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On oscillatory behavior of two-dimensional time scale systems
- Özkan Öztürk^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-018-1475-4
© The Author(s) 2018
- Received: 25 April 2017
- Accepted: 7 January 2018
- Published: 15 January 2018
Abstract
This paper deals with long-time behaviors of nonoscillatory solutions of a system of first-order dynamic equations on time scales. Some well-known fixed point theorems and double improper integrals are used to prove the main results.
Keywords
- nonoscillation
- long-time behavior
- two-dimensional system
- asymptotic properties
- time scale systems
MSC
- 34N05
- 39A10
- 39A13
1 Introduction
This paper helps readers to understand the significance of the nonoscillation theory in a more general context. In fact, nonoscillation plays a very important role in understanding the long-time behavior of solutions of a system for several reasons, for example, stability and control theories. Two-dimensional systems of first-order equations have several real-life applications in engineering. For example, Bartolini et al. [4–6] considered a second-order system in order to control uncertain nonlinear systems by some control techniques, for example, sliding mode and approximate linearization. In addition to nonoscillatory solutions of second-order systems, periodic and subharmonic solutions were also considered in [7–9], and important results were obtained.
Theorem 1
Let X be a Banach space such that Y is a closed, nonempty, convex, and bounded subset of X. Suppose also \(F:Y\rightarrow Y\) is a compact operator. Then F has a fixed point.
Theorem 2
If \((Y, \leq)\) is a complete lattice and \(F:Y\rightarrow Y\) is order-preserving, then F has a fixed point. As a matter of fact, the set of fixed points of F is a complete lattice.
2 Main results
2.1 Existence in \(S^{-}\)
Theorem 3
Let \(B(t_{0},\infty)<\infty\). Then \(S^{-}_{B, B}\neq\emptyset\) if and only if \(I_{1}<\infty\), provided that \(k<0\) and \(l>0\).
Proof
Theorem 4
Suppose \(B(t_{0})<\infty\). \(S^{-}_{B,0}\neq\emptyset\) if and only if \(I_{1}<\infty\), where \(k=0\) and \(l>0\).
Proof
Theorem 5
Suppose \(A(t_{0})<\infty\). \(S^{-}_{0,B}\neq\emptyset\) if and only if \(I_{2}<\infty\) for \(m>0\).
Proof
Theorem 6
Suppose \(A(t_{0})<\infty\). \(S^{-}_{0,0}\neq\emptyset\) if \(I_{1}<\infty\) and \(I_{2}=\infty\) for \(k=0\), \(l<0\), and \(m>0\), provided that f is odd.
Proof
2.2 Nonexistence in \(S^{-}\)
Theorem 7
Let \(B(t_{0},\infty)<\infty\). If \(I_{3}=\infty\), then \(S^{-}_{B,B}= \emptyset\).
Proof
Theorem 8
Suppose \(B(t_{0},\infty)<\infty\). If \(I_{4}=\infty\), then \(S^{-}_{B,0}=\emptyset\).
Proof
The following theorem can be proven similarly to the previous theorems.
Theorem 9
Let \(A(t_{0},\infty)<\infty\). If \(I_{3}=\infty\), then \(S^{-}_{0,B}=\emptyset\).
3 Examples
Making a statement without examples can make the results muddy, whereas examples make results clearer and give more information to readers. Therefore, we give the following examples for validating our claims.
Theorem 10
([2, Theorem 1.79])
Example 1
Example 2
4 Conclusion
Existence in \(\pmb{S^{-}}\)
\(S^{-}_{B,B} \) | ≠∅ | \(B(t_{0},\infty)<\infty\) | \(I_{1}<\infty\), k<0, l>0 |
\(S^{-}_{B,0} \) | ≠∅ | \(B(t_{0},\infty)<\infty\) | \(I_{1}<\infty\), k = 0, l>0 |
\(S^{-}_{0,B} \) | ≠∅ | \(A(t_{0},\infty)<\infty\) | \(I_{2}<\infty\), m>0 |
\(S^{-}_{0,0} \) | ≠∅ | \(A(t_{0},\infty)<\infty\) | \(I_{1}<\infty\), \(I_{2}=\infty\), k = 0, l<0, m>0 |
Nonexistence in \(\pmb{S^{-}}\)
\(S^{-}_{B,B} \) | =∅ | \(B(t_{0},\infty)<\infty\) | \(I_{3}=\infty \) |
\(S^{-}_{0,B} \) | =∅ | \(A(t_{0},\infty)<\infty\) | \(I_{3}=\infty \) |
\(S^{-}_{B,0} \) | =∅ | \(B(t_{0},\infty)<\infty\) | \(I_{4}=\infty \) |
Declarations
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The author declares that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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