Oscillation of a perturbed nonlinear third order functional differential equation

In this paper, the authors present some new results on the oscillatory and asymptotic behavior of solutions of the perturbed nonlinear third order functional differential equation ( b(t) ( a(t)(x′(t))α )′)′ + p(t) f (x(τ(t))) = h(t, x(t), x(τ(t)), x′(t)). In addition to other conditions, the authors assume that u f (u) > 0 for u 6= 0 and f is increasing. Examples to illustrate the results are included.

Corresponding author.Email: John-Graef@utc.eduBy a solution of (1.1) we mean a function x(t) whose quasi-derivatives a(t)(x (t)) α and (a(t)(x (t)) α ) are continuous on [T x , ∞), T x ≥ t 0 , and which satisfies Eq. (1.1) on [T x , ∞).We consider only those solutions x(t) of (1.1) that satisfy sup {|x(t)| : t ≥ T} > 0 for all T ≥ T x .A solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros, and nonoscillatory otherwise.Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.
In two very nice papers Baculíková and Džurina studied the oscillatory and asymptotic behavior of solutions of some third order nonlinear delay differential equations.In [1], they considered the equation (b(t)(x (t)) α ) + p(t) f (x(τ(t))) = 0 under the same covering assumptions as those above and assumed that In [2], they considered the equation and (1. (1.4) Notice that condition (1.3) implies that q is small in that we must have Condition (1.4) requires q to be small is some sense relative to b and a.
Our goal here is to establish oscillation results for equation (1.1) without imposing a "smallness" condition on the perturbation term.We also present some results on the boundedness and oscillatory behavior of a special case of (1.1), namely, where β and γ are the ratios of odd positive integers with β > γ and e : [t 0 , ∞) → R is a continuous function.As was done in [1,2], we will use a comparison approach.

Oscillation of equation (1.1)
We assume that there exists a positive continuous function q For any t 1 ≥ t 0 , we set We also assume that there are functions ξ, η and set and In some of our results we will also ask that Our first oscillation result is contained in the following theorem.
Proof.Let x(t) be a nonoscillatory solution of equation (1.1).Without loss of generality we may assume that x(t) and x(τ(t)) are positive and condition (2.2) holds for t ≥ t 1 for some t 1 ≥ t 0 .If x(t) is eventually negative, a similar proof holds.From our assumptions and equation (1.1), we see that for all t ≥ t 1 .
It is easy to see that we need to consider the following two cases: We will first examine Case (I).For t ≥ t 2 , we see that Integrating this inequality from t 2 to τ(t) ≥ t 2 , we have where y(t) = b(t) (a(t)(x (t)) α ) .Using (2.9) in (2.8) and applying (H3), we obtain It follows from [11, Corollary 1] that the corresponding differential equation (2.6) also has a positive solution.This contradiction completes the proof for Case (I).
For Case (II), it is easy to see that Hence, Setting u = τ(t) and v = ξ(t) in the above inequality, we obtain From (2.10) and (2.11) we see that (2.12) Using (2.12) in equation (2.8), we have It folows from [11, Corollary 1] that the corresponding differential equation (2.7) also has a positive solution, which is a contradiction.This completes the proof of the theorem.
The next two corollaries follow immediately from known oscillation criteria for first order delay differential equations; fo example, see [ then equation (1.1) is oscillatory.
The following example illustrates the above results.
Example 2.4.Consider the equation Here a(t , and α = 3.Let p(t) and q(t) be positive continuous functions with Q(t) = p(t) − q(t) positive for all large t.Now, and are oscillatory, then equation (2.15) is oscillatory.
Instead of condition (2.2), we assume that there exists a function (2.16) and we set (2.17) We can then obtain the following theorem. where Dividing by a(t) and integrating from τ(t) to ρ(τ(t)), we obtain for all large t.Using (2.20) in (2.8) and proceeding as in the proof of Case (II) in Theorem 2.1, we arrive at the desired contradiction.This completes the proof of the theorem.
To illustrate this result we have the following example.

Boundedness and oscillation of equation (1.5)
In order to obtain our results in this section, we need the following lemma.
Lemma 3.1 (Young's inequality).Let X and Y be nonnegative, n > 1, and and equality holds if and only if Y = X n−1 .
Theorem 3.2.In addition to condition (H1), assume that and Then every nonoscillatory solution of equation (1.5) is bounded.
Proof.Let x(t) be a nonoscillatory solution of equation (1.5) such that x(t) > 0 for t ≥ t 1 for some t 1 ≥ t 0 .Applying (3.1) to [q(s)x γ (s) − p(s)x β (s)] with , and m = β β − γ , we obtain From equation (1.5) we then have where c 2 = a(t 1 )(x (t 1 )) α .From condition (3.2) and (3.3), there exists a constant C such that Integrating this inequality from t 1 to t and using condition (3.2), we arrive at the desired conclusion.
The following result is concerned with the oscillation of equation (1.5).