On the Convergence Analysis of Modified Hybrid Steepest-Descent Methods with Variable Parameters

Abstract. Assume that F is a nonlinear operator on a real Hilbert spaceH which is η-strongly monotone and κ-Lipschitzian on a nonempty closed convex subset C of H. Assume also that C is the intersection of the fixed point sets of a finite number of nonexpansive mappings on H. We construct an iterative algorithm with variable parameters which generates a sequence {xn} from an arbitrary initial point x0 ∈ H. The sequence {xn} is shown to converge in norm to the unique solution u ∗ of the variational inequality 〈F (u∗), v − u∗ 〉 ≥ 0, ∀ v ∈ C. Applications to constrained generalized pseudoinverse are included. Key Words: Iterative algorithms, modified hybrid steepest-descent methods with variable parameters, convergence, nonexpansive mappings, Hilbert space, constrained generalized pseu-doinverses. 2 1