Nonoscillation Criteria for Two-Dimensional Time-Scale Systems

Abstract We study the existence and nonexistence of nonoscillatory solutions of a two-dimensional systemof first-order dynamic equations on time scales. Our approach is based on the Knaster and Schauder fixed point theorems and some certain integral conditions. Examples are given to illustrate some of our main results.


Introduction
In this paper, we study on the asymptotic behavior of solutions of the nonlinear system of the rst-order dynamic equations x ∆ (t) = a(t)f (y(t)) y ∆ (t) = −b(t)g(x(t)), (1.1) where f , g ∈ C(R, R) are nondecreasing such that uf (u) > , ug(u) > for u ≠ and a, b ∈ C rd [t , ∞) T , R + . Whenever we write t ≥ t , we mean that t ∈ [t , ∞) T := [t , ∞) ∩ T. A time scale, denoted by T, is a closed subset of real numbers. An excellent introduction of time scales calculus can be found in [2,3] by Bohner and Peterson. Throughout this paper, we assume that T is unbounded above. We call (x, y) a proper solution if it is de ned on [t , ∞) T and sup{|x(s)|, |y(s)| : s ∈ [t, ∞) T } > for t ≥ t . A solution (x, y) of (1.1) is said to be nonoscillatory if the component functions x and y are both nonoscillatory, i.e., either eventually positive or eventually negative. Otherwise, it is said to be oscillatory. Throughout this paper, without loss of generality, we assume that x is eventually positive. Our results can be shown for that x is eventually negative similarly. If T = R and T = Z, equation (1.1) turns out to be system of rst-order di erential equations and di erence equations x ′ = a(t)f (y(t)) y ′ = −b(t)g(x(t)) see [9], ∆xn = an f (yn) ∆yn = −bn g(xn) see [10], respectively. Oscillation and nonoscillation criteria for two-dimensional time scale systems have been studied by [1], [5], [8], [11,12].
One can easily show that any nonoscillatory solution (x, y) of system (1.1) belongs to one of the following classes: where M is the set of all nonoscillatory solutions of system (1.1). In this paper, we only focus on the existence and nonexistence of solutions of system (1.1) The set up of this paper is as follows. In Section 1, we give preliminary lemmas that are used in the proofs of our main theorems. In Section 2, we introduce the subclasses that are obtained by using system (1.1) and show the existence of nonoscillatory solutions of system (1.1) by using the Knaster and Schauder xed point theorems and certain improper integrals. In Section 3, we show the nonexistence of such solutions by relaxing the monotonicity condition on the functions f and g. We nalize the paper by giving some examples and a conclusion.
The following lemma is shown in [1]. For convenience, let us set The following lemma shows the existence and nonexistence of nonoscillatory solutions of system (1.1) by using convergence/divergence of Y(t) and Z(t). Proof. Here we only prove part (c) because (d) can be shown similarly. Suppose that Y(t ) < ∞ and Z(t ) = ∞. So assume that there exists a nonoscillatory solution (x, y) of system (1.1) in M + such that xy > eventually. Without loss of generality, assume that x(t) > for t ≥ t . Then by monotonicity of x and g, there exists a number k > such that g(x(t)) ≥ k for t ≥ t . Integrating the second equation of system (1.1) from t to t gives us As t → ∞, it follows that y(t) → −∞. But this contradicts that y is eventually positive. Proof is by contradiction.
The following two lemmas are related with the rst component function of any nonoscillatory solutions of (1.1) when Y(t ) < ∞. Proof. Suppose that Y(t ) < ∞ and (x, y) is a nonoscillatory solution of system (1.1). Then by Lemma 1.1, x and y are themselves nonoscillatory. Without loss of generality, assume that there exists t ≥ t such that x(t) > for t ≥ t . If (x, y) ∈ M − , then by the rst equation of system (1.1), x ∆ (t) < for t ≥ t . Therefore, the Brought to you by | Missouri University of Science and Technology Authenticated Download Date | 4/24/19 6:14 PM limit of x exists. So let us show that the assertion follows if (x, y) ∈ M + . Suppose (x, y) ∈ M + . Then from the rst equation of system (1.1), we have x ∆ (t) > for t ≥ t . Hence two possibilities might happen: The limit of the component function x exists or blows up. Now let us show that lim t→∞ x(t) = ∞ cannot happen. Integrating the rst equation of system (1.1) from t to t and using the monotonicity of y and f yield t t a(s)∆s.
Taking the limit as t → ∞, it follows that x has a nite limit. This completes the proof.
Proof. Suppose that Y(t ) < ∞ and (x, y) is a nonoscillatory solution of system (1.1). Without loss of generality, let us assume that x is eventually positive. Then by Lemma 1.3, we have x(t) ≤ d for t ≥ t and for some d > .
If y(t) > for t ≥ t , then x is eventually increasing by the rst equation of system (1.1). So for large t, the assertion follows. If y(t) < for t ≥ t , then integrating the rst equation of system (1.1) from t to ∞ and the monotonicity of f and y give Setting c = −f (y(t )) > on the last inequality proves the assertion.
According to Lemma 1.2 (c), we assume Y(t ) < ∞ and Z(t ) = ∞ from now on. Let (x, y) be a nonoscillatory solution of system (1.1) such that the component function x of the solution (x, y) is eventually positive. Then by the second equation of system (1.1), we have y < and eventually decreasing. Then for d < , we have y → d or y → −∞. In view of Lemma 1.3, x has a nite limit. So in light of this information, we obtain the following lemma.

