Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators. (English) Zbl 1441.34006
Summary: 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 for the 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.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34G20 | Nonlinear differential equations in abstract spaces |
| 34H05 | Control problems involving ordinary differential equations |
| 93B05 | Controllability |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
Keywords:
approximate controllability; fractional evolution equations; nonlocal conditions; \(\alpha\)-order solution operator; \(\alpha\)-order resolvent operatorReferences:
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