Continuous nonlinear perturbations of linear accretive operators in banach spaces

AbstractLet A be a linear, closed, densely defined m-accretive operator from a Banach space X to itself, and let T(t), t ⩾ 0, be the semigroup of operators which has −A as its infinitesimal generator. Let B be a nonlinear, continuous, everywhere defined accretive operator from X to itself, and let S(t), t ⩾ 0, be the semigroup of nonlinear operators which has −B as its infinitesimal generator. It is shown that for all x ϵ X, t ⩾ 0, limn → ∞(T(tn) S(tn))nx = U(t)x exists, U(t)x = T(t)x − ∝0t T(t − s) BU(s) x ds, and U(t), t ⩾ 0, is a strongly continuous semigroup of nonlinear contractions on X. It is shown also that −(A + B) is the infinitesimal generator of U(t), t ⩾ 0, and A + B is m-accretive on X