Subordination Principle for a Class of Fractional Order Differential Equations

The fractional order differential equation u'(t) = Au(t) + gamma(DtAu)-Au-alpha(t) + f(t), t > 0, u(0) = a is an element of 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 is an element of (0, 1), gamma > 0 and integral 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.