Geometric characteristics for convergence and asymptotics of successive approximations of equations with smooth operators
We discuss the problem of characterizing the possible asymptotic behaviour of the iterates of a sufficiently smooth nonlinear operator acting in a Banach space in small neighbourhoods of a fixed point. It turns out that under natural conditions, for the most part of initial approximations these iterates tend to "lie down" along a finite-dimensional subspace generated by the leading (peripherical) eigensubspaces of the Fréchet derivative at the fixed point and moreover the asymptotic behaviour of...