Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 149, pp. 1-13.

Title: Monotone iterative method for  semilinear impulsive
       evolution equations of mixed type in Banach spaces

Authors: Pengyu Chen (Northwest Normal Univ., Lanzhou, China)
         Jia Mu (Northwest Normal Univ., Lanzhou, China)

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

Submitted August 4, 2010. Published October 21, 2010.
Math Subject Classifications: 34K30, 34K45, 35F25.
Key Words: Initial value problem; lower and upper solution;
           impulsive integro-differential evolution equation;
           C0-semigroup; cone.