Nonlinear Inverse Problems for Fractional Differential Equations with Sectorial Operators
Abstract
Abstract A nonlinear inverse problem for an equation in a Banach space, resolved with respect to the oldest Gerasimov–Caputo fractional derivative, is considered. A linear closed operator in the linear part of the equation generates an analytic resolving family of operators. Nonlinear operator in the equation depends on minor fractional derivatives, fractional integrals and an unknown element, which depends on time. The equation is endowed by the Cauchy conditions and an overdetermination condition with an abstract linear bounded operator. By the method of a contraction mapping in a metric space theorems on the unique solvability in the sense of generalized solution and of smooth solution are proved. Abstract results were used for research of a nonlinear inverse problem for a system of Navier–Stokes type with time-fractional Gerasimov–Caputo derivatives and integrals.