  We consider a class of inverse problems defined by a nonlinear map from
parameter or model functions to the data. We assume that solutions exist. The
space of model functions is a Banach space which is smooth and uniformly
convex; however, the data space can be an arbitrary Banach space. We study
sequences of parameter functions generated by a nonlinear Landweber iteration
and conditions under which these strongly converge, locally, to the solutions
within an appropriate distance. We express the conditions for convergence in
terms of H\"{o}lder stability of the inverse maps, which ties naturally to the
analysis of inverse problems.
