Asymptotics and oscillation of nth-order nonlinear differential equations with p-Laplacian like operators

This paper is concerned with nth-order nonlinear differential equations of the form (a(t)|x(n–1)(t)|p–2x(n–1)(t))′ + r(t)|x(n–1)(t)|p–2x(n–1)(t) + q(t)|x(g(t))|p–2x(g(t)) = 0 with n ≥ 2. By discussing the signs of ith-order derivatives of eventually positive solutions, for i = 1, . . . ,n – 1, and using the generalized Riccati technique and integral averaging technique, we derive new criteria for oscillation and asymptotic behavior of the equation. Our results generalize and improve many existing results in the literature.


Introduction
In this paper, we study the nth-order nonlinear differential equation with Laplacian and deviating argument 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 ∈ [t  , ∞). Throughout this paper, we assume the following: (H) a(t) ∈ C  ([t  , ∞), (, ∞)), r(t), q(t) ∈ C([t  , ∞), R), q(t) ≥ , and a (t) + r(t) ≥ ; (H) p >  is a real number, g(t) ∈ C([t  , ∞), R) such that lim t→∞ g(t) = ∞. Asymptotics and oscillation of (.) and related equations have been discussed by many authors; see [-] and the references therein. In particular in , Zhang et al. [] established oscillation criteria for (.) when n ≥  is even via the integral averaging technique and two kinds of functions H(t, s) and H * (t, s), employed the comparison technique to discuss the oscillation of (.) when n ≥  is even and the oscillation and asymptotic behavior of (.) when n ≥  is odd.
By imposing some additional assumptions, in the present paper we shall discuss the signs of ith-order derivatives of eventually positive solutions for i = , . . . , n -, and we establish concrete criteria for the asymptotics and oscillation of (.) for both the evenorder and the odd-order cases, where the deviating arguments may be retarded, advanced, or mixed. Our results will generalize and improve those in [] and many other papers for the even-order case and develop new results for the odd-order case.
By a solution of (.) we mean a nontrivial real-valued function x(t) ∈ C n- ([t  , ∞), R) such that a(t)|x (n-) (t)| p- x n- (t) ∈ C  ([t  , ∞), R), which satisfies (.). Our attention is restricted to those solutions of (.) that satisfy sup{|x(t)| : t ≥ t x } >  for any t x ≥ t  . A solution x(t) of (.) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.
This paper is organized as follows: After this introduction, we present the main results in Section , followed by illustrative examples in Section . We then introduce some preliminary lemmas in Section , which are used to prove the main results in Section . The conclusion is drawn in Section .

Main results
In this section, we present our main results which provide conditions for every solution of (.) to be oscillatory on [t  , ∞) or convergent to  as t → ∞. In order to state the main theorems, we need the following notation.
For t, T ∈ R such that t ≥ T, we define where z ∈ C  (R, (, ∞)) is to be given in Theorem ., z + (t) = max{z (t), }, and [ Furthermore, for sufficiently large T ∈ R, one of the following conditions is satisfied:

Then:
(i) every solution x(t) of (.) is either oscillatory or tends to zero as t → ∞ when n is odd, and (ii) every solution x(t) of (.) is oscillatory when n is even.
The results in the second theorem hold only for g(t) ≥ t or g(t) ≤ t with g (t) > .
Theorem . Let p > . Assume that (H)-(H) and (.) hold. If either (.) is satisfied, or (.) is satisfied and for sufficiently large T ∈ R, then: (i) every solution x(t) of (.) is either oscillatory or tends to zero as t → ∞ when n is odd, and (ii) every solution x(t) of (.) is oscillatory when n is even.
The results in the third theorem hold only for p ≥ .
Theorem . Let p ≥ . Assume that (H)-(H) and (.) hold and either (.) or (.) is satisfied. Furthermore, for sufficiently large T ∈ R, there exists a z ∈ C  (R, (, ∞)) such that one of the following conditions is satisfied: Then: (i) every solution x(t) of (.) is either oscillatory or tends to zero as t → ∞ when n is odd, and (ii) every solution x(t) of (.) is oscillatory when n is even.

Examples
In this section, we give two examples to illustrate our main results.
By Theorem ., every solution x(t) of (.) is oscillatory or tends to  as t → ∞ when n is odd, and (.) is oscillatory when n is even.
Remark . In Example  above, we see that a(t) =  t is strictly decreasing on [, ∞); but in [], we see that its condition (H) requires a (t) > , so the results in [] are not applicable to this example.

Example  Consider the equation
where n ≥ , t ∈ [, ∞). Here we have: Hence Condition (a) of Theorem . is satisfied. By Theorem ., every solution x(t) of (.) is oscillatory or tends to zero as t → ∞ when n is odd, and (.) is oscillatory when n is even.
Remark . In Example  above, we see that r(t) = - <  and g(t) = t +  ≥ t on [, ∞). But in [], we see that its condition (H) requires r(t) > , and in [], its condition (H) requires g(t) ≤ t, so the results in [, ] are not applicable to this example.

Preliminary lemmas
In this section, we present several technical lemmas which will be used in the proofs of the main results. The first one is on the signs of derivatives of certain classes of functions. The next lemma concerns the signs of derivatives of eventually positive solutions of (.). In particular, we derive conditions for the following inequalities to hold eventually: Proof If x(t) is an eventually positive solution of (.), then by (H), there exists a t  ∈ [t  , ∞) such that From (.) and (H), we have which implies that for t ≥ t  , Thus, x (n) (t) <  eventually.
From Lemma . with m = n -, there exist a t  ∈ [t  , ∞) T and an integer l,  ≤ l ≤ m such that (.) and (.) are satisfied. That is, and x(t) is eventually monotone. When n is even, by Lemma ., l must be an odd number. By (.) and (.), we can get x (t) > . Hence In this case, we claim that l = m = n -. Otherwise, we obtain the odd integer l ≤ m - = n -. By (.), we get This means It follows from (.) that there exist a T ≥ t  and b >  such that x(g(t)) ≥ b for t ≥ T. From (.) and (.), and x (n-) (t) >  we have If (.) holds, by integrating (.) from T to t with t ≥ T we obtain which contradicts the fact that x (n-) (t) >  for t ∈ [t  , ∞). Hence, l = m = n - and (.) holds. If (.) holds, by integrating (.) from t to u with T ≤ t ≤ u we obtain Taking the limit as u → ∞, we have Taking the limit as v → ∞, we obtain Since x (n-) (t) > , integrating the above inequality from T to t with t ≥ T, we get Taking t → ∞, we obtain which contradicts (.). Hence, l = m = n - and (.) holds.
When n is odd, by Lemma ., l must be an even integer. By (.) and (.), we have either x (t) >  or x (t) < . That means lim t→∞ x(t) = c ≥ . We claim that if lim t→∞ x(t) = , then l = m = n -. Otherwise, there is the even number l ≤ m - = n - such that (.) and (.) hold. By a similar argument to above, we can reach a contradiction to (.) or (.). This completes the proof.
Noting that (a(t) exp( a(τ ) dτ )(x (n-) (t)) p- ) <  for t ≥ T, it follows from (.) that Integrating the inequality above from T to t for t ≥ T, we get It is easy to see by induction that

Proofs of main results
In this section, we give proofs for our main results by employing generalized Riccati techniques and integral averaging techniques.
Proof of Theorem . Suppose to the contrary that (.) has a nonoscillatory solution x(t). Without loss of generality, we may assume that x(t) is eventually positive. Then, by (H)-(H), there exists a T ∈ [t  , ∞) such that for t ≥ T, x(t) > , and Lemmas . and . hold.