On the equations of evolution in a Banach space

such as was treated in the previous papers [3] and [4]. However, we shall show that we can replace the strong continuous differentiability of A(t)A(s)~ by its Holder continuity by means of a slight change of the proof. It is quite clear that the differentiability of A(t)A(s)~ is not necessary for the construction of the formal fundamental solution U(t, s) of (0. 1'). In the previous papers, however, we used the differentiability essentially in appearance when we proved that the formal fundamental solution was really the desired one. So it is in this part that the modification of the proof is required. The inhomogeneous equation (0. 1) can be treated similarly. Next, we shall give a generalization of a theorem of Solomijak concerning a perturbed equation