Extremal mild solutions to Hilfer evolution equations with non-instantaneous impulses and nonlocal conditions

Abstract

In this article, we study the existence of extremal mild solutions for a class of Hilfer fractional evolution equation with non-instantaneous impulses and nonlocal conditions in ordered Banach spaces. The definition of mild solutions for our concerned problem was given based on a $$C_0$$ -semigroup $$W(\cdot )$$ generated by the operator $$-A$$ and probability density function. By means of monotone iterative technique and the method of lower and upper solutions, the existence of extremal mild solutions between lower and upper mild solutions for nonlinear Hilfer evolution equations with non-instantaneous impulses and nonlocal conditions is obtained under the situation that the corresponding $$C_0$$ -semigroup $$W(\cdot )$$ and non-instantaneous impulsive function $$\gamma _k$$ are compact, $$W(\cdot )$$ is not compact and $$\gamma _k$$ is compact, $$W(\cdot )$$ and $$\gamma _k$$ are not compact, respectively. At the end, in order to illustrate our results, three concrete examples are given.