Hybrid Method with Perturbation for Lipschitzian Pseudocontractions

Abstract

Assume that F is a nonlinear operator which is Lipschitzian and strongly monotone on a nonempty closed convex subset C of a real Hilbert space H. Assume also that $\Omega$ is the intersection of the fixed point sets of a finite number of Lipschitzian pseudocontractive self-mappings on C. By combining hybrid steepest-descent method, Mann’s iteration method and projection method, we devise a hybrid iterative algorithm with perturbation F, which generates two sequences from an arbitrary initial point $x_0\in H$. These two sequences are shown to converge in norm to the same point $P_{\Omega }{x}_0$ under very mild assumptions.