In this paper, the general Caputo‐type fractional differential operator introduced by Pr. Anatoly N. Kochubei is applied to the linear theory of viscoelasticity. Firstly, using the general Caputo‐type derivative, a generalized linear viscoelastic constitutive equation is proposed for the first time. Secondly, the momentum equation for the plane Couette flow of viscoelastic fluid with the constitutive relation is given as an integrodifferential equation and the analytical solution of the equation is established by employing the separation of variables method. Lastly, for special cases of the general constitutive relation, the analytical solutions are obtained in terms of the Mittag‐Leffler functions.

The present paper deals with initial value problems for the fractional evolution equations involving the Caputo fractional derivative. By deriving a property of the three-parametric Mittag-Leffler function and using the Schauder fixed point theorem, new sufficient conditions for existence and uniqueness of mild solutions are established. Primary 26A33; secondary 34K37; 34A08 MSC: