# Periodic Boundary Value Problems for Nonlinear Second Order Differential Equations with Impulses - Part I

## Irena Rachunkova and Milan Tvrdy

Irena Rachunkova, Department of Math., Palacky University, 779 00 OLOMOUC, Tomkova 40, Czech Republic, e-mail: rachunko@risc.upol.cz;
Milan Tvrdy, Mathematical Institute, Acad. Sci.of the Czech Republic, 115 67 PRAHA 1, Zitna 25, Czech Republic, e-mail: tvrdy@math.cas.cz, http://www.math.cas.cz/~tvrdy/;

Summary: The paper deals with the nonlinear impulsive periodic boundary value problem $u''=f(t,u,u'); u(t_i+)=J_i(u(t_i)), u'(t_i+)=M_i(u'(t_i)), i=1,2,..., m; u(0)=u(T), u'(0)=u'(T).$ We establish the existence results which rely on the presence of a~well ordered pair $(\sigma_1,\sigma_2)$ of lower/upper functions $(\sigma_1\le\sigma_2$ on $[0,T]$) associated with the problem. In contrast to previous papers investigating such problems, the monotonicity of the impulse functions $J_i, M_i$ is not required here.

Keywords:  Second order nonlinear ordinary differential equation with impulses, periodic solutions, lower and upper functions, Leray-Schauder topological degree, a priori estimates.

Mathematics Subject Classification 2000: 34B37, 34B15, 34C25.