On a problem of T. Kato

Let X be a Banach space with norm 11 11 such that its dual space X* is uniformly convex. We denote by F the duality map from X into X * , that is, F(x) is the unique element of X * satisfying For a closed convex set C in X we use the notation I Following T. Kato we say that a multivalued operator A in X is accretive ifaccretive (hypermaximal accretive in the terminology of F. Browder) if R (I + ilA) = X for all il > 0. We recall first a result of T. Kato [ S ]. THEOREM 0. Let A be m-accretive; then for each x E D (A) the equation du-+ A o u (t) 3 O a.e. o n (0, +a> , u (0) = x, dt (1) has a unique solution u on [0, + co) such that (2) u (t) E D (A) for all t E [O, +cq) ,