Ya. Alber, S. Guerre-Delabriere and L. Zelenko, “The Principle of Weakly Contractive Maps in Metric Spaces,” Communications on Applied Nonlinear Analysis, Vol. 5, No. 1, 1998, pp. 45-68.

has been cited by the following article:

• TITLE: Stochastic Approximation Method for Fixed Point Problems

  AUTHORS: Ya. I. Alber, C. E. Chidume, Jinlu Li

  KEYWORDS: Hilbert Spaces; Stochastic Approximation Algorithm; Weakly Contractive Operators; Nonexpansive Operators; Fixed Points; Convergence in Mean Square; Convergence Almost Sure (a.s.); Nonasymptotic Estimates of Convergence Rate

  JOURNAL NAME: Applied Mathematics, Vol.3 No.12A, December 31, 2012

  ABSTRACT: We study iterative processes of stochastic approximation for finding fixed points of weakly contractive and nonexpansive operators in Hilbert spaces under the condition that operators are given with random errors. We prove mean square convergence and convergence almost sure (a.s.) of iterative approximations and establish both asymptotic and nonasymptotic estimates of the convergence rate in degenerate and non-degenerate cases. Previously the stochastic approximation algorithms were studied mainly for optimization problems.