Smoothing methods and semismooth methods for nondifferentiable operator equations

We consider superlinearly convergent analogues of Newton methods for nondif ferentiable operator equations in function spaces. The superlinear converge nce analysis of semismooth methods for nondifferentiable equations describe d by a locally Lipschitzian operator in R-n is based on Rademacher's theore m which does not hold in function spaces. We introduce a concept of slant d ifferentiability and use it to study superlinear convergence of smoothing m ethods and semismooth methods in a uni ed framework. We show that a functio n is slantly differentiable at a point if and only if it is Lipschitz conti nuous at that point. An application to the Dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed.