Third order differential equations with delay

In this paper, we study the oscillation and asymptotic properties of solutions of certain nonlinear third order differential equations with delay. In particular, we extend results of I. Mojsej (Nonlinear Analysis 68, 2008) and we improve conditions on the property B of N. Parhi and S. Padhi (Indian J. Pure Appl. Math. 33, 2002). Some examples are considered to illustrate our main results.


P. Liška
We will denote by (L, g) and (L A , g) the linear versions of equations (N, g) and (N A , g), respectively, i.e. 1 p(t) 1 r(t) x + q(t)x g(t) = 0 (L, g) and the adjoint equation 1 r(t) Further, we denote by (L) and (L A ) the corresponding linear equations without the delay.Prototypes of equations (L, g) and (L A , g) are The asymptotic behaviour of solutions of special types of the above equations have been studied by many authors.This paper benefits mostly from work of Kusano and Naito [8] and from papers written by Cecchi, Došlá, Marini [4,5], Akin-Bohner, Došlá, Lawrence [3] or Mojsej [9], see also references there.Some other results are given in papers [2,6,10] or recently in [1].The extensive survey can be found in the excellent book [11], see also references there.The equation (E±) has been studied in [12].
The aim of the paper is to extend some results from the paper by I. Mojsej [9] and to study the influence of the delayed argument on the oscillation of equations (N, g) and (N A , g).Some examples are considered to illustrate our results.
If x is a solution of (N, g) then functions x [1]   are called quasiderivatives of x.Similarly, we can proceed for (N A , g).
A solution x of (N, g) is said to be proper if it exists on the interval [a, ∞) and satisfies the condition sup{|x(s)| : t ≤ s < ∞} > 0 for any t ≥ a.
A proper solution is called oscillatory or nonoscillatory according to whether it does or does not have arbitrarily large zeros.Similar definitions hold for (N A , g).Following [7], we define property A and property B by the following way.
Definition 1.1.The equation (N, g) is said to have property A if any proper solution x of (N, g) is either oscillatory or satisfies Definition 1.2.The equation (N A , g) is said to have property B if any proper solution z of (N A , g) is either oscillatory or satisfies The notation y(t) ↓ 0 (y(t) ↑ ∞) means that y monotonically decreases to zero as t → ∞ (y monotonically increases to ∞ as t → ∞).
It is clear, that (N, g) or (L, g) has property A if and only if all nonoscillatory solutions of (N, g), or (L, g), respectively, belong to the class N 0 and lim t→∞ x [i] (t) = 0, i = 0, 1, 2. Similarly, (N A , g) or (L A , g) has property B if and only if all nonoscillatory solutions of (N A , g), or (L A , g), respectively, belong to the class M 3 and lim t→∞ z [i] (t) = ∞, i = 0, 1, 2.
We will study the relationship between property A for (L, g) and for (N, g) and property B for (L A , g) and (N A , g).Our results complete recent ones in [9].As a consequence, an equivalence result for property A for (L, g) and for property B for (L A , g) is obtained.The paper is completed by some examples, which illustrate the role of function g.

Results about relationship between the oscillation and properties A and B for linear equations without delay can be summarized as follows.
Theorem A ([4]).The following assertions are equivalent: We can reformulate Theorem 3.1 in [9] as follows.

P. Liška
Theorem C. Consider (N, g) and function τ(t) such that If equation (L) has property A, then equation (N, g) has property A.
In particular, for (L, g) we have the following result.In [12] there are criteria for the equation (E−) to have property B, which can be summarized as follows.

Theorem D. The equation (E−) has property B if any of the following conditions
(2.3) Our aim is the extension of Theorem C for the equation (N A , g) and property B. In particular, the question is whether or not we can complete the diagram in Corollary 2.2 with the last implication.

Main results
First we prove a slight modification of Theorem 2.1 from [3].
Then every solution z of (N A , g) from the class M 3 satisfies Third order differential equations with delay 5 Proof.We rewrite (N A , g) as a system x (t) = q(t) z g(t) λ sgn z g(t) . (3.2) Let z(t) be a solution of (N A , g) from the class M 3 .Then the vector z(t), y(t), x(t) , where y(t) = 1 p(t) z (t) and x(t) = 1 r(t) y (t), is a solution of system (3.2) such that sgn x(t) = sgn y(t) = sgn z(t) for large t.
We prove that lim There exists T ≥ a such that x(t) > 0, y(t) > 0, z(t) > 0 for t ≥ T. As y(t) is eventually increasing, there exists T 1 ≥ T and K > 0 such that Using the assumption ∞ p(t) dt = ∞ we get lim t→∞ z(t) = ∞.
Since x(t) is eventually increasing, there exists T 2 ≥ T 1 and L > 0 such that and integrating in [T 2 , t] Using the assumption ∞ r(t) dt = ∞ we get lim t→∞ y(t) = ∞.
Integrating the first equation of (3.2) from T 1 to g(t) and using (3.3) we obtain Using the third equation of (3.2) and (3.3), there exists T 2 ≥ T 1 such that Integrating the last inequality from T 2 to t gives In order to the equation (N A , g) having the property B we establish sufficient condition for M 1 = ∅.To this aim the following lemma will be needed.
Proof.Since (2.1) and (3.8) hold, then by using Lemma 3.4, where functions r and p are exchanged, we get As g(t) < t and 0 < λ ≤ 1, we have (2) Let us consider the equations (N, g) and (N A , g) with symmetrical operator, i.e. r(t) = p(t) x + q(t) x g(t) 1/λ sgn x g(t) = 0 (S, g) and 1 p(t) Further, we denote (S) and (S A ) corresponding linear equations without the deviating argument, i.e. equations (S, g) and (S A , g), where g(t) = t and λ = 1.
Let τ satisfy (2.1).We have hence all these equations have property A or property B, respectively.In the book [11], see Section 6.3, or in [12] oscillation of equations (4.4) and (4.5) has been investigated in the terms of property Ā and property B. There are given some sufficient conditions for equation (4.4) to have property Ā and for equation (4.5) to have property B. In general, property Ā is weaker than property A and means that every nonoscillatory solution of (4.4) is in the class N 0 .
Observe that equations (4.7) appear in [11,12], where various criteria are used to verify that equations of the type (4.4) have property Ā.As far as property B is concerned, conditions P. Liška from [12] are summarized in Theorem D. The first condition can not be applied and condition (2.3)
dτ ds dt = ∞.Under our assumptions Theorem 1 from [8] reads as follows.Theorem B. (i) If equation (L, g) has property A, then equation (L) has property A. (ii) If equation (L A , g) has property B, then equation (L A ) has property B.

Theorem 3 . 5 .
t → ∞ and using(3.5)we get the contradiction.The main result is the following extension of Theorem C to property B. Let (2.1) and (3.5) hold and assume that

( 1 ) 2 . 4 . 1 .
dτ ds dt = ∞ , which, due to Theorem A, means that the equation (L A ) has property B.Moreover, assumption (3.8) implies that (3.1) holds, so by Theorem 3.1, every solution z(t) of (N A , g) from the class M 3 satisfies lim t→∞ z [i] (t) = ∞, for i = 0, 1, 2. According to Theorem 3.3 the condition (3.5) implies that M 1 = ∅, thus (N A , g) has property B. Now we can complete Corollary 2.Corollary Let (2.1), (3.5) and (3.8) hold.Then (L) has property A k s + 3 (L, g) has property A (L A ) has property B K S (L A , g) has property B It follows from Theorem 3.5 and Corollary 2.2.