Nonmonotone impulse effects in second-order periodic boundary value problems

We deal with the nonlinear impulsive periodic boundary value problemu″=f(t,u,u′),u(ti+)=Ji(u(ti)),u′(ti+)=Mi(u′(ti)),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(σ1,σ2)of lower/upper functions(σ1≤σ2 on [0,T])associated with the problem. In contrast to previous papers investigating such problems, the monotonicity of the impulse functionsJi,Miis not required here.