Nonlinear boundary value problem for nonlinear second order differential equations with impulses
The paper deals with the impulsive nonlinear boundary value problem \[ u''(t) = f(t,u(t),u'(t)) \quad\mbox{for a.e.}\ t \in [0,T], \] \[ u(t_j+) = J_j(u(t_j)),\quad u'(t_j+) = M_j(u'(t_j)),\quad j = 1,\ldots,m, \] \[ g_1(u(0),u(T)) = 0, \quad g_2(u'(0),u'(T)) = 0,...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2005-05-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=218 |
| _version_ | 1827951410741772288 |
|---|---|
| author | Jan Tomeček |
| author_facet | Jan Tomeček |
| author_sort | Jan Tomeček |
| collection | DOAJ |
| description | The paper deals with the impulsive nonlinear boundary value problem
\[
u''(t) = f(t,u(t),u'(t)) \quad\mbox{for a.e.}\ t \in [0,T],
\]
\[
u(t_j+) = J_j(u(t_j)),\quad u'(t_j+) = M_j(u'(t_j)),\quad j = 1,\ldots,m,
\]
\[
g_1(u(0),u(T)) = 0, \quad g_2(u'(0),u'(T)) = 0,
\]
where $f \in Car([0,T]\times\mathbb{R}^{2})$, $g_1$, $g_2 \in C(\mathbb{R}^2)$, $J_j$, $M_j \in C(\mathbb{R})$. An existence theorem is proved for non-ordered lower and upper functions. Proofs are based on the Leray–Schauder degree and on the method of a priori estimates. |
| first_indexed | 2024-04-09T13:41:20Z |
| format | Article |
| id | doaj.art-2b509fa9bad5416786ea0cb817de7b8b |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:41:20Z |
| publishDate | 2005-05-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-2b509fa9bad5416786ea0cb817de7b8b2023-05-09T07:52:57ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752005-05-0120051012210.14232/ejqtde.2005.1.10218Nonlinear boundary value problem for nonlinear second order differential equations with impulsesJan Tomeček0Palacky University, Olomouc, Czech RepublicThe paper deals with the impulsive nonlinear boundary value problem \[ u''(t) = f(t,u(t),u'(t)) \quad\mbox{for a.e.}\ t \in [0,T], \] \[ u(t_j+) = J_j(u(t_j)),\quad u'(t_j+) = M_j(u'(t_j)),\quad j = 1,\ldots,m, \] \[ g_1(u(0),u(T)) = 0, \quad g_2(u'(0),u'(T)) = 0, \] where $f \in Car([0,T]\times\mathbb{R}^{2})$, $g_1$, $g_2 \in C(\mathbb{R}^2)$, $J_j$, $M_j \in C(\mathbb{R})$. An existence theorem is proved for non-ordered lower and upper functions. Proofs are based on the Leray–Schauder degree and on the method of a priori estimates.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=218 |
| spellingShingle | Jan Tomeček Nonlinear boundary value problem for nonlinear second order differential equations with impulses Electronic Journal of Qualitative Theory of Differential Equations |
| title | Nonlinear boundary value problem for nonlinear second order differential equations with impulses |
| title_full | Nonlinear boundary value problem for nonlinear second order differential equations with impulses |
| title_fullStr | Nonlinear boundary value problem for nonlinear second order differential equations with impulses |
| title_full_unstemmed | Nonlinear boundary value problem for nonlinear second order differential equations with impulses |
| title_short | Nonlinear boundary value problem for nonlinear second order differential equations with impulses |
| title_sort | nonlinear boundary value problem for nonlinear second order differential equations with impulses |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=218 |
| work_keys_str_mv | AT jantomecek nonlinearboundaryvalueproblemfornonlinearsecondorderdifferentialequationswithimpulses |