Oscillation of trinomial differential equations with positive and negative terms

In the paper, we offer a new technique for investigation of properties of trinomial differential equations with positive and negative terms ( b(t) ( a(t)x′(t) )′)′ + p(t) f (x(τ(t)))− q(t)h(x(σ(t))) = 0. We offer criteria for every solution to be oscillatory. We support our results with illustrative examples.


Introduction
We consider the third order trinomial differential equation with positive and negative terms b(t) a(t)x (t) + p(t) f (x(τ(t))) − q(t)h(x(σ(t))) = 0, (E) where (H 1 ) a(t), b(t), p(t), q(t), τ(t), σ(t) ∈ C([t 0 , ∞)) are positive; (H 2 ) f (u), h(u) ∈ C(R), u f (u) > 0, uh(u) > 0 for u = 0, h is bounded, f is nondecreasing; We consider the canonical case of (E), that is Corresponding author.Email: blanka.baculikova@tuke.skand throughout the paper we assume that (H 6 ) By a solution of (E) we understand a function x(t) with both quasi-derivatives a(t)x (t), b(t) (a(t)x (t)) continuous on [T x , ∞)), T x ≥ t 0 , which satisfies Eq. (E) on [T x , ∞).We consider only those solutions x(t) of (E) which satisfy sup{|x(t)| : t ≥ T} > 0 for all T ≥ T x .A solution of (E) is said to be oscillatory if it has arbitrarily large zeros, and otherwise it is called nonoscillatory.Equation (E) is said to be oscillatory if all its solutions are oscillatory.
We reveal that the solutions' spaces of (E 1 ) and (E 2 ) are absolutely different.If we denote by N the set of all nonoscillatory solutions of considered equations, then for (E 1 ) the set N has the following decomposition where positive solution On the other hand, for (E 2 ) the set N has the following reduction with positive solution Consequently, the nonoscillatory solutions' space of (E) with positive and negative part is unclear.
Another method frequently used in the oscillation theory of trinomial differential equations is to omit one term.And so, if we omit the negative part of (E), we are led to the differential inequality But it is well known that properties of the corresponding differential equation (E 1 ) are connected with the opposite differential inequality.Similarly omitting the positive term of (E) yields the differential inequality which is again opposite to those that we need.So there is only a limited number of papers dealing (E) with positive and negative parts.In this paper we use a method that overcomes those difficulties appearing due to negative and positive terms of (E).

Main results
In this paper we reduce the investigation of trinomial equations to oscillation of a suitable first order differential equation.We establish a new comparison method for investigating properties of trinomial differential equations with positive and negative terms.We denote Let the first order delay differential equation be oscillatory.Then every solution of (E) either oscillates or converges to zero as t → ∞.
Proof.Assume that (E) possesses a nonoscillatory solution x(t).Without loss of generality we may assume that x(t) is eventually positive.We introduce the auxiliary function Note that condition (H 6 ) and the fact that h(u) is bounded implies that w(t) exists for all t and so the definition of w(t) is correct.Moreover, Therefore, condition (H 5 ) together with a modification of Kiguradze's lemma [5,6] imply that either eventually, let us say for t ≥ t 1 .First assume that w(t) ∈ N 2 .Using the fact that b(t)(a(t)w (t)) is decreasing, we have Using the last estimate and properties of w(t) and x(t) one can see that du ds, which in view of (2.3) and (H 3 ) ensures that z(t) = b(t)(a(t)w (t)) is a positive solution of the differential inequality It follows from Theorem 1 in [8] that the corresponding differential equation (E 0 ) also has a positive solution.A contradiction and the case w(t) ∈ N 2 is impossible.Now we assume that w(t) ∈ N 0 .Since w(t) is positive and decreasing, there exists lim t→∞ w(t) = 2 ≥ 0. It follows from (2.2) that lim t→∞ x(t) = 2 .If we assume that > 0, then x(τ(t)) ≥ > 0, eventually.An integration of (2.3) yields Integrating from t to ∞ and then from t 1 to ∞ one gets which contradicts to (2.1) and the proof is complete.
For a special case of (E) we have the following easily verifiable criterion.
either oscillates or converges to zero.
Proof.Theorem 2.1.1 in [7] guarantees oscillation (E 0 ) with f (u) = u.The assertion of the corollary now follows from Theorem 2.1.
As a matter of fact we are able to provide a general criterion for the studied property of (E).Then every solution of (E) either oscillates or converges to zero.
Proof.By Theorem 2.1 it is sufficient to show that (E 0 ) is oscillatory.Assume on the contrary that (E 0 ) possesses a nonoscillatory, let us say positive solution y(t).It follows from (E 0 ) that y (t) < 0. Thus, there exists lim t→∞ y(t) = c ≥ 0. An integration of (E 0 ) from τ(t) to t provides p(s) f (J(s)) ds.
The last inequality together with (P 2 ) implies that c = 0 and what is more, p(s) f (J(s)) ds.
Taking limit superior on both sides, we get a contradiction with (P 2 ).
For the function f (u) = u β we immediately get the following corollary.
either oscillates or converges to zero.
Employing the additional condition, we achieve the oscillation of (E).We use the auxiliary function and we use the notation and in the rest of this paper, we assume that Theorem 2.6.Let (2.5) hold and (E 0 ) be oscillatory.Let the first order delay differential equation y (t) + p(t) f (I(t)) f y(η(t)) = 0 (E 5 ) be oscillatory.Then (E) is oscillatory.
Proof.Assume that (E) has a positive solution x(t).Let w(t) be defined by (2.2).Proceeding exactly as in the proof of Theorem 2.1, we verify that w(t) ∈ N 0 and there exists a finite lim t→∞ w(t) = c ≥ 0. By (H 5 ), this implies lim t→∞ b(t) (a(t)w (t)) = lim t→∞ a(t)w (t) = 0. Taking (2.2) into account, we see that We introduce another auxiliary function On the other hand, an integration of (a(t)z (t)) < (a(t)x (t)) from t to ∞ in view of (2.6) yields Using the monotonicity of y(t) = b(t)(a(t)z (t)) , the last inequality implies Dividing by a(t) and integrating form τ(t) to ξ(τ(t)), we have Setting into (2.8), one can see that y(t) = b(t)(a(t)z (t)) is a positive solution of differential inequality y (t) + p(t) f (I(t)) f y(η(t)) ≤ 0.
It follows from Theorem 1 in [8] that the corresponding differential equation (E 5 ) also has a positive solution.A contradiction and thus the case z(t) ∈ N 2 is also impossible and we conclude that (E) is oscillatory.
Remark 2.7.Employing sufficient conditions for oscillation of (E 5 ) together with those for (E 0 ), we obtain oscillatory criteria for (E).
The following results are obvious.

Comparison with existing results
The results obtained provide a new technique for studying oscillation and asymptotic properties of trinomial third order differential equations with positive and negative terms via oscillation of a suitable first order equations.

Corollary 2 . 2 . 1 )
Assume that (2.1) holds and Then every solution of the trinomial differential equation b