Existence of nonoscillatory solutions to nonlinear higher-order neutral dynamic equations

We investigate the existence of different types of nonoscillatory solutions to a class of higher-order nonlinear neutral dynamic equations on a time scale. Two examples are provided to show the significance of the conclusions.


Introduction
The time scale theory has been introduced and developed rapidly since 1988; see, for instance, [1-4, 7, 8]. Afterwards, many scholars were concerned with the oscillation of dynamic equations on time scales and they obtained abundant achievements. Besides, some research on the existence and asymptotic behavior of nonoscillatory solutions to dynamic equations on time scales has been also improved recently, we refer the reader to [5,6,[9][10][11][12][13][14][15].
Since 2007, numerous researchers have investigated the existence of nonoscillatory solutions to several classes of nonlinear neutral dynamic equations

r(t) x(t) + p(t)x g(t)
and ∞ t 0 1/r(t) t < ∞. Similar to the results in [15], there are two types of asymptotic behavior of eventually positive solutions to (2). Later on, Deng and Wang [5] considered (2) with another condition ∞ t 0 1/r(t) t = ∞, and summarized four types of eventually positive solutions to (2). It is clear to see that the asymptotic behavior of eventually positive solutions in [5] are more complex than that of [6]. From [5,6], we can see the fact that the existence and asymptotic behavior of nonoscillatory solutions are greatly different for various kinds of the integral convergence and divergence of the reciprocals of the coefficients r i in equations. To find a general relationship between these factors, some researches have been performed.

Auxiliary results
We denote all continuous functions mapping [T 0 , ∞) T into R by C([T 0 , ∞) T , R). Then define a Banach space for simplicity, and we just consider the eventually positive solutions to (4). Now, a lemma is presented to show the relationship between the functions z and x. The proof is similar to the one in [5, Lemma 2.3] and so is omitted.

Lemma 2.1
Suppose that x is an eventually positive solution to (4) and lim t→∞ z(t)/R λ (t) = a for λ = 0, 1. Then we have Next, we divide all eventually positive solutions to (4) into four groups. (4), then one of the following four cases holds:

Theorem 2.2 If x is an eventually positive solution to
S is the set of all eventually positive solutions of (4), and b is a positive constant.
Proof Assume that x is an eventually positive solution to (4). From (C2) and (C3), there exist a t 1 ∈ [t 0 , ∞) T and a p 1 satisfying |p 0 | < p 1 < 1 such that x(t) > 0, x(g(t)) > 0, x(h(t)) > 0, and |p(t)| ≤ p 1 for t ∈ [t 1 , ∞) T . For t ∈ [t 1 , ∞) T , according to (4) and (C4), we have which means that R n-1 is strictly decreasing on [t 1 , ∞) T . Moreover, it follows that If there exists a T ∈ [t 1 , ∞) T such that R n-2 (T, x(T)) ≤ 0, then by (6) we know that R n-2 is eventually negative. Otherwise, we arrive at R n-2 (t, x(t)) > 0 for all t ∈ [t 1 , ∞) T . Hence, R n-2 is always eventually monotonic. Letting t be replaced by s and integrating (6) from t 1 to t, t ∈ [σ (t 1 ), ∞) T , by (C1) we obtain which means that R n-2 is upper bounded. When n = 3, we see that r 2 z is eventually monotonic and upper bounded. When n ≥ 4, since r 2 R n-3 is eventually monotonic, it follows that r 2 R n-3 and R n-3 are eventually positive or eventually negative. Thus, R n-3 is eventually monotonic.
Since R n-2 is upper bounded, there exist a constant c 1 and a t 2 Substituting s for t and integrating (7) from which implies that R n-3 is upper bounded. When n = 4, we see that r 3 z is eventually monotonic and upper bounded. By analogy, for all n ≥ 3, it always satisfies the requirement that r n-1 z is eventually monotonic and upper bounded. Then we need to consider two cases.
Case 1. r n-1 z is eventually strictly decreasing. We can claim that Otherwise, there exist a constant c 2 < 0 and a t 3 ∈ [t 2 , ∞) T such that Letting t be replaced by s and integrating (9) as t → ∞. Then we get p 0 ∈ (-1, 0], and thus there exists a t 4 ∈ [t 3 , ∞) T such that x(t) < -p(t)x(g(t)) < p 1 x(g(t)) for t ∈ [t 4 , ∞) T . In view of (C3), there exists a positive integer N satisfying c k ∈ [t 4 , ∞) T for all k ≥ N . Moreover, for any k ≥ N + 1, which means that lim k→∞ x(c k ) = 0 and thus lim k→∞ z(c k ) = 0. It is in contradiction with lim t→∞ z(t) = -∞. Therefore, (8) holds.
If L 1 = 0, since r n-1 z and z are both eventually positive, then it follows that z is eventually strictly increasing. From the above, we know that lim t→∞ z(t) < 0 does not hold. Therefore, we get Case 2. r n-1 z is eventually strictly increasing, which means that r n-1 z is eventually positive or eventually negative.
If r n-1 z is eventually positive, since it is also upper bounded, then there exists a constant Substituting s for t and integrating (10) from t 3 to t, t ∈ [σ (t 3 ), ∞) T , we have If r n-1 z is eventually negative, then it follows that lim t→∞ r n-1 (t)z (t) ≤ 0. From the above, it means that lim t→∞ r n-1 (t)z (t) = 0 and so z (t) < 0 for t ∈ [t 2 , ∞) T . Moreover, we get Employing the L'Hôpital's rule in [3, Theorem 1.120], we obtain By virtue of Lemma 2.1, it is clear that one of the cases (A1)-(A4) holds. This completes the proof.

Main results
In this section, the existence of eventually positive solutions to (4) is presented. Now, we show a sufficient and necessary condition for the type A(∞, b).

Theorem 3.1 Equation (4) has an eventually positive solution x ∈ A(∞, b) if and only if there exists a constant K > 0 such that
where b is a positive constant.
Proof Suppose that x is an eventually positive solution to (4) satisfying x ∈ A(∞, b). By Lemma 2.1 we claim that If lim t→∞ z(t) < ∞, then it will cause lim t→∞ x(t) < ∞, which contradicts x ∈ A (∞, b).
In the following, the sufficient conditions for the types A(b, 0) and A(∞, 0) are given in Theorems 3.2 and 3.3, respectively.

Theorem 3.2 If there exists a constant K > 0 such that
then (4) has an eventually positive solution x ∈ A(b, 0), where b is a positive constant.
Proof Suppose that there exists a constant K > 0 such that (14) holds. Similarly as the proof of the sufficiency in Theorem 3.1, we consider two cases. Case 1. 0 ≤ p 0 < 1. When p 0 > 0, taking p 1 chosen in Theorem 3.1, then there exists a T 0 ∈ [t 0 , ∞) T such that When p 0 = 0, choose p 1 such that |p(t)| ≤ p 1 ≤ 1/13 for t ∈ [T 0 , ∞) T . There also exists a (5), and the operators U 2 and V 2 : Ω 2 → BC 0 [T 0 , ∞) T as follows: Similarly, there exists an x ∈ Ω 2 such that ( Since (15), we obtain 0 < lim t→∞ z(t) < ∞. By Lemma 2.1, it follows that Similarly, we get a conclusion as in Case 1. The proof is complete.

Theorem 3.3 If there exists a constant M > 0 such that |p(t)R(t)| ≤ M for t
then (4) has an eventually positive solution x ∈ A(∞, 0).