The Method of Lower and Upper Solutions for Sobolev Type Hilfer Fractional Evolution Equations

The purpose of this paper is concerned with the existence of extremal mild solutions for Sobolev type Hilfer fractional evolution equations with nonlocal conditions in an ordered Banach spaces E. By using monotone iterative technique coupled with the method of lower and upper solutions, with the help of the theory of propagation family as well as the theory of the measure of noncompactness and Sadovskii's fixed point theorem, we obtain some existence results of extremal mild solutions for Hilfer fractional evolution equations. Finally, an example is provided to show the feasibility of the theory discussed in this paper.