Asymptotic behavior of even-order damped differential equations with p-Laplacian like operators and deviating arguments

We study the asymptotic properties of the solutions of a class of even-order damped differential equations with p-Laplacian like operators, delayed and advanced arguments. We present new theorems that improve and complement related contributions reported in the literature. Several examples are provided to illustrate the practicability, maneuverability, and efficiency of the results obtained. An open problem is proposed.


Introduction
In this paper, we study the asymptotic behavior of a class of even-order damped differential equations with p-Laplacian like operators and deviating arguments a(t) x (n-) (t) p- x (n-) (t) + r(t) x (n-) (t) p- x (n-) (t) + q(t) x g(t) p- x g(t) = , (.) where t ∈ I := [t  , ∞), t  ∈ (, ∞), n ≥  is an even integer, p >  is a constant, a ∈ C  (I, (, ∞)), r, q, g ∈ C(I, R), r(t) ≥ , a (t) + r(t) ≥ , q(t) > , and lim t→∞ g(t) = ∞. As pointed out by Hale [], differential equations have applications in the natural sciences, engineering technology, and automatic control. In particular, equation (.) has numerous applications in continuum mechanics as seen from Agarwal et al. [] and Zhang et al. []. As usual, by a solution of (.) we mean a continuous function x ∈ C n- ([T x , ∞), R) which has the property that a|x (n-) | p- x (n-) ∈ C  ([T x , ∞), R) and satisfies (.) on [T x , ∞). We consider only those extendable solutions of (.) that satisfy condition sup{|x(t)| : t ≥ T ≥ T x } >  and we tacitly assume that (.) possesses such solutions. A solution of (.) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. Equation (.) is oscillatory if all its solutions oscillate.
There has been a growing interest in the study of the oscillatory and asymptotic behavior of various classes of differential equations during the past decades; we refer the reader to [-] and the references cited therein. In the following, we briefly review several related results that have motivated the work in this paper. Zhang et al. [], Liu et al. [], and Zhang et al. [] considered the oscillation of (.) under the assumptions that Assuming that γ >  is a quotient of odd positive integers, Erbe et al.
[] studied a secondorder damped differential equation and established some oscillation results in the case where (.) holds and

Rogovchenko and Tuncay [] and Saker et al.
[] investigated the oscillation of a secondorder damped differential equation and they obtained several sufficient conditions which ensure that every solution x of (.) is either oscillatory or satisfies lim t→∞ x(t) = . Zhang [] considered oscillatory behavior of (.) in the case when (.) is satisfied and So far, the study of the asymptotic behavior of equation (.) when the integral in (.) is finite, i.e., has received considerably less attention in the literature. Hence, our objective in this paper is not only to analyze the asymptotic properties of (.) in the case where (.) holds, but also to derive new asymptotic tests for (.) under the assumption that The new theorems obtained improve and complement the relevant results reported in [, , , , , , ]. As is customary, all functional inequalities considered in the sequel are supposed to hold for all t large enough. Without loss of generality, we deal only with positive solutions of (.) since, if x is a solution, so is -x. For a compact presentation of our results, we use the following notation: , where the meaning of ρ and will be specified later.

Lemmas
To establish our main results, we make use of the following auxiliary lemmas.

Asymptotic results via the Riccati method
If, for some constant λ  ∈ (, ), We consider each of the two cases separately.
Case I. Assume first that case () holds. For t ≥ t  , we define the function ω by Then ω(t) >  for all t ≥ t  and .
Let u := x . It follows from Lemma . that, for some constant M >  and for all sufficiently large t, Thus, we deduce that . From (.) and (.), we obtain Hence, we have Using the inequality where C > , y ≥ , and D + := max(, D) (see Fišnarová and Mařík ([], Lemma ) for details), we get Integrating this inequality from t  to t, we obtain Hence, for all s ≥ t ≥ t  , Integrating this inequality from t to ι, we obtain Taking ι → ∞ and using the fact that lim ι→∞ x (n-) (ι) ≥  and the definition of δ, we have Hence, by (.) and (.), we get On the other hand, by Lemma ., we have, for every λ ∈ (, ) and for all sufficiently large t, t) and integrating the resulting inequality from t  to t, we obtain Using inequalities (.), (.), and the definition of ϕ, we have which contradicts (.). This completes the proof.
Assume n =  and let the definition of ω in (.) be replaced by Then we have the following result.
Observe, however, that if γ = , then If there exists a function m ∈ C  (I, (, ∞)) such that and, for some constant λ  ∈ (, ), then the conclusion of Theorem . remains intact.
Proof Assume that x is an eventually positive solution of (.) that satisfies (.). Similar analysis to that in Zhang et al. ([], Lemma .) leads to the conclusion that, for all t ≥ t  , there exist two possible cases () and () (as those in the proof of Theorem .), where t  ≥ t  is sufficiently large. Assume first that case () holds. We define the function ω by , t ≥ t  .
With a similar proof as that of Case I in Theorem ., one arrives at a contradiction with condition (.). Assume, instead, that case () holds. Define the function v as in (.). As in the proof of Theorem ., we obtain (.), (.), (.), and (.). On the other hand, we derive from (.) that
Using the latter inequality and (.), we have which implies that x (n-) /m is nondecreasing. Hence, it follows from (.) that Thus, by (.) and (.), we have The remaining proof is similar to that of Case II in Theorem ., and hence is omitted.

Remark . The optional function m satisfying condition (.) exists and can be constructed by taking m(t) := δ(t).
Assume n =  and let ω be as follows: Then we obtain the following result that leads to the conclusion of Theorem ..
Theorem . Let conditions (.) and (.) be satisfied and n = . Assume that there exists a function ρ ∈ C  (I, (, ∞)) such that If there exists a function m ∈ C  (I, (, ∞)) such that (.) is satisfied and then the conclusion of Theorem . remains intact.
Example . For t ≥ , consider a second-order advanced differential equation with damping where q  >  is a constant. Let t  = , p = , a(t) = t  , r(t) = t/, q(t) = q  , g(t) = t, ρ(t) = , and m(t) = δ(t) = t -/ /. Similar analysis to that in Example . implies that condition (.) holds and condition (.) is satisfied for q  >  √ . Thus, by Theorem ., equation Observe that the results reported in [, ] cannot be applied to equation (.) since g(t) > t.
In the next theorem, we consider equation (.) under the assumptions that (.) holds and and Note that condition (.) is also satisfied in this case.
Similarly, we have the following criterion for (.) in the case when n = .
Theorem . Let (.) and (.) be satisfied and n = . Suppose that there exists a function ρ ∈ C  (I, (, ∞)) such that (.) holds. If and there exists a function ξ ∈ C  (I, (, ∞)) such that (.) holds and lim sup then the conclusion of Theorem . remains intact.
The following example is provided to show that our results are sharp for the secondorder Euler differential equation (t  x (t)) + q  x(t) = , q  > .
Example . For t ≥ , consider a second-order differential equation with damping where r  ≥  and q  >  are constants. Let t  = , p = , a(t) = t  , r(t) = r  , q(t) = q  , g(t) = t, and ρ(t) = . Then h + (t) =  and so condition (.) is satisfied. It is not hard to verify that  ≤ E(t  , t) ≤ e r  , e -r  t - ≤ δ(t) ≤ t - , and A(t) = /t. Then condition (.) is satisfied for all sufficiently large t and, for q  > /, Finally, the following example is given to present an open problem of this paper.