See the book of A. Bensoussane et al. for different notions of solutions in section  : 3 Nonhomogeneous linear evolution equations, especially Proposition 3.1 and 3.2 where it is proved that some of these notions coincide.

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäuser, 2007.

See the book of A. Bensoussane et al. for different notions of solutions in section: 3 Nonhomogeneous linear evolution equations, especially Proposition 3.1 and 3.2 where it is proved that some of these notions coincide.

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäuser, 2007.