Existence of Nonoscillatory Solutions in M −
The following theorems show the existence of nonoscillatory solutions in subclasses of M − given in Lemma 1.5.
By integrating the second equation from t to t, using inequality (2.2) with c = c and the monotonicity of g, we have So as t → ∞, the assertion follows since y has a nite limit. (For the case x < eventually, the proof can be shown similarly with c < .) Conversely, suppose that (2.1) holds for some c > . (For the case c < can be shown similarly.) Then there exist t ≥ t and d > such that where c = −f (− d). Let X be the space of all continuous and bounded functions on [t , ∞) T with the norm y = sup Let Ω be the subset of X such that It is easy to see that T maps into itself. Indeed, we have Then the Lebesque dominated convergence theorem and the continuity of g give (Tyn) − (Ty) → as n → ∞, i.e., T is continuous. Also since it follows that T(Ω) is relatively compact. Then by the Schauder Fixed point theorem, there existsȳ ∈ Ω such thatȳ = Tȳ. So as t → ∞, we haveȳ(t) → − d < . Settinḡ Integrating the rst equation from t to ∞ gives (2.4) By integrating the second equation from t to t and using (2.4, ) we get By setting c = d > and d = c < and taking the limit of the last inequality as t → ∞, the assertion follows. (The case x < eventually can be done similarly with c > and d < .) where c < and d > . (The case c > and d < can be done similarly.) Let X be the set of all all bounded and continuous functions endowed with the norm y = sup t∈[t ,∞) T |y(t)|. Clearly (X, · · · ) is a Banach space, see [4]. De ne a subset Ω of X such that De ne an operator F : Ω → X such that Second, we show that F is continuous on Ω. Let yn be a sequence in Ω such that yn → y ∈ Ω =Ω. Then By the Lebesque dominated convergence theorem and the continuity of f and g, it follows that F is continuous.
and therefore F is equibounded and equicontinuous, i.e., relatively compact. So by the Schauder xed point theorem, there existsȳ ∈ X such that Proof. Suppose that there exists a nonoscillatory solution (x, y) ∈ M − B,∞ such that x > eventually, x(t) → c and y(t) → −∞ as t → ∞, where < c < ∞. Because of the monotonicity of x and the fact that x has a nite limit, there exist t ≥ t and c > such that

