Monotone Iterative Method for Semilinear Impulsive Evolution Equations of Mixed Type 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 initial value problem u′(t) + Au(t) = f(t, u(t), Tu(t)), t ∈ J, t 6= tk, ∆u|t=tk = Ik(u(tk)), k = 1, 2, . . . , m, u(0) = x0, where A : D(A) ⊂ E → E is a closed linear operator and −A generates a strongly continuous 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 strongly continuous semigroup.