Convergence of the Kato approximants for evolution equations involving functional perturbations

The existence of a unique strong solution of the nonlinear abstract functional differential equation $\text{u′(t) + A(t)u(t) = F(t,u}{}_{\mathrm{t}}\text{), u}{}_0\text{= φεC}{}^1\text{(¦?r,0¦,X),tε¦0, T¦}$, (E) is established. X is a Banach space with uniformly convex dual space and, for $\text{t? ¦0, T¦, A(t)}$ is m-accretive and satisfies a time dependence condition suitable for applications to partial differential equations. The function F satisfies a Lipschitz condition. The novelty of the paper is that the solution u(t) of (E) is shown to be the uniform limit (as n → ∞) of the sequence u_n(t), where the functions u_n(t) are continuously differentiate solutions of approximating equations involving the Yosida approximants. Thus, a straightforward approximation scheme is now available for such equations, in parallel with the approach involving the use of nonlinear evolution operator theory.