Monotone iterative method for the periodic boundary value problems of impulsive evolution equations in Banach spaces

We use a monotone iterative method in the presence of lower and upper solutions to discuss the existence and uniqueness of mild solutions for the boundary value problem of impulsive evolution equation in an ordered Banach space Eu′(t)+Au(t)=f(t,u(t),Fu(t),Gu(t)),t∈J,t≠tk,Δu|t=tk=Ik(u(tk)),k=1,2,⋯,m,u(0)=u(ω),where A: D(A) ⊂ E → E is a closed linear operator and −A generates a C0-semigroup T(t)(t ≥ 0) in E. Under wide monotonicity conditions and the non-compactness measure condition of the nonlinearity f, we obtain the existence of extremal mild solutions and a unique mild solution between lower and upper solutions requiring only that −A generate a C0-semigroup.