Oscillatory Properties of the Solutions of First Order Linear Neutral Differential Equations with "Maxima"

Differential equations with maxima are special type of differential equations that contain the maximum. Though differential equations with maxima are often met in applications, the qualitative theory of these equations is relatively little developed. The existence of periodic solutions of these equations is considered in [1]. Our aim is to obtain new sufficient conditions for the oscillation of all the solutions of equation (1). By the solution of equation (1) we mean a continuous function x : [tx, ∞) → R such that x(t) p(t)x(σ(t)) is continuously differentiable and x(t) satisfies equation (1) for all sufficiently large t tx . The vanishing solutions of , q t q t dt 0 0 0 ! = 3 _ _ i i # , , lim t R 0 t 3 2 ! v a a = "3 _ i max x t p t x t q t x s 0 , t t v + + = a _ _ _ ` _ _ i i ij i i 9 7 C A all large t will be excluded from our consideration. The solution of (1) is called oscillatory if it has arbitrarily large zeros in [tx , ∞) and it is called nonoscillatory otherwise. The nonoscillating solutions of (1) are characterized as being eventually positive or eventually negative. Let us define the function


Introduction
We consider a first order linear neutral functional differential equation of the form , t Ն t 0 (1) Let the following conditions regarding functions p, σ, q be always assumed to hold: t → ∞ p : [t 0 , ∞) → R is continuous (2) σ : [t 0 , ∞) → R is continuous and stricly increasing, Differential equations with maxima are special type of differential equations that contain the maximum. Though differential equations with maxima are often met in applications, the qualitative theory of these equations is relatively little developed. The existence of periodic solutions of these equations is considered in [1]. Our aim is to obtain new sufficient conditions for the oscillation of all the solutions of equation (1). By the solution of equation (1) we mean a continuous function x : [t x , ∞) → R such that x(t) ϩ ϩ p(t)x(σ (t)) is continuously differentiable and x(t) satisfies equation (1) for all sufficiently large t Ն t x . The vanishing solutions of all large t will be excluded from our consideration. The solution of (1) is called oscillatory if it has arbitrarily large zeros in [t x , ∞) and it is called nonoscillatory otherwise. The nonoscillating solutions of (1) are characterized as being eventually positive or eventually negative. Let us define the function where t Ϫ τ ϭ σ (t). Then, from equation (1) we get .
The following lemmas will be useful in the proof of the main results. Lemmas can be found in [2,3].

Lemma 1.1 We consider equation
.
Then the following statements are true: the inequality a) has no eventually negative solution, or inequality c) has no eventually positive solution, d) has no eventually negative solution, and e) the following equation has only oscillatory solutions.

Lemma 1.2 Let the conditions (2) -(4) hold. Then the following statements are true:
a)If p(t) Յ Ϫ1 and (i) x(t) is an eventually positive solution of (1), then the function z(t) is an eventually decreasing function and z(t) Ͻ 0 eventually, (ii) x(t) is an eventually negative solution of (1), then the function z(t) is an eventually increasing function and z(t) Ͼ 0 eventually. b)If Ϫ1 Յ p(t) Յ 0 and (i) x(t) is an eventually positive solution of (1), then the function z(t) is an eventually increasing function and z(t) Ͼ 0 eventually, (ii) x(t) is an eventually negative solution of (1), then the function z(t) is an eventually decreasing function and z(t) Ͻ 0 eventually. # Proof. Let us suppose that equation (1) has a nonoscillatory solution x(t). Let x(t) Ͻ 0. From Lemma (1.2) we have that z(t) Ͼ Ͼ 0. Then, from (5) we get the inequality

Main results
from which we have and, finally . (8) Next, from Lemma (1.2) we know that z(t) is an eventually increasing function, then for sufficiently large t we have Then and From the last inequality and (8) we can deduce that . (9) Then from (6) and from (9) it follows that only the positive function z(t) satisfies the inequality (10) But from (7) and from Lemma (1.1) it follows that the last inequality (10) has no positive solutions. The contradiction obtained shows that equation (1) has no negative solutions, which is a contradiction.
Let x(t) Ͼ 0 From Lemma (1.2) it follows that z(t) Ͻ 0 As above, we obtain the estimate (11) Then from (6) and from (11) it follows that only the negative function satisfies the inequality (12) But from (7) and from Lemma (1.1) it follows that the last inequality (12) has no negative solutions. Hence equation (1) has no positive solutions, which was a contradiction.
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