Abstract
Initial problems for semilinear differential equations in Banach spaces with fractional Caputo derivative are studied. Firstly the unique solvability of the Cauchy problem for the semilinear equation solved with respect to the fractional derivative is researched, when the linear operator in the equation generates a resolving family of operators which is analytic in a sector. Then equation with degenerate operator at the Caputo derivative is considered in the case of the generation of an analytic in a sector degenerate resolving family of operators by the linear part of the equation. The unique solvability sufficient conditions for the Cauchy problem and for the Showalter–Sidorov problem are found. Abstract results are applied to the research of initial boundary value problems for a class of time-fractional order partial differential equations.
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