A point-of-attraction result for Newton's method with point-based approximations

Point-based approximations (PBAs) permit the construction and analysis of Newton-type methods for functions that may not be differentiable. A convergence theorem of Kantorovich type exists for these methods. Here we establish a point-of-attraction result for the same class of methods, exhibiting conditions on a solution so that for starting points sufficiently close to that solution the Kantorovich conditions will hold and, therefore, all of the consequences of the convergence theorem will follow. We discuss application of these results to equations on monotone graphs, demonstrate the construction of PBAs for such problems and give a small demonstration example. Keywords: Newton's methodmonotone graphvariational conditionvariational inequalitynonsmooth equationsnormal manifoldnormal mapAMS Subject Classifications:: Primary, 49M15Secondary, 90C33 Acknowledgements This material is based on work supported in part by the Air Force Research Laboratory under Award No. FA9550-07-1-0389 and in part by the National Science Foundation under Grant DMS-0807893. Any opinions, findings and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the sponsoring agencies.