On a new Newton-Mysovskii-type theorem with applications to inexact Newton-like methods and their discretizations

Abstract

In this manuscript we first introduce a new Newton-Mysovskii-type theorem to study the convergence of inexact Newton-like methods to a locally unique solution of a nonlinear operator equation with a nondifferentiable term. We then study their discretized versions in connection with the mesh independence principle. This principle asserts that the behaviour of the discretized process is asymptotically the same as that for the original iteration and consequently, the number of steps required by the two processes to converge to within a given tolerance is essentially the same. So far this result has been proved by others using Newton's method for certain classes of boundary value problems and even more generally by considering a Lipschitz uniform discretization. In some of our earlier papers we extended these results to include Newton-like methods under more general conditions. However, all previous results assume that the iterates can be computed exactly. This is not true in general. That is why we use perturbed Newton-like methods and even more general conditions. Our results, on the one hand, extend and on the other hand, make more practical and applicable all previous results. In particular, we solve a nonlinear integral equation appearing in radioactive transfer.