#### Abstract

We consider the nonlinear dynamic system . We establish some necessary and sufficient conditions for the existence of oscillatory and nonoscillatory solutions with special asymptotic properties for the system. We generalize the known results in the literature. Some examples are included to illustrate the results.

#### 1. Introduction

In this paper we investigate the nonlinear two-dimensional dynamic system: where is a nonnegative, rd-continuous function which is defined for . Here, is a time scale unbounded from above. We assume throughout that is a continuous function with for , and is continuous as a function of with sign property for and .

By the solution of system (1.1), we mean a pair of nontrivial real-valued functions which has property and satisfies system (1.1) for . Our attention is restricted to those solutions of system (1.1) which exist on some half-line and satisfy for any . As usual, a continuous real-valued function defined on is said to be oscillatory if it has arbitrarily large zeros, otherwise it is said to be nonoscillatory. A solution of system (1.1) is called oscillatory if both and are oscillatory functions, and otherwise it will be called nonoscillatory. System (1.1) is called oscillatory if its solutions are oscillatory.

The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. thesis in 1990 in order to unify continuous and discrete analysis (see [1]). Not only can this theory of the so-called “dynamic equations” unify the theories of differential equations and difference equations, but also extend these classical cases to cases “in between”, for example, to the so-called -difference equations and can be applied on other different types of time scales. Since Hilger formed the definition of derivatives and integral on time scales, several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [2] and references cited therein. A book on the subject of time scales (see [3]) summarizes and organizes much of time scale calculus. The reader is referred to Chapter 1 in [3] for the necessary time scale definitions and notations used throughout this paper.

The system (1.1) includes two-dimensional linear/nonlinear differential and difference systems, which were investigated in the literature, see for example [4–9] and the references therein.

On the other hand, the system (1.1) reduces to some important second-order dynamic equations in the particular case, for example where is rd-continuous on . In recent years there has been much research activity concerning the oscillation and nonoscillation of solutions of dynamic equation (1.2) on time scales. We refer the reader to the recent papers [10–13] and the references therein. However, most of previous studies for the system (1.1) have been restricted to the case where , for example [4–8, 14–18] and the references therein. Erbe and Mert [14, 17] obtained some oscillation results for the system (1.1). Fu and Lin [15] obtained some oscillation and nonoscillation criteria for the linear dynamic system (1.1).

Since there are few works about oscillation and nonoscillation of dynamic systems on time scales (see [15]), motivated by [9, 14, 15], in this paper we investigate oscillatory and nonoscillatory properties for the system (1.1) in the case of general in which and are not necessarily separable. In the next section, by means of appropriate hypotheses on and fixed point theorem, we establish some new sufficient and necessary conditions for the existence of nonoscillatory solutions with special asymptotic properties for the system (1.1). In Section 3, we obtain sufficient and necessary conditions for all solutions of the system (1.1) to be oscillatory via the results in Section 2 and some inequality techniques without using Riccati transformation. Our results not only unify the known results of differential and difference systems but also extend and improve the existing results of dynamic systems on time scales in the literature.

#### 2. Nonoscillation Results

In this section, we generalize and improve some results of [7–9, 15, 18]. Some necessary and sufficient conditions are given for the system (1.1) to admit the existence of nonoscillatory solutions with special asymptotic properties. These results will be used for the next section. Additional hypotheses on and are needed for this purpose.(H_{1}) For any positive constant and with , there exist positive constants and , depending possibly on and , such that implies
(H_{2}) There exists a positive constant such that for .(H_{3}) For any positive constant and with , there exist positive constants and , depending possibly on and , such that implies
where is a positive nondecreasing function.

For convenience, we will employ the following notation:

Theorem 2.1. * Assume that is nondecreasing and that (H _{1}) holds. Then system (1.1) has a nonoscillatory solution such that and if and only if for any positive constant *

*Proof. * Suppose that is a nonoscillatory solution of (1.1) such that and . Without loss of generality, we assume that . Then there exist and positive constants and such that for . Condition (H_{1}) implies that
for and some constant . It follows from the second equation in (1.1) that
Let and noting that , we have
Thus, from (2.5), (2.7) and the first equation in (1.1), we obtain that
which implies that (2.4) holds.

Conversely, suppose that (2.4) holds, we may assume that . In view of (H_{1}), there is a constant such that implies
Since (2.4) holds, we can choose large enough such that
Let be the Banach space of all real-valued rd-continuous functions on endowed with the norm . We defined a bounded, convex, and closed subset of as
Define an operator as follows:
Now we show that satisfies the assumptions of Schauder's fixed-point theorem (see [19, Corollary 6]).(i) We will show that for any . In fact, for any and , in view of (2.10), we get
Similarly, we can prove that for any and . Hence, for any .(ii) We prove that is a completely continuous mapping. First, we consider the continuity of . Let and as , then and as for any . Consequently, by the continuity of and , for any , we have
From (2.9), we obtain that
On the other hand, from (2.12) we have
for and
for . Therefore, from (2.16) and (2.17), we have
Referring to Chapter 5 in [20], we see that the Lebesgue dominated convergence theorem holds for the integral on time scales. Then, from (2.14) and (2.15), (2.18) yields , which implies that is continuous on .

