Asymptotic Behaviour of a Two-Dimensional Differential System with a Finite Number of Nonconstant Delays under the Conditions of Instability
The asymptotic behaviour of a real two-dimensional differential system ∑𝑥′(𝑡)=𝖠(𝑡)𝑥(𝑡)+𝑚𝑘=1𝖡𝑘(𝑡)𝑥(𝜃𝑘(𝑡))+ℎ(𝑡,𝑥(𝑡),𝑥(𝜃1(𝑡)),…,𝑥(𝜃𝑚(𝑡))) with unbounded nonconstant delays 𝑡−𝜃𝑘(𝑡)≥0 satisfying lim𝑡→∞𝜃𝑘(𝑡)=∞ is studied under the assumption of instability. Here, 𝖠, 𝖡𝑘, and ℎ are supposed to be matrix...
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/952601 |
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| author | Zdeněk Šmarda Josef Rebenda |
| author_facet | Zdeněk Šmarda Josef Rebenda |
| author_sort | Zdeněk Šmarda |
| collection | DOAJ |
| description | The asymptotic behaviour of a real two-dimensional differential system
∑𝑥′(𝑡)=𝖠(𝑡)𝑥(𝑡)+𝑚𝑘=1𝖡𝑘(𝑡)𝑥(𝜃𝑘(𝑡))+ℎ(𝑡,𝑥(𝑡),𝑥(𝜃1(𝑡)),…,𝑥(𝜃𝑚(𝑡))) with unbounded nonconstant delays 𝑡−𝜃𝑘(𝑡)≥0 satisfying lim𝑡→∞𝜃𝑘(𝑡)=∞ is studied under the assumption of instability. Here, 𝖠, 𝖡𝑘, and ℎ are supposed to be matrix functions and a vector function. The conditions for the instable properties of solutions and the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the considered real system to one equation with complex-valued coefficients. Asymptotic properties are studied by means of a Lyapunov-Krasovskii functional and the suitable Ważewski topological principle. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one constant or nonconstant delay were studied. |
| format | Article |
| id | doaj-art-e8edfc0e45164ff39ac2781420951dc9 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
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| series | Abstract and Applied Analysis |
| spelling | doaj-art-e8edfc0e45164ff39ac2781420951dc92025-02-03T01:27:51ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/952601952601Asymptotic Behaviour of a Two-Dimensional Differential System with a Finite Number of Nonconstant Delays under the Conditions of InstabilityZdeněk Šmarda0Josef Rebenda1Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 616 00 Brno, Czech RepublicDepartment of Mathematics and Statistics, Faculty of Science, Masaryk University, 611 37 Brno, Czech RepublicThe asymptotic behaviour of a real two-dimensional differential system ∑𝑥′(𝑡)=𝖠(𝑡)𝑥(𝑡)+𝑚𝑘=1𝖡𝑘(𝑡)𝑥(𝜃𝑘(𝑡))+ℎ(𝑡,𝑥(𝑡),𝑥(𝜃1(𝑡)),…,𝑥(𝜃𝑚(𝑡))) with unbounded nonconstant delays 𝑡−𝜃𝑘(𝑡)≥0 satisfying lim𝑡→∞𝜃𝑘(𝑡)=∞ is studied under the assumption of instability. Here, 𝖠, 𝖡𝑘, and ℎ are supposed to be matrix functions and a vector function. The conditions for the instable properties of solutions and the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the considered real system to one equation with complex-valued coefficients. Asymptotic properties are studied by means of a Lyapunov-Krasovskii functional and the suitable Ważewski topological principle. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one constant or nonconstant delay were studied.http://dx.doi.org/10.1155/2012/952601 |
| spellingShingle | Zdeněk Šmarda Josef Rebenda Asymptotic Behaviour of a Two-Dimensional Differential System with a Finite Number of Nonconstant Delays under the Conditions of Instability Abstract and Applied Analysis |
| title | Asymptotic Behaviour of a Two-Dimensional Differential System with a Finite Number of Nonconstant Delays under the Conditions of Instability |
| title_full | Asymptotic Behaviour of a Two-Dimensional Differential System with a Finite Number of Nonconstant Delays under the Conditions of Instability |
| title_fullStr | Asymptotic Behaviour of a Two-Dimensional Differential System with a Finite Number of Nonconstant Delays under the Conditions of Instability |
| title_full_unstemmed | Asymptotic Behaviour of a Two-Dimensional Differential System with a Finite Number of Nonconstant Delays under the Conditions of Instability |
| title_short | Asymptotic Behaviour of a Two-Dimensional Differential System with a Finite Number of Nonconstant Delays under the Conditions of Instability |
| title_sort | asymptotic behaviour of a two dimensional differential system with a finite number of nonconstant delays under the conditions of instability |
| url | http://dx.doi.org/10.1155/2012/952601 |
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