The purpose of this paper is to
construct a nonlinear semi-group determined by a given (multi-valued) nonlinear
operator A in a Banach space X, and to investigate the differentiability of this
semi-group. The semi-group treated in this paper is the semigroup {T(t);t ≧ 0} of
nonlinear operators in X such that for each τ > 0,{T(t);0 ≦ t ≦ τ} is equi-Lipschitz
continuous on bounded sets. In order that an operator A in X determine such a
semi-group {T(t);t ≧ 0} on D(A) with (d∕dt)T(t)x ∈ AT(t)x for almost all t ≧ 0 and
x ∈ D(A), it is required that X have a uniformly convex dual, A be dissipative in a
local sense, I − λA,λ positive and small, satisfy a range condition and an
injectiveness condition, and finally the family of operators (I −λA)−n,n = 1,2,3, be
locally equi-bounded.
