Properties of the third order trinomial functional differential equations

The purpose of the paper is to study asymptotic properties of the third-order delay differential equation ( r2(t) ( r1(t) ( y′(t) )γ)′)′ + p(t) ( y′(t) )γ + q(t) f (y (τ(t))) = 0. (E) Employing comparison principles with a suitable first order delay differential equation we shall establish criteria for all nonoscillatory solutions of (E) to converge to zero, while oscillation of a couple of first order delay differential equations yields oscillation of (E). An example is provided to illustrate the main results.


Introduction
In this paper, we are dealing with the oscillation and asymptotic behavior of solutions of the third-order nonlinear delay differential equation By a solution of (E), we mean a function y(t) such that r 2 (t) r 1 (t) (y (t)) γ ∈ C 1 [T y , ∞) for a certain T y ≥ t 0 and y(t) satisfies (E) on the half-line [T y , ∞).Our attention is restricted to only such extendable solutions y(t) of (E) which satisfy sup{|y(t)| : t ≥ T} > 0 for all T ≥ T y .Further, we make a standing hypothesis that (E) possesses such a solution.As customary, a solution y(t) of (E) is said to be oscillatory if it has arbitrarily large zeros on [T y , ∞) and otherwise it is called to be nonoscillatory.Equation (E) itself is called oscillatory if all of its solutions are oscillatory.
As is well known, differential equations of third order have long been considered as valuable tools in the modeling of many phenomena in different areas of applied mathematics and physics.Indeed, it is worthwhile to mention their use in the study of entry-flow phenomenon [11], the propagation of electrical pulses in the nerve of a squid approximated by the famous Nagumo's equation [16], the feedback nuclear reactor problem [23] and so on.
Hence, a great deal of work has been done in recent decades and the investigation of oscillatory and asymptotic properties for these equations has taken the shape of a well-developed theory turned mainly toward functional differential equations.In fact, the development of oscillation theory for the third order differential equations began in 1961 with the appearance of the work of Hanan [10] and Lazer [15].Since then, many authors contributed to the subject studying different classes of equations and applying various techniques, see, for instance, .A systematic survey of the most significant efforts in this theory can be found in the excellent monographs of Swanson [21], Greguš [9] and the very recent-one of Padhi and Pati [19].
In fact, determination of trinomial delay differential equations of third order often depends on the close related second order differential equation.The case when this associated equation is oscillatory was object of research in [7].Taking under the assumption the nonoscillation of the corresponding auxiliary equation, special cases of (E) has been considered in many papers.
The partial case of (E), namely y (t) + p(t)y (t) + q(t)y(τ(t)) = 0 has been studied e.g., by present authors [6], Parhi and Padhi [17,18].Series of articles [3][4][5] deal with the case of (E) when γ = 1, i.e., By means of a generalized Riccati transformation and integral averaging technique, authors have established some sufficient conditions which ensure that any solution of (E ) oscillates or converges to zero.Further oscillation criteria have been obtained by establishing a useful comparison principle with either first or second order delay differential inequality, given in [1].
Another approach of investigation (E ), which depends on the sign of a particular functional, was proposed in [8] as a generalization of known results for ordinary case [15].In spite of a substantial number of existing papers on asymptotic behavior of solutions of third order trinomial equation (E ), many interesting questions regarding oscillatory properties remain without answers.More exactly, existing literature does not provide any criteria which directly ensure oscillation of (E ) as well as condition In view of the above motivation, our purpose in this paper is to extend the technique presented in [6] to cover also more general differential equation (E).We stress that our criteria does not require any condition on the function r 1 (t).
As convenient, all functional inequalities considered in this paper are assumed to hold eventually, that is, they are satisfied for all sufficiently large t.
We say that (E) has the property (P) if all of its nonoscillatory solutions y(t) satisfy the condition y(t)y (t) < 0. (1.1) As will be shown, the properties of (E) are closely connected with the positive solutions of the auxiliary second-order differential equation as the following theorem says.
Theorem 1.1.Let (V) possess a positive solution v(t).Then the operator Proof.It is straightforward to see that Corollary 1.2.If v(t) is a positive solution of (V), then (E) can be written as the binomial equation because such equations (as will be shown later) have simpler structure of possible nonoscillatory solution.
In what follows, we first investigate the properties of the positive solutions of (V) and then, instead of studying properties of the trinomial equation (E), we will study the behavior of its pertaining binomial representation (E c ).
The following result is a consequence of Sturm's comparison theorem and guarantees the existence of a nonoscillatory solution of (V).
Then (V) posseses a positive solution.
To be sure that (V) possesses a positive solution, in what follows, we will assume that (1.5) holds.
For our next purposes, the following lemma will be useful.
then another linearly independent solution of (V) is given by Really, taking (1.6) into account, it is easy to see that Bringing together all the previous results, it is reasonable to conclude the following.
Lemma 1.5.Let (1.5) hold.Then the trinomial equation (E) can be written in its binomial form (E c ).
From now on, we are prepared to study the properties of (E) with the help of its equivalent representation (E c ).In view of familiar Kiguradze's lemma [12], we have the following structure of nonoscillatory solutions of (E).Lemma 1.6.Let (1.5) hold and assume that v(t) is such positive solution of (V) that satisfies (1.3).If (1.4) is satisfied, then every positive solution of (E c ) is either of degree 0, that is or of degree 2, that is, (1.8) In the case when (1.4) fails, there may exists one extra class, that is (1.9) If we denote the classes of positive solutions of (E c ) satisfying (1.7), (1.8) and (1.9) by N 0 , N 2 and N * , respectively, Then the set N of all positive solutions of (E c ) (as well as (E)) has the following decomposition N = N 0 ∪ N 2 provided that both (1.3) and (1.4) hold and

