Almost oscillatory three-dimensional dynamical system
© Akin-Bohner et al; licensee Springer. 2012
Received: 5 March 2012
Accepted: 17 April 2012
Published: 17 April 2012
In this article, we investigate oscillation and asymptotic properties for 3D systems of dynamic equations. We show the role of nonlinearities and we apply our results to the adjoint dynamic systems.
2010 Mathematics Subject Classification: 39A10
KeywordsTime scales Oscillation Three-dimensional dynamical system
Here, we consider only unbounded time scales. The theory of time scales is initiated by Stefan Hilger  his PhD dissertation in 1988 in order to unify continuous and discrete analysis. The theory of dynamic equations on time scales helps us not only to avoid proving results twice but also to extend them for other time scales such as the set of all integer multiples of a number h > 0, the set of all integer powers of a number q > 1. We refer readers the books by Bohner and Peterson [2, 3] for an excellent introduction with applications and advances in dynamic equations.
is considered on time scales by Akin-Bohner et al. , where α, β, γ are ratios of odd positive integers. The continuous version of a system similar to system (1) has been considered by Chanturia  and the discrete version of system (1) by Schmeidel [6, 7] see also references therein. System (4) with α = β = 1 can be written as a third-order difference equation with quasi-differences those oscillatory and asymptotic properties have been investigated in [8, 9].
First, we study the case λ = 1. We will obtain asymptotic properties of nonoscillatory solutions of system (1) and we will establish oscillation criteria for system (1). Then we consider the case λ = -1 and we extend our results proved in . In the last section, we apply our results to adjoint systems and we discuss the role of nonlinearities. Our results are new also for the difference systems.
A proper solution of system (1) is said to be oscillatory if all of its components x, y, z are oscillatory, i.e., neither eventually positive nor eventually negative. Otherwise, proper solution is said to be nonoscillatory. Obviously, if one component of a solution (x, y, z) is eventually of one sign, then all its components are eventually of one sign and so nonoscillatory solutions have all components nonoscillatory.
Remark 1.1. The terminology used in the above definitions is not unified in the literature. The terminology Property A and Property B are due to [5, 10]. As can be noticed in [, p. 126] in a picturesque way, Property A and Property B state that every solution which may oscillate, does oscillate. Some authors use a different terminology--the system or higher order equation is almost oscillatory, or strongly oscillatory.
Changing order of integration is used in our main results. The proof of following lemma can be found in the article by Akin-Bohner and Sun .
2 Nonoscillatory solutions of system (1): case λ = 1
Throughout this and the following section, we consider the system (1), when λ = 1.
In this section, we study asymptotic properties of nonoscillatory solutions which we use in the following section. The following lemma is the analogy of a lemma in .
Then every nonoscillatory solution of system (1) with λ = 1 is of either Type (a) or Type (c).
Proof. Let (x, y, z) be a nonoscillatory solution of system (1). Without loss of generality, we assume that x(t) > 0 for . Then we have y(t) and z(t) are monotone for t ≥ T. Since y is monotonic, we have either y(t) < 0 or y(t) > 0 for all t ≥ T. Similarly, z(t) < 0 or z(t) > 0 for all t ≥ T.
First, let z(t) > 0 for t ≥ T. Suppose y(t) < 0 for large t. Since z is positive and increasing, there exists such that for large t. From here and integrating the second equation we get , thus y(t) → ∞, which is a contradiction with the fact that y(t) < 0. Therefore, this case is not possible, and so (x, y, z) is of Type (a).
Now let z(t) < 0 for t ≥ T. Suppose that y(t) < 0 for large t. It implies that x is eventually positive decreasing. Integrating the first equation, we get x(t) → ∞, which is a contradiction with the boundedness of x. Therefore, this case is not possible, and so (x, y, z) is of Type (c).
The proof for the case when x(t) < 0 for large t is analog. □
Solutions of Type (a) are sometimes called strongly monotone solutions (see e.g., ).
This implies that by (2).
From the above inequality, (2) implies . □
Then any Type (a) solution (x, y, z) of system (1) with λ = 1 satisfies (5).
As t → ∞, by (8). By Lemma 2.2, the proof is completed. □
We conclude this section with the property Type (c) solution of (1) which will be needed in the following section.
and therefore (2) implies that . But this contradicts the fact that y(t) > 0 for t ≥ T1. Therefore, . □
3 Almost oscillatory system: case λ = 1
We now establish conditions that system (1) is almost oscillatory.
Observe that by Lemmas 2.1 and 2.2, every bounded solution is either Type (c) or oscillatory. Hence, if system (1) is almost oscillatory, then every bounded solution is oscillatory.
