Abstract
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 α-order solution operator and α-order resolvent operator. An example is given to illustrate the feasibility of our theoretical results.
Acknowledgements
The authors would like to thank the anonymous referee for his/her carefully reading the manuscript and very important comments and suggestions that improved the results and quality of this paper. Research was supported by National Natural Science Foundations of China (No. 11501455 and 11661071), the Science Research Project for Colleges and Universities of Gansu Province (No. 2019B-047), Project of NWNU-LKQN2019-3, Project of NWNU-LKQN2019-13 and Doctoral Research Fund of Northwest Normal University (No. 6014/0002020209).
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