Abstract:
The issues of unique solvability in the sense of mild solutions of
the Cauchy problem for quasilinear equations in Banach spaces solved
with respect to the highest fractional Gerasimov–Caputo derivative, with a sectorial
operator in the linear part, are investigated. The existence and uniqueness of a global
mild solution in the case of Lipschitzian nonlinear mapping, depending
on several minor fractional derivatives of Gerasimov–Caputo, as well
as the local existence and uniqueness of a mild solution for locally Lipschitzian nonlinear mapping, are proved. Using the abstract results obtained, the Cauchy problem for a class of partial differential equations
in a half-space is investigated.
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