t) are polynomials, and lim t→+∞ |z(t)|e −α max t = 0. Denote by m 0 the greatest degree of A j (t), B j (t), j = 1, . . ., n.Then we have We have β 1 < β j for all j 2. Therefore there exists a sufficiently large m such that the inequality R 1 So, the function y has an infinite set of roots and extrema in R + , with maxima and minima of y bounded away from zero uniformly.By the mean value theorem, all the derivatives of y possess these properties (and w does, since y (4m) (t) = w(t)).Therefore x is oscillatory.
ds 0. However, F(0) > 0. Thus, the characteristic function of equation (3.2) has a real root.).As is noted above, in this case the equation ) .The function u = ψ(u, v) can be interpreted as a surface, in the space Ouvw, which is the boundary of the region of oscillation.Its graph created by a computer is presented on Fig. 3.1.Theorem 3.2 now obtains a geometric sense: equation (3.2) is oscillatory if and only if the point (µr α+2 , α, q) is above the surface u = ψ(v, w).
Putting p = 0 in (3.2), we get the equation Evidently, q = 0 for (3.5).Therefore the criterion of oscillation is simplified.
Suppose M(v 0 , u) ∈ D. Then u > u 0 and f (ζ) > 0 for all ζ ∈ R, i.e., the function f has no real roots.Suppose M(v 0 , u) ∈ D. Then u u 0 .Using (3.8), we get f (ζ * ) 0. Since lim ζ→+0 f (ζ) = u > 0, we see that the function f has a real root.Since the point M 0 on the curve ϕ is taken arbitrarily, the lemma is proved.Thus the fact that µr 2 is on the axis Ou for u > 2ζ 0 e −ζ 0 corresponds to the oscillation of equation (3.7) under the conditions of Corollary 3.6.
Remark 3.7.The set of oscillation for equation (3.7) is the complement to the set of positiveness for the fundamental solution.The common boundary of the sets is the curve u = ϕ(v) obtained in paper [16], which is devoted to the study of the positiveness of the fundamental solution for equation (3.7).

Nonautonomous equations
Using Lemma 2.3 and the results of Section 3, we can obtain oscillation conditions for some classes of nonautonomous equations with distributed delay.Proof.First let us prove that ∞ 0 a(s) ds = ∞.We have lim t→∞ h(t) = ∞ and g(t) h(t), hence lim t→∞ g(t) = ∞.Therefore, if the function a is summable on the real positive semiaxis, then p = lim t→∞ t g(t) a(s) ds = 0.But this is impossible, since the axis Ov is not included in D. Denote ϕ(t) = t 0 a(s) ds.The function ϕ is a continuous and increasing R + -onto-R + map.Hence there exists the inverse function ϕ −1 defined on R + .By the change of variables (analogous to that applied in [15]) τ = ϕ(t), ζ = ϕ(s), x ϕ −1 (τ) = y(τ), equation (2.1) is reduced to the form where a(s) ds. Since a(s) ds = p, and Corollary 4.3 can be applied to equation (4.2).This implies that every solution of equation (4.2) oscillates.We have x(t) = y(ϕ(t)), so every solution of equation (2.1) also oscillates.h(t) K(t, s) ds = 1 and lim t→∞ (t − g(t)) = m > 1/e, then equation (2.1) is oscillatory.
Proof.Assume that there exists a definite-sign solution v = v(t) of equation (2.1).Thus, by virtue of the equation, there exists T > 0 such that for all t T the inequalities v(t) > 0 and v(t) 0 hold.Hence By Lemma 2.3, the equation ẋ(t) + x(t − m) = 0 has a definite-sign solution.Therefore m 1/e.This contradiction completes the proof.
Proof.Let us prove that ∞ 0 ρ(s) ds = ∞ under the conditions of Theorem 4.8.Since lim t→∞ h(t) = ∞ and g(t) h(t), we obtain lim t→∞ g(t) = ∞.Assume that e for sufficiently large t.Hence the assumption is not true.

Theorem 3 . 1 .
Equation (3.1) is oscillatory if and only if the function F has no real roots.

Remark 4 . 6 .
The oscillation region D defined by Theorem 4.5 is sharp, since Theorem 4.5 coincides with Theorem 3.5 in the case of constant coefficients and delays.

Table 3 .
1: Criteria of the oscillation of solutions for equation (3.5).

Theorem 4.1. For
equation (2.1), suppose that K(t, s) k(t − s) 0, where k is a locally summable function, lim t→∞ (t − h(t)) = r, lim t→∞ (t − g(t)) = p, r > p If the function K(t, s) is bounded below by a nonzero constant, then oscillation conditions for equation (2.1) can be conveniently formulated in terms of the belonging of a given point to the set D. For (2.1), suppose that K(t, s) µ, lim t→∞ (t − h(t)) = r, lim t→∞ (t − g(t)) = p, r > p 0, and the point µ(r − p) 2 , p r−p belongs to D. Then equation (2.1) is oscillatory.For (4.1), suppose that K(t, s) µ, lim t→∞(t − h(t)) = r, and µr 2 > 2ζ 0 e −ζ 0 , where ζ 0 is the positive root of the equation 1 − ζ 2 = e −ζ .Then equation (4.1) is oscillatory.The nearer equation (2.1) is to the autonomous equation (3.1), the sharper Corollaries 4.2-4.4are.For equation (3.1) sufficient oscillation conditions become necessary and sufficient.The three propositions stated below (Theorem 4.5, Theorem 4.8, Condition 4.11) can be regarded as different variants of Condition 1.1.Each of them has its area of application.) ds = r, and the point (r − p) 2 , p r−p belongs to the set D. Then equation (2.1) is oscillatory.
+ to R + bijectively, equation (2.1) is also oscillatory.Let us show that the constant 1/e is sharp in the inequality (4.3) .
then equation (2.1) is oscillatory.We will show that the constant 1/e in Condition 4.11 is also sharp.α5Analog of Condition 1.2 for equations with distributed delayLet t ∈ R + .Define E(t) = {s : h(s) t g(s)}.Corollary 5.4.