  We analyze backward step control globalization for finding zeros of
G\^ateaux-differentiable functions that map from a Banach space to a Hilbert
space. The results include global convergence to a distinctive solution
characterized by propagating the initial guess by a generalized Newton flow
with guaranteed bounds on the discrete nonlinear residual norm decrease and an
(also numerically) easily controllable asymptotic linear residual convergence
rate. The convergence theory can be exploited to construct efficient numerical
methods, which we demonstrate for the case of a Krylov-Newton method and an
approximation-by-discretization multilevel framework. Both approaches optimize
the asymptotic linear residual convergence rate, either over the Krylov
subspace or through adaptive discretization, which in turn yields practical and
efficient stopping criteria and refinement strategies that balance the
nonlinear residuals with the relative residuals of the linear systems. We apply
these methods to the class of nonlinear elliptic boundary value problems and
present numerical results for the Carrier equation and the minimum surface
equation.
