Existence and Approximate Controllability of Fractional Evolution Equations with Nonlocal Conditions Via Resolvent Operators

• Published in: Fractional calculus & applied analysis Vol. 23; no. 1; pp. 268 - 291
• Main Authors: Chen, Pengyu, Zhang, Xuping, Li, Yongxiang
• Format: Journal Article
• Language: English
• Published: Warsaw Versita 01.02.2020; De Gruyter; Nature Publishing Group
• Subjects: 35R11; 93B05; Abstract Harmonic Analysis; Analysis; approximate controllability; Banach spaces; Controllability; Evolution; Fixed points (mathematics); fractional evolution equations; Functional Analysis; Integral Transforms; Mathematical analysis; Mathematics; nonlocal conditions; Operational Calculus; order resolvent operator; order solution operator; Primary 26A33; Research Paper; Secondary 34K37; Stability; fractional evolution equations; approximate controllability; order solution operator; Primary 26A33; Secondary 34K37, 35R11, 93B05; order resolvent operator; nonlocal conditions
• ISSN: 1311-0454; 1314-2224
• DOI: 10.1515/fca-2020-0011

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.