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 $$displaylines{ u'(t)+Au(t)= f(t,u(t),Tu(t)),quad tin J,; t eq t_k,cr Delta u |_{t=t_k}=I_k(u(t_k)) ,quad k=1,2,dots ,m,cr u(0)=x_0, }$$ where $A:D(A)subset E o E$ is a closed linear operator and $-A$ generates a strongly continuous semigroup $T(t)(tgeq 0)$ 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.