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: | |
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| 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 |
| Summary: | 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. |
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| ISSN: | 1417-3875 1417-3875 |