a(τ)f (y(τ))∆τ and taking limit as t → ∞, we have that there exists a nonoscillatory solution in
Integrating the rst equation from t to t gives us So by taking the limit as t → ∞, we have ∞ t a(s)|f (y(s))|∆s < ∞. (2.7) The monotonicity of g, (2.6) and integrating the second equation from t to t yield Ω of X such that First, we need to show that F : Ω → Ω is an increasing mapping into itself. It is obvious that it is an increasing mapping and since (2.9), it follows that F : Ω → Ω. Then by the Knaster xed point theorem, there existsx ∈ Ω such that By taking the derivative of (2.12) and the fact that f is an odd function, we havē (x(τ))∆τ and using the monotonicity of g givē So we have thatx(t) > andȳ(t) < for t ≥ t , andx(t) → d andȳ(t) → −∞ as t → ∞. This completes the proof.
for some c > and any d > (c < and d < ), where f is an odd function, then M − ,∞ ≠ ∅.
It is clear that (Ω, ≤) is complete. De ne an operator F : Ω → X such that It is clear that F is an increasing mapping. We also need to show that F : Ω → Ω. By (2.13), the monotonicity of g and the fact that x ∈ Ω, we have for t ≥ t and any d > . So by setting f ( ) = d , we have Then by the Knaster xed point theorem, there existsx ∈ Ω such thatx = Fx. Settinḡ using the fact thatx ∈ Ω and taking the limit ofx andȳ as t → ∞, the proof is complete. (The case c < and d < can be shown similarly.) Brought to you by | Missouri University of Science and Technology Authenticated Download Date | 4/24/19 6:14 PM

Nonexistence of Nonoscillatory Solutions in M −
In the previous section, we used the monotonicity of the functions f and g in order to show the existence of nonoscillatory solutions of system (1.1). Nonexistence of such solutions in M − ,B , M − B,B , and M − B,∞ directly follows from Theorems 2.1 -2.3. In this section, we relax this condition by assuming that there exist positive constants F and G such that in order to get the emptiness of those subclasses. The following theorems show the nonexistence of such solutions in the subclasses of M − given in Lemma 1.5.
Proof. Assume that there exists a solution (x, y) ∈ M − such that x > eventually, x → and y → −∞ as t → ∞. By Lemma 1.4, there exist c > and t ≥ t such that By integrating the second equation from t to t, and using (3.1) and (3.3), there exist t ≥ t and G > such that By integrating the rst equation from t to t, and using (3.4) and (3.1), there exist t ≥ t and F > such that As t → ∞, it contradicts to (3.2). So the assertion follows. Proof is by contradiction. (3.8) By integrating the second equation from t to t, and by using (3.1) and (3.8), we have that there exists G > such that So as t → ∞, it contradicts to (3.6). Proof is by contradiction. Proof. Suppose (3.9) holds and that there exists a nonoscillatory (x, y) solution of (1.1) in M − B,∞ such that x > eventually, x(t) → c > and y(t) → −∞ as t → ∞. Since x has a nite limit, there exist t ≥ t such that c ≤ x(t) for t ≥ t . Integrating the rst equation from t to t gives (3.10) By taking the limit of (3.10) as t → ∞, we have ∞ t a(s)|f (y(s))|∆s < ∞. (3.11) By integrating the second equation from t to t, using (3.1) and the fact that x(t) ≥ c for t ≥ t , we have that there exist t ≥ t and G > such that By (3.12) and the fact that f is an odd function, there exist t ≥ t and F > such that By taking the limit of the last inequality as t → ∞ and by (3.11), we obtain a contradiction. So the assertion follows.

Examples
In this section, we give some examples in order to highlight our main results.
One can easily show that T a(s)∆s = (q − ) So as s → ∞, we have that One can also show So as T → ∞, we have

Conclusions
In this paper, we consider the case Y(t ) < ∞ and Z(t ) = ∞ in order to show the existence and nonexistence of nonoscillatory solutions in M − . When we have the case Y(t ) = ∞ and Z(t ) < ∞, (5.1) we know from Lemma 1.2(d) that all nonoscillatory solutions belong to M + . So as a future work, we will consider the case (5.1) in order to show the existence and nonexistence of nonoscillatory solutions in M + . Another open problem is to extend our main results to the delay equation x ∆ (t) = a(t)f (y(t)) y ∆ (t) = −b(t)g(x(τ(t))), (5.2) where τ : T → T is an increasing function such that τ(t) < t and τ(t) → ∞ as t → ∞. Even though the system x ∆ (t) = a(t)f (y(t)) y ∆ (t) = −b(t)g(x(t − τ)), where τ > , is considered in [11], it is not valid for all time scales, such as T = q N , where q > .