Convergence of the Kato approximants for evolution equations involving functional perturbations (Q788183)

This paper is concerned with the nonlinear functional differential equation \[ (*)\quad u'(t)+A(t)u(t)=F(t,u_ t),\quad 0\leq t\leq T,\quad u_ 0=\phi \in C^ 1([-r,0];X) \] in a uniformly smooth Banach space X. Here each A(t) is m-accretive and F is Lipschitzian. It is shown that under a certain time dependence condition on A(t) the unique solution to (*) is the uniform limit of the sequence \(\{u_ n\}\) where each \(u_ n\), \(n=1,2,...\), is the solution of (*) with A(t) replaced by its Yosida approximation \(A_ n(t)=A(t)(I+(1/n)A(t))^{-1}.\)