Nonmonotone impulse effects in second-order periodic boundary value problems

Abstract

We deal with the nonlinear impulsive periodic boundary value problem $u^{″}=f\left(t,u,u^{\prime }\right)$, $u\left(t_i+\right)={\text{J}}_i\left(u\left(t_i\right)\right)$, $u^{\prime }\left(t_i+\right)={\text{M}}_i\left(u^{\prime }\left(t_i\right)\right)$, $i=1,2,\dots ,m$, $u\left(0\right)=u\left(T\right)$, $u^{\prime }\left(0\right)=u^{\prime }\left(T\right)$. We establish the existence results which rely on the presence of a well-ordered pair $\left({\sigma }_1,{\sigma }_2\right)$ of lower/upper functions $\left({\sigma }_1\le {\sigma }_2\hspace{0.17em}\text{on}\hspace{0.17em}\left[0,T\right]\right)$ associated with the problem. In contrast to previous papers investigating such problems, the monotonicity of the impulse functions ${\text{J}}_i$, ${\text{M}}_i$ is not required here.