Nonlinear boundary value problem for nonlinear second order differential equations with impulses

• Main Author: Jan Tomecek
• Format: Article
• Language: English
• Published: University of Szeged 2005-05-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=218

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.