Existence of solutions for the semilinear abstract Cauchy problem of fractional order. (English) Zbl 1498.34165
Summary: In this paper we establish the existence of solutions for the nonlinear abstract Cauchy problem of order \(\alpha \in (1, 2)\), where the fractional derivative is considered in the sense of Caputo. The autonomous and nonautonomous cases are studied. We assume the existence of an \(\alpha\)-resolvent family for the homogeneous linear problem. By using this \(\alpha\)-resolvent family and appropriate conditions on the forcing function, we study the existence of classical solutions of the nonhomogeneus semilinear problem. The non-autonomous problem is discussed as a perturbation of the autonomous case. We establish a variation of the constants formula for the nonautonomous and nonhomogeneous equation.
MSC:
| 34G20 | Nonlinear differential equations in abstract spaces |
| 35G25 | Initial value problems for nonlinear higher-order PDEs |
| 47D09 | Operator sine and cosine functions and higher-order Cauchy problems |
| 26A33 | Fractional derivatives and integrals |
Keywords:
fractional differential equations in abstract spaces; classical solutions; mild solutions; semilinear differential equations; bounded semivariation operator valued functions; resolvent familiesReferences:
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