Impulsive system of ODEs with general linear boundary conditions
The paper provides an operator representation for a problem which consists of a system of ordinary differential equations of the first order with impulses at fixed times and with general linear boundary conditions \begin{gather} z'(t) = A(t)z(t) + f(t,z(t)) \textrm{ for a.e. }t \in [a,b] \subs...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2013-05-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=2301 |
| _version_ | 1797830672735272960 |
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| author | Irena Rachůnková Jan Tomeček |
| author_facet | Irena Rachůnková Jan Tomeček |
| author_sort | Irena Rachůnková |
| collection | DOAJ |
| description | The paper provides an operator representation for a problem which consists of a system of ordinary differential equations of the first order with impulses at fixed times and with general linear boundary conditions
\begin{gather}
z'(t) = A(t)z(t) + f(t,z(t)) \textrm{ for a.e. }t \in [a,b] \subset \mathbb{R}, \\
z(t_i+) - z(t_i) = J_i(z(t_i)), \quad i = 1,\ldots,p,\\
\ell(z) = c_0, \quad c_0 \in \mathbb{R}^n.
\end{gather}
Here $p,n \in N$, $a < t_1 < \ldots < t_p < b$, $A \in L^1([a,b];\mathbb{R}^{n\times n})$, $f \in \operatorname{Car}([a,b]\times\mathbb{R}^n;\mathbb{R}^n)$, $J_i \in C(\mathbb{R}^n;\mathbb{R}^n)$, $i=1,\ldots,p$, and $\ell$ is a linear bounded operator on the space of left-continuous regulated functions on interval $[a,b]$. The operator $\ell$ is expressed by means of the Kurzweil-Stieltjes integral and covers all linear boundary conditions for solutions of the above system subject to impulse conditions. The representation, which is based on the Green matrix to a corresponding linear homogeneous problem, leads to an existence principle for the original problem. A special case of the $n$-th order scalar differential equation is discussed. This approach can be also used for analogical problems with state-dependent impulses. |
| first_indexed | 2024-04-09T13:39:48Z |
| format | Article |
| id | doaj.art-bf0059e697654369a3435d453a17d19e |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:39:48Z |
| publishDate | 2013-05-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-bf0059e697654369a3435d453a17d19e2023-05-09T07:53:03ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752013-05-0120132511610.14232/ejqtde.2013.1.252301Impulsive system of ODEs with general linear boundary conditionsIrena Rachůnková0Jan Tomeček1Palacky University, Olomouc, Czech RepublicPalacky University, Olomouc, Czech RepublicThe paper provides an operator representation for a problem which consists of a system of ordinary differential equations of the first order with impulses at fixed times and with general linear boundary conditions \begin{gather} z'(t) = A(t)z(t) + f(t,z(t)) \textrm{ for a.e. }t \in [a,b] \subset \mathbb{R}, \\ z(t_i+) - z(t_i) = J_i(z(t_i)), \quad i = 1,\ldots,p,\\ \ell(z) = c_0, \quad c_0 \in \mathbb{R}^n. \end{gather} Here $p,n \in N$, $a < t_1 < \ldots < t_p < b$, $A \in L^1([a,b];\mathbb{R}^{n\times n})$, $f \in \operatorname{Car}([a,b]\times\mathbb{R}^n;\mathbb{R}^n)$, $J_i \in C(\mathbb{R}^n;\mathbb{R}^n)$, $i=1,\ldots,p$, and $\ell$ is a linear bounded operator on the space of left-continuous regulated functions on interval $[a,b]$. The operator $\ell$ is expressed by means of the Kurzweil-Stieltjes integral and covers all linear boundary conditions for solutions of the above system subject to impulse conditions. The representation, which is based on the Green matrix to a corresponding linear homogeneous problem, leads to an existence principle for the original problem. A special case of the $n$-th order scalar differential equation is discussed. This approach can be also used for analogical problems with state-dependent impulses.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=2301system of impulsive differential equationsfixed impulsesgeneral linear boundary conditionn-th order odegreen matrixkurzweil-stieltjes integral |
| spellingShingle | Irena Rachůnková Jan Tomeček Impulsive system of ODEs with general linear boundary conditions Electronic Journal of Qualitative Theory of Differential Equations system of impulsive differential equations fixed impulses general linear boundary condition n-th order ode green matrix kurzweil-stieltjes integral |
| title | Impulsive system of ODEs with general linear boundary conditions |
| title_full | Impulsive system of ODEs with general linear boundary conditions |
| title_fullStr | Impulsive system of ODEs with general linear boundary conditions |
| title_full_unstemmed | Impulsive system of ODEs with general linear boundary conditions |
| title_short | Impulsive system of ODEs with general linear boundary conditions |
| title_sort | impulsive system of odes with general linear boundary conditions |
| topic | system of impulsive differential equations fixed impulses general linear boundary condition n-th order ode green matrix kurzweil-stieltjes integral |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=2301 |
| work_keys_str_mv | AT irenarachunkova impulsivesystemofodeswithgenerallinearboundaryconditions AT jantomecek impulsivesystemofodeswithgenerallinearboundaryconditions |