Local convergence of some iterative methods for generalized equations. (Q1428240)

The authors propose a combination of Newton and secant methods for solving the generalized equation: \(0\in f(x)+g(x)+F(x)\). Here \(f\) is Fréchet differentiable in a neighborhood of a solution \(x^*\), \(g\) is Fréchet differentiable \textit{at} \(x^*\) and \(F\) is a set-valued map acting in Banach spaces. An iterative sequence \(\{x_k, k=1, 2, \ldots \}\) is defined via \[ 0\in f(x_k)+g(x_k) (\nabla f(x_k) + [x_{k-1},x_k; g])(x_{k+1}-x_k)+F(x_{k+1}),~k=0, 1, \ldots, \] where \([x,y;g]\) is the \textit{first order divided difference of } \(g\) on the points \(x\) and \(y\). Super-linear convergence, as well as quadratic convergence of other variants of the method, is established.