Periodic solutions of delay impulsive differential equations

We study the following semilinear impulsive differential equation with delay: u′(t)+Au(t)=f(t,u(t),ut),t>0,t≠ti,u(s)=ϕ(s),s∈[−r,0],Δu(ti)=Ii(u(ti)),i=1,2,…,00r>0 is a constant and ut(s)=u(t+s)ut(s)=u(t+s), s∈[−r,0]s∈[−r,0]. Here, Δu(ti)=u(ti+)−u(ti−) constitutes an impulsive condition, which can be used to model more physical phenomena than the traditional initial value problems. We assume that f(t,u,w)f(t,u,w) is TT-periodic in tt and then prove with some compactness conditions that if solutions of the equation are ultimately bounded, then the differential equation has a TT-periodic solution. The new results obtained here extend some results in this area for differential equations without impulsive conditions or without delays.