Canonical form
Since (E c ) is in a canonical form, the set of all positive solutions of (E c ) is given by Now we are prepared to provide criteria for property (P) of (E) and later also for oscillation of (E).
Let us denote Theorem 2.1.Let (1.5) hold and assume that v(t) is such positive solution of (V) that (1.3) and (1.4) are satisfied.If the first order nonlinear differential equation is oscillatory, then (E) has property (P).
Proof.Assume that (E) has an eventually positive solution y(t).Then y(t) is also solution of (E c ).It follows from Lemma 1.6 that y(t) is either of degree 2 or degree 0. If y(t) ∈ N 2 , then by making use of the fact that Integrating from t 1 to t, we are led to Hence, Combining the last inequality together with (E c ), we obtain Therefore, it is clear that z(t) is a positive solution of differential inequality On the other hand, in view of Theorem 1 of Philos [20], the corresponding differential equation (E P ) also has a positive solution.This is a contradiction and we conclude that y(t) is of degree 0 and the first two inequalities of (1.7) implies property (P) of equation (E).
Employing criteria for oscillation of (E P ) we immediately get criteria for property (P) of (E).
Corollary 2.2.Let (1.5) hold and assume that v(t) is such positive solution of (V) that (1.3) and (1.4) are satisfied.Let f (u) = u γ .Assume that then (E) has the property (P).
The sufficient conditions for oscillation of (E P ) in previous corollaries are recalled from [14], [13] and [22], respectively.Now, we enhance our results to ensure stronger asymptotic behavior of the nonoscillatory solutions of (E).We impose an additional condition on the coefficients of (E) to guarantee that every solution of (E) either oscillates or tends to zero as t → ∞.Lemma 2.6.Assume that equation (E) posseses property (P).If then every nonoscillatory solution of (E) tends to zero as t → ∞.
Proof.Let y(t) be an eventually positive solution of (E).Recall (E) possesses property (P), iff y(t)y (t) < 0. It is clear that there exists a lim t→∞ y(t) = ≥ 0. Assume for contradiction > 0. On the other hand, y(t) is also a solution of (E c ) of degree 0. Using (H 4 ) in (E c ), we have Then, integration of the previous inequality from t to ∞ leads to Integrating the last inequality from t to ∞, we conclude Integrating once more the last inequality from t to ∞, we obtain Letting t → ∞ and using (2.2), it is easy to see that lim t→∞ y(t) = −∞, which contradicts the fact that y(t) is a positive solution of (E c ).Therefore, we deduce that = 0.The proof is complete.
Requiring oscillation of another suitable first order differential equation, we can obtain even oscillation of (E).
Theorem 2.7.Let (1.5) hold and assume that v(t) is such positive solution of (V) that (1.3) and (1.4) are satisfied.Suppose that there exists a function ξ If both the first-order delay differential equations (E P ) and are oscillatory, then (E) is oscillatory.
Oscillation of trinomial equations 9 Proof.Let y(t) be an eventually positive solution of (E).It follows from Lemma 1.6 that either y(t) ∈ N 0 or y(t) ∈ N 2 .In view of the proof of Theorem 2.1, it is known that oscillation of (E P ) eliminates all solutions of degree 2. Therefore, y(t) is of degree 0. An integration of (E c ) from t fo ξ(t) yields Then Integrating the above inequality from t to ξ(t) once more, we have Finally, integration from t to ∞ leads us Let us denote the right-hand side of the above inequality by z(t).Then y(t) ≥ z(t) > 0 and it is easy to verify that 0 Consequently, Theorem 1 of Philos [20] implies that the corresponding differential equation (E 0 ) has also a positive solution z(t), which contradicts our assumption.We conclude that also N 0 = ∅ and thus, (E) is oscillatory.The proof is complete.
Proof.To ensure oscillation of (E), assume for the sake of contradiction that y(t) is a positive solution of (E).Then y(t) is also solution of (E c ).Using result of Theorem 2.7, oscillation of (E P ) and (E 0 ) guaranties that classes N 0 and N 2 are empty.So assume that y(t) ∈ N * .Therefore, y(t) is decreasing and integration from t to ∞ yields y(t) ≥ − ∞ t v(s) r 1 (s) q(s)v(s) f (P(τ(s))) ds ≤ 0.
Repeating integration from t 1 to t, we get r 1 (t) v(t) y (t) γ + f (L) Finally, integrating once more,

Summary
In this paper, we have extended the technique presented in [6] to cover a more general differential equation (E).Easily verifiable criteria are established to complement other known results for the case γ = 1.We point out that our main theorems do not require any restricted conditions to coefficient r 1 (t) and can ensure oscillation of all solutions of (E).

u t 1 q
(s)v(s) f (P(τ(s))) ds du our assumption.The proof is complete.