Monotone Iterative Technique for the Initial Value Problems of Impulsive Evolution Equations in Ordered Banach Spaces

Abstract

This paper deals with the existence and uniqueness of mild solutions for the initial value problems of abstract impulsive evolution equations in an ordered Banach space $E:u^{\prime }\left(t\right)+Au\left(t\right)=f(t,u(t),Gu(t\left)\right)$, $t\in [0,a]$, $t\ne t_k$, $\Delta u{|}_{t=t_k}=I_k\left(u\right(t_k\left)\right)$, $0<t_1<t_2<\cdots <t_m<a$, $u\left(0\right)=u_0$, where $A:D\left(A\right)\subset E\to E$ is a closed linear operator, and $f:[0,a]\times E\times E\to E$ is a nonlinear mapping. Under wide monotone conditions and measure of noncompactness conditions of nonlinearity $f$, some existence and uniqueness results are obtained by using a monotone iterative technique in the presence of lower and upper solutions.