Picard iteration for nonsmooth equations (Q2770162)

The authors analyze generalized Newton methods for solving \(F(x)=0\) for Lipschitz continuous maps F from the viewpoint of Picard iteration. According to the Rademacher theorem, F is differentiable almost everywhere and a generalized derivative as given by \textit{F. H. Clarke} [Optimization and nonsmooth analysis (1983; Zbl 0582.49001); Proposition 2.6.2] can be defined. It is proven that the radius of the weak Jacobian (RGJ) for a Picard iteration function is equal to its least Lipschitz constant. Linear convergence or superlinear convergence results are obtained provided the the RGJ of the Picard iteration function at the solution point is less than one or equal to zero, respectively. These results are applied to analyze the generalized Newton method for piecewise \(C^1\) functions and a splitting method for certain nonlinear partial differential equations.