Picard Iterations for Solution of Nonlinear Equations in Certain Banach Spaces

Let E be a real uniformly smooth Banach space and let A: D A.;E¬E be locally Lipschitzian and strongly quasi-accretive. It is proved that a Picard recursion process converges strongly to the unique solution of the equation Axsf, fgR A., with the convergence being at least as fast as a geometric progression. Related results deal with the convergence of Picard iterations to the fixed point of locally Lipschitzian strong hemicontractions T and to the solutions of nonlinear equations of the forms xqAxsf and xyl Axsf, where A is an accretive-type map.