Nonlinear Accretive Mappings in Banach Spaces: The Solvability and a Solution Algorithm

Abstract

This paper deals with a nonlinear equation $Fx = u$, where F is a mapping of a Banach space ${\bf X}$ into itself, and is not necessarily differentiable, and u is a given arbitrary point in ${\bf X}$. The $\lambda $-functional is introduced for the concrete and practical characterization of accretive mappings in Banach spaces. In terms of the $\lambda $-functional, conditions on F which guarantee the solvability of $Fx = u$ are derived. These results are applied to investigate the solvability of an equation $\psi (x,v) = u$. An iterative method to solve $Fx = u$ is also proposed and a detailed study of the convergence of this method is given.