Our first result is
Then system (1) with λ = 1 is almost oscillatory.
and so (10) implies . This contradicts the assumptions on z. Therefore, solutions of Type (c) are not possible. If (x, y, z) is a Type (a), then from Lemma 2.2 and (10), we obtain (5). □
Remark 3.1. Theorem 3.1 extends [, Theorem 2] for difference systems where it is proved that every bounded solution of (1) with λ = 1 is oscillatory.
Then every nonoscillatory solution of (1) with λ = 1 is a strongly monotone solution.
and (8) holds, then system (1) with λ = 1 is almost oscillatory.
Passing t → ∞ and using (12), we get a contradiction with the boundedness of y.
The second statement follows from Theorem 2.1. □
then every nonoscillatory solution of (1) with λ = 1 is a strongly monotone solution. In addition, if (8) holds, then system (1) with λ = 1 is almost oscillatory.
Proof. By Lemma 2.1 any nonoscillatory solution of system (1) is either of Type (a) or (c) for . We show that a nonoscillatory solution of system (1) of Type (c) cannot occur. Assume that there exists a nonoscillatory solution (x, y, z) of system (1) of Type (c) for t ≥ T. Without loss of generality, we assume that x(t) > 0 for t ≥ T.
but this contradicts (15) as t → ∞. □
Remark 3.2. Theorem 3.3 extends [, Theorem 2.4] stated for differential systems.
4 Almost oscillatory system: case λ = -1
In this section, we study the case λ = -1.
We start with the classification of solutions of system (1) with λ = -1. This is an analogue of Kiguradze lemma.
Then every nonoscillatory solution of system (1) with λ = -1 is either Type (a) or Type (b).
Proof. It is similar as the proof [, Lemma 3.2], so the details are omitted. □
Solution of Type (b) are sometimes called Kneser solutions (see e.g., ).
and so (2) implies . This contradicts our assumptions about the nature of x and therefore . In a similar way, we can show that . □
Theorem 4.1. Assume (10). Then system (1) with λ = -1 is almost oscillatory.
and passing t → ∞ we get a contradiction with the boundedness of z. □
Similarly, as in [, Theorem 3.1] for system (4) the following holds.
then every Kneser solution of system (1) with λ = -1 satisfies.
Proof. Without loss of generality, assume that (x, y, z) is a Kneser solution of system (1) such that x(t) > 0 for t ≥ t0, t0 ∈ [0,∞). From the first equation of system (1), x is nonincreasing, and . By Lemma 4.2, . We now show that , and so we let . Then there exists T1≥ t0, T1 ∈ [0, ∞) and L > 0 such that h(x(t)) > L β for t ≥ T1.
If (17), then , which is a contradiction.
This implies that , a contradiction. □
then every nonoscillatory solution of (1) with λ = -1 is a Kneser solution.
In addition, if (17) holds, then system (1) with λ = -1 is almost oscillatory.
Proof. Suppose (x, y, z) is a nonoscillatory solution of system (1). By Lemma 4.1, each nonoscillatory solution of system (1) is either Type (a) or Type (b). Without loss of generality, we assume that x(t) > 0 for .
which yields (21). Hence, passing t → ∞ in (4), we get a contradiction. Therefore, system (1) cannot have Type (a) solutions. So, every nonoscillatory solution is of Type (b). By Lemma 4.2 and Theorem 4.2, we have (6). So this completes the proof. □
Remark 4.1. Theorem 4.3 extends [, Theorem 1] stated for difference systems.
5 Conclusion and application to adjoint systems
In this section, we consider (4) where λ = ± 1 and we summarize the above results.
We conclude that if (10) holds, then, independently on the nonlinearities, (1) is almost oscillatory for λ = ± 1. If (16) holds, then, in general, almost oscillation depends on the sign of λ and on the types of nonlinearities.
The terminology "adjoint system" is used due to the fact that the corresponding matrices of systems (22) and (23) are adjoint.
Applying our results from Sections 3 and 4 we get conditions when both systems are almost oscillatory.
then linear adjoint systems (22) and (23) are almost oscillatory.
Thus, interchanging a and b, we get from Theorem 4.1 that every Kneser solution of (23) satisfies (6). This together with [, Theorem 4.4] result that (23) is almost oscillatory. □
Then both systems (22) and (23) are almost oscillatory.
Proof. It follows from Theorems 3.2 and 4.3. □
The second author is supported by Grant P201/11/0768 of the Grant Agency of the Czech Republic.
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