Subordination Principle for a Class of Fractional Order Differential Equations

The fractional order differential equation (u\u27(t)=Au(t)+gamma D_t^{alpha} Au(t)+f(t), t>0), (u(0)=ain X) is studied, where (A) is an operator generating a strongly continuous one-parameter semigroup on a Banach space (X), (D_t^{alpha}) is the Riemann–Liouville fractional derivative of order (alpha in (0,1)), (gamma>0) and (f) is an (X)-valued function. Equations of this type appear in the modeling of unidirectional viscoelastic flows. Well-posedness is proven, and a subordination identity is obtained relating the solution operator of the considered problem and the (C_{0})-semigroup, generated by the operator (A). As an example, the Rayleigh–Stokes problem for a generalized second-grade fluid is considered