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

J. Tomecek, Palacky University, Olomouc, Czech Republic

E. J. Qualitative Theory of Diff. Equ., No. 10. (2005), pp. 1-22.

Abstract: The paper deals with the impulsive nonlinear boundary value problem

\begin{displaymath}
u''(t) = f(t,u(t),u'(t)) \quad\mbox{for a.~e.}\ t \in [0,T],
\end{displaymath}
\begin{displaymath}
u(t_j+) = J_j(u(t_j)),\quad u'(t_j+) = M_j(u'(t_j)),\quad j = 1,\ldots,m,
\end{displaymath}
\begin{displaymath}
g_1(u(0),u(T)) = 0, \quad g_2(u'(0),u'(T)) = 0,
\end{displaymath}

where $f \in Car([0,T]\times\rr^{2})$, $g_1$, $g_2 \in C(\rr^2)$, $J_j$, $M_j \in C(\rr)$. 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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