Nonoscillatory Solutions to Higher-Order Nonlinear Neutral Dynamic Equations

For a class of nonlinear higher-order neutral dynamic equations on a time scale, we analyze the existence and asymptotic behavior of nonoscillatory solutions on the basis of hypotheses that allow applications to equations with different integral convergence and divergence of the reciprocal of the coefficients. Two examples are presented to demonstrate the efficiency of new results.


Introduction
In this article, we investigate the existence and asymptotic behavior of nonoscillatory solutions to a class of dynamic equations on a time scale T R n (t, x(t)) + f (t, x(h(t))) = 0, where sup T = ∞, t ∈ [t 0 , ∞) T with t 0 ∈ T, n ≥ 3, and Definition 1.As is customary in this field, a solution of Equation (1) is termed nonoscillatory provided that x is either eventually positive or eventually negative; otherwise, it is said to be oscillatory.
In the special case when n = 2, Equation (1) reduces to a dynamic equation r(t)(x(t) + p(t)x (g(t))) ∆ ∆ + f (t, x (h(t))) = 0, which was examined by Deng and Wang [11] and Gao and Wang [13].The different assumptions ∞ t 0 1/r(t)∆t = ∞ in [11] and 1/r(t)∆t < ∞ in [13] cause a phenomenon that the asymptotic behavior of nonoscillatory solutions to Equation ( 3) is greatly different.Moreover, it is clear that the asymptotic behavior is more complicated assuming that To find a more general rule of the existence and asymptotic behavior of nonoscillatory solutions to Equation (1), Qiu [19] considered Equation (1) in the special case where n = 3, namely, to divide the eventually positive solutions of Equation ( 4) into five groups, and presented some existence conditions of them, respectively.Qiu and Wang [20] were concerned with Equation (1) under the conditions It shows that there exist only two cases that lim t→∞ x(t) = b > 0 and lim t→∞ x(t) = 0, where x is assumed to be an eventually positive solution of Equation (1).Furthermore, this result can be extended to [13] when n = 2 and [24] when n = 1.
For Equation (1), it is significant to continue to investigate more general cases of the convergence and divergence of the integrals Throughout, we assume that the following hypotheses are satisfied: then define where is supposed to hold.
In view of the results established in [11,13], it is not difficult to see that the existence and asymptotic behavior of nonoscillatory solutions to Equation (1) are more complex than those in [20].Therefore, the criteria obtained in this article develop and improve some known conclusions reported in the references.Finally, we present two examples to demonstrate the versatility of new results.

Auxiliary Results
To establish criteria for the existence of nonoscillatory solutions to Equation (1), we need a Banach space and Krasnoselskii's fixed point theorem as follows.
Lemma 1. (Krasnoselskii's fixed point theorem) Let Ω be a bounded, convex, and closed subset of a Banach space X.Assume that there are two operators U, S : Ω → X such that U is contractive, S is completely continuous, and Ux + Sy ∈ Ω for all x, y ∈ Ω.Then, U + S has a fixed point in Ω.
Define z(t) = x(t) + p(t)x(g(t)).Without loss of generality, we consider only the eventually positive solutions of Equation (1).Then, we have the following lemma (see [12] (Lemma 2.3) and [22] (Lemma 2.1)).Lemma 2. Let x be an eventually positive solution of Equation (1) and where λ = 1 only if condition (5) holds.Suppose that a is finite.Then, For the sake of simplicity, we give a classification to divide all eventually positive solutions of Equation ( 1) into four types.Theorem 1.Let x be an eventually positive solution of Equation (1).Then, there are four possible types for x: Proof.Let x be an eventually positive solution of Equation (1).By virtue of (C2) and (C3), there exist a Then, we need to consider two cases.
Case 1. Suppose first that R ∆ n−2 is eventually negative.Then, where −∞ ≤ L 2 < 0. Hence, there exist a constant c 1 < 0 and a Integrating inequality (7 n−3 is negative and R n−3 is strictly decreasing for large t.When n = 3, z is nonoscillatory.We can declare that lim where 0 ≤ L 0 < ∞.Do not assume it; that is, lim t→∞ z(t) < 0.Then, we have p 0 ∈ (−1, 0] and so there exists a which yields lim k→∞ x(c k ) = 0 and lim k→∞ z(c k ) = 0.This contradicts the assumption, and so equality (8) holds. When If there is a which yields Integrating inequality (10) where 0 ≤ L 2 < ∞.We consider the following two cases: Integrating inequality (11) from t 2 to t, t ∈ [σ(t 2 ), ∞) T , we arrive at .
By virtue of (C1), r 2 (t)R ∆ n−3 (t, x(t)) → ∞ as t → ∞, which implies that R ∆ n−3 is positive and R n−3 is strictly increasing for large t.Thus, R n−3 is nonoscillatory.When n = 3, R n−3 = z.As before, we have lim where 0 ) is nonoscillatory, and R n−4 is eventually monotonic.If n = 4, then R n−4 = z, and equality (12) holds.When n ≥ 5, it follows that R n−5 is eventually monotonic similarly.Analogously, for n ≥ 3, it follows that equality (12) always holds.
where −∞ < L 1 ≤ ∞.Moreover, r 2 R ∆ n−3 is strictly increasing for large t.It follows that R ∆ n−3 is nonoscillatory.Thus, R n−3 is always eventually monotonic and nonoscillatory.Similarly as before, we deduce that lim t→∞ z(t) = L 0 ≥ 0 when n ≥ 3. When Integrating inequality (13) from t 3 to t, t ∈ [σ(t 3 ), ∞) T , we have When n = 3, R n−3 = z, and so z is upper bounded.When n ≥ 4, there exist a constant Similarly, we see that R n−4 is upper bounded.If n = 4, then R n−4 = z, and thus z is upper bounded.Analogously, for n ≥ 3, we deduce that z is always upper bounded.Hence, 0 ≤ L 0 < ∞.
The proof is complete.

Main Results
We establish several criteria for the existence of various types of eventually positive solutions of Equation (1).Firstly, suppose that which means that condition (5) is not satisfied.
Theorem 2. Let condition (14) be fulfilled.Then, Equation (1) has an eventually positive solution x satisfying for some constant K > 0, where b > 0 is a constant.
Remark 1. Actually, the assumption (14) in Theorem 2 is not needed in the sufficiency of its proof.Thus, we obtain a corollary as follows.
Corollary 1. Assume that condition (15) is fulfilled for some constant K > 0.Then, Equation (1) has an eventually positive solution x satisfying lim t→∞ x(t) = b, where b > 0 is a constant.

Now, we let
where S stands for the set containing all eventually positive solutions of Equation ( 1).Then, a lemma is presented as follows.
Lemma 3. Let x be an eventually positive solution of Equation (1) such that lim t→∞ x(t) = ∞.Then, condition (5) is satisfied, and x ∈ A(0) or x ∈ A(b), where b > 0 is a constant.
Theorem 3. Equation (1) has an eventually positive solution which is in A(b) iff for some constant K > 0, where b > 0 is a constant.
Remark 2. It is not easy to establish the sufficient and necessary conditions which guarantee that Equation (1) has an eventually positive solution x satisfying lim t→∞ x(t) = 0. We refer the reader to [20] (Theorems 3.2 and 3.3) for sufficient conditions to ensure it.