OSCILLATION CRITERIA FOR SECOND-ORDER QUASI-LINEAR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATION

. In this work, new suﬃcient conditions for oscillation of solution of second order neutral delay diﬀerential equation are established. One objective of our paper is to further simplify and complement some results which were published lately in the literature. In order to support our results, we introduce illustrating examples.


1.
Introduction. In numerous applications in biology, electrical engineering and physiology, the dependence on the past appears naturally. Differential equations with delay appears in modeling of these natural phenomena , see [4,15]. Whereas, delay differential equations or retarded equations, shape a class of mathematical models which let the systems rate of change to not depend only on its present state, but its past history also. The differential equation is called a neutral delay, If the delayed argument occurs in the highest derivative of the state variable. In recent decades, there has been great interest in the study of neutral differential equations. The significant practical importance and theoretical interest for qualitative study of the neutral differential equations is due to the appearance of differential equations with neutral delay appear in mechanics in problem of oscillating masses connected to an elastic bar, in solution of the problems of variations with delay in the time and the problems of electric networks which containing transmission lines without loss, as an example, lines of transmission without loss are used to interconnect switching circuits in computers with high speed, see [7]. Lately, there were a many articles dedicated to the oscillation of solutions of the differential equations with delay or neutral delay, see [1,2,3,5,6,8,10,11,12,14,17,19].
One of these equations has got a lot of attention [9,14,20], the quasi-linear neutral delay equation where t ≥ t 0 and provided that the following are satisfied: (h 1 ) r and p are positive functions, p (t) ∈ [0, 1), α is a quotient of odd positive integers and and has the properties z (t) and r (t) [z (t)] α are continuously differentiable for all t ∈ [t x , ∞). In this study, we take into account only the solutions of (1) which satisfy sup{|x(t)| : t ≥ t} > 0 for any t ≥ t x . If x (t) has arbitrary large zeros, then x (t) is called oscillatory, otherwise it is called non-oscillatory.
Actually, under the assumption η (t 0 ) = ∞, there are many studies in the literature that have been concerned with the oscillation and nonoscillation criteria of solutions of Eq. (1), see for example [2,10,17,19].
For the assumption η (t 0 ) < ∞, Liu et al. [9] obtained some sufficient conditions which guarantee that all solution of Eq. (1) is either oscillatory or tends to zero.
In the previous results, we note the following: -Liu's conditions in Theorem 1.1 guarantee that all solution of Eq. (1) is either oscillatory or tends to zero; -To apply the Theorems 1.1, 1.2 and 1.3, we must ensure that the functions ρ (t) and τ (t) nondecreasing; -Theorems 1.1, 1.2 and 1.3 includes extra conditions (4), (8) and (10). In this paper, we establish a new criteria of oscillation for the quasi-linear neutral Eq. (1). One objective of our paper is to further simplify, improve and complement Theorems 1.1, 1.2 and 1.3. Firstly, we improve Theorem 1.1 so that the condition guarantee that all solution of Eq. (1) is oscillatory, and without imposing restrictions on the derivatives of ρ (t) and τ (t). As well, we simplify Theorems 1.2 and 1.3 by obtaining a new criteria ensure oscillation of (1) without check the excess conditions.
2. Main results. Let us show the following notations: then (1) is oscillatory.
Proof. Contrary to what is required, we will assume that Eq. (1) has non-oscillatory solution. Moreover, we suppose that x (t) is a positive solution of (1) (the case x (t) < 0 is taught by the same way). Thus, there exists a t 1 ≥ t 0 such that x (τ (t)) > 0 and x (σ (t)) > 0 for all t ≥ t 1 . Since x (t) > 0, we have that z (t) > 0 and Then the function r (t) [z (t)] α is strictly decreasing on [t 0 , ∞) and of one sign. Assume first that z (t) < 0 for t ≥ t 1 . Hence,
Theorem 2.2. Assume that the delay equation is oscillatory and Then (1) is oscillatory.
Proof. We can proceed exactly as in the proof of the Theorem 2.1. Then, z (t) is of one sign eventually. Now, we let z (t) < 0 for all t ≥ t 1 . Integrating (15) from t 1 to t, we get Since σ (t) > 0, we obtain G (s) ds and so Hence, we have that z is a positive solution of the inequality In view of [18, Lemma 1], we see that the first-order delay differential Eq. (25) has a positive solution. This means that there is no positive solution of Eq. (1), a contradiction. Next, we assume that z (t) > 0 for all t ≥ t 1 . Therefore, we get that (26) lead to (24), then the remaining part of this proof is similar to that of proof of Theorem 2.1. Thus, the proof is completed.
On the other hand, in [16], it was pointed out that the Eq. (25) when β/α > 1, still may have a nonoscillatory solution even though Tang [18] have studied the oscillation behavior of solutions of Eq. (25) when β/α > 1. In the following, by using the results of [6] and [18], we will obtain a new criteria for oscillation of solutions of (1).