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

## 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:

In this paper, using the lower/upper functions argument, we establish new existence results for the nonlinear impulsive periodic boundary value problem

$u''=f(t,u,u'), u(t_i+)=J_i(u(t_i)), u'(t_i+)=M_i(u'(t_i)), i=1,2,...,m, u(0)=u(T), u'(0)=u'(T),$

where  f   fulfils the Caratheodory conditions on  $[0,T]\times R^2$  and   $J_i, M_i$   are continuous on   R. The main goal of the paper is that the lower/upper functions   $\sigma_1 / \sigma_2$   associated with the problem are not well-ordered, i.e.   $\sigma_1\not\le\sigma_2$  on [0,T].

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 .