EXISTENCE AND APPROXIMATE CONTROLLABILITY OF FRACTIONAL EVOLUTION EQUATIONS WITH NONLOCAL CONDITIONS VIA RESOLVENT OPERATORS

In this article, we are concerned with the existence of mild solutions as well as approximate controllability for a class of fractional evolution equations with nonlocal conditions in Banach spaces. Sufficient conditions of existence of mild solutions and approximate controllability for the desired problem are presented by introducing a new Green's function and constructing a control function involving Gramian controllability operator. The discussions are based on Schauder's fixed point theorem as well as the theory of alpha-order solution operator and alpha-order resolvent operator. An example is given to illustrate the feasibility of our theoretical results.