Next, we show that is uniformly cauchy. In fact, for any , take and such that
Then for any and , we have
This means that is uniformly cauchy.

Finally, we prove that is equicontinuous on for any . Without loss of generality, we set . For any , we have for and
for .

Now, we see that for any , there exists such that when with , for any . This means that is equicontinuous on for any . By Arzela-Ascoli theorem (see [19, Lemma 4]), is relatively compact. From the above, we have proved that is a completely continuous mapping.

By Schauder’s fixed point theorem, there exists such that . Therefore, we have
Set
Then and . On the other hand,
which implies that and . The proof is complete.

Corollary 2.2. * Suppose that is nondecreasing and that (H _{1}) and (H_{2}) hold. Then system (1.1) has a nonoscillatory solution such that and if and only if for some *

Theorem 2.3. * Suppose that and is nondecreasing. Suppose further that (H _{3}) holds. Then system (1.1) has a nonoscillatory solution such that and if and only if for some *

*Proof. * Suppose that is a nonoscillatory solution of (1.1) such that and . We may assume that . Hence, there exist and positive constant such that and for . By condition (H_{3}), there exists a constant such that for . According to the second equation in (1.1), we have
which implies that (2.26) holds with .

Conversely, Let (2.26) holds for some , where . By (H_{3}), there exists a constant such that implies for . Take so large that
where . We introduce be the partially ordered Banach space of all real-valued and rd-continuous functions with the norm , and the usual pointwise ordering .

Define
It is easy to see that is a bounded, convex, and closed subset of . Let us further define an operator as follows:

Since it can be shown that is continuous and sends into a relatively compact subset of , the Schauder’s fixed point theorem ensures that the existence of an such that , this is
Set
Then and . On the other hand, by L'Hôpital's Rule (see [15, Lemma 2.11]), we have
and . The proof is complete.

*Remark 2.4. *Theorems 2.1 and 2.3 extend and improve essentially the known results of [7–9, 15, 18].

#### 3. Oscillation Results

In this section, we need some additional conditions to guarantee that the system (1.1) has oscillatory solutions.(H_{4}) There exists a continuous nondecreasing function such that
and for some constants and with .(H_{5}) There exists a continuous nondecreasing function being a constant, such that
and for some positive constant and .

Theorem 3.1. *Suppose that and is nondecreasing. Suppose further that (H _{1}), (H_{2}) and (H_{4}) hold. Then system (1.1) is oscillatory if and only if for all *

*Proof. * If (3.3) does not hold, by Theorem 2.1, system (1.1) has a nonoscillatory solution such that and .

Conversely, suppose that (3.3) holds and that (1.1) has a nonoscillatory solution for . We may assume that for , where . Since , it is easy to show that . From the second equation in (1.1), we have . Hence, . It follows from the first equation in (1.1) that , and by Theorem 2.1. Integrating the second equation in (1.1) from to yields that
By (3.4), (H_{2}) and in view of nondecreasing , it follows that
for .

Since (H_{4}) holds and , there is and such that
for . From (3.5) and (3.6), we get
Integrating (3.7) from to , we have
Since , and are nondecreasing, we obtain
By (H_{4}), (3.8) and (3.9), we get
which contradicts (3.3) when . The proof is complete.

Theorem 3.2. *Suppose that and (H _{5}) holds. Suppose further that is nondecreasing in for each fixed and is nondecreasing. Then system (1.1) is oscillatory if and only if for all *

* Proof. * If (3.11) does not hold, by Theorem 2.3, system (1.1) has a nonoscillatory solution such that and .

Conversely, suppose that (3.11) holds and that system (1.1) has a nonoscillatory solution for . We assume that for , where . Then by same argument in the proof of Theorem 3.1, we have eventually. We claim that (3.11) implies . In fact, if , then for . According to the first equation in (1.1), we get
Integrating the second equation in (1.1) from to , we have
which is a contradiction. Hence, .

By (H_{5}), we have
From (3.14), it follows
Hence,
In view of (H_{5}) and (3.11), this is a contradiction. The proof is complete.

*Remark 3.3. * Theorems 3.1 and 3.2 improve the existing results of [15, 18].

*Example 3.4. * Consider the system:
where and are constants as well as .

Let
It is easy to see that is increasing and for ,
For , we have
that is, (2.25) holds. By Corollary 2.2, system (3.17) has a nonoscillatory solution such that and .

On the other hand, For , we obtain
Hence, (2.26) holds. By Theorem 2.3, system (3.17) has a nonoscillatory solution such that and .

*Example 3.5. * Consider the system:
where and .

Let
Obviously, is increasing in for fixed , and taking , we have
On the other hand, we obtain
that is, (3.3) holds. Hence, system (3.22) is oscillatory by Theorem 3.1.

*Example 3.6. * Consider the system:
on a time scale which contains only isolated points and is unbounded above. Here, , is a nonnegative rd-continuous function on .

We take , then all conditions of Theorem 3.2 are satisfied. Hence, system (3.26) is oscillatory if and only if
On the other hand, system (3.26) can be written in the Emden-Fowler equation:
Since we do not assume that is a quotient of odd positive integers, (3.28) includes the equation studied in [21]. Theorem 3.2 generalizes and improves Theorem 7 of [21].

#### Acknowledgment

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have lead to the present improved version of the original paper.