The method of lower and upper solution for Hilfer evolution equations with non-instantaneous impulses

In this article, we study the existence of mild solutions for a class of Hilfer fractional evolution equations with non-instantaneous impulses in ordered Banach spaces. The definition of mild solutions for our problem was given based on a C-0-semigroup W(.) generated by the operator - A and probability density function. By means of monotone iterative technique and the method of lower and upper, the existence of extremal mild solutions between lower and upper mild solutions for nonlinear evolution equation with non-instantaneous impulses is obtained under the situation that the corresponding C-0-semigroup W(.) and non-instantaneous impulsive function gamma(k) are compact, W(.) is not compact and gamma(k) is compact, W(.) and gamma(k) are not compact, respectively. At last, two examples are given to illustrate the abstract results.