Positive Solutions for the Initial Value Problems of Impulsive Evolution Equations

Abstract:

      This paper deals with the existence of $e$-positive mild solutions (see Definition 1) for the initial value problem of impulsive evolution equation with noncompact semigroup $$\left\{\begin{array}{ll} u'(t) Au(t)=f(t,\ u(t)),\ t\in [0,\ \infty),\ t\neq t_{k},\\[6pt] \triangle u|_{t=t_{k}}=I_{k}(u(t_{k})),\ k=1,2,\ldots,\\[6pt] u(0)=x_{0}\end{array}\right.$$ in an ordered Banach space $E$. By using operator semigroup theory and monotonic iterative technique, without any hypothesis on the impulsive functions, an existence result of $e$-positive mild solutions is obtained under weaker measure of noncompactness condition on nonlinearity of $f$. Particularly, an existence result without using measure of noncompaceness condition is presented in ordered and weakly sequentially complete Banach spaces, which is very convenient for application. An example is given to illustrate that our results are more valuable.