Asymptotic convergence of nonlinear contraction semigroups in Hilbert space

Let S be a contraction semigroup on a closed convex subset C of a Hilbert space. If the generator of S satisfies a strengthened monotonicity condition then the weak lim_{t → ∞}S(t)x exists for all x in C. As one consequence, the method of steepest descent converges weakly for convex functions in Hilbert space; and it converges strongly for even convex functions.