Nonmonotone impulse effects in second-order periodic boundary value problems

Abstract

We deal with the nonlinear impulsive periodic boundary value problem $u''= f(t,u,u')$, $u(t_i+)=\mathrm{J}_i(u(t_i))$, $u'(t_i+)=\mathrm{M}_i(u'(t_i))$, $i=1,2,\dotsc,m$, $u(0)=u(T)$, $u'(0)= u'(T)$. We establish the existence results which rely on the presence of a well-ordered pair $(\sigma_1,\sigma_2)$ of lower/upper functions $(\sigma_1\le\sigma_2 \text{ on } [0,T])$ associated with the problem. In contrast to previous papers investigating such problems, the monotonicity of the impulse functions $\mathrm{J}_i$, $\mathrm{M}_i$ is not required here.