Mild solutions for abstract fractional differential equations

Abstract

We propose a unified functional analytic approach to derive a variation of constants formula for a wide class of fractional differential equations using results on (a, k)-regularized families of bounded and linear operators, which covers as particular cases the theories of C ₀-semigroups and cosine families. Using this approach we study the existence of mild solutions to fractional differential equation with nonlocal conditions. We also investigate the asymptotic behaviour of mild solutions to abstract composite fractional relaxation equations. We include in our analysis the Basset and Bagley–Torvik equations.

Keywords:

• linear and semilinear evolution equations
• regularized operator families
• mild solutions

AMS Subject Classifications::

• Primary: 45N05
• Secondary: 43A60

Acknowledgements

The first author is partially financed by Fondecyt Grant number 1100485.