Exponential inequalities for Mann’s stochastic algorithm

Abstract

In this article, we investigate the problem of approximating the fixed point for some function using a Mann iterative process with random errors. After establishing some exponential inequalities, we prove the complete convergence of Mann’s algorithm toward the fixed point and deduce a confidence interval for this one. In addition, we establish the convergence rate of Mann’s algorithm. Several numerical examples are sketched to illustrate the performance of the proposed algorithm.

Keywords:

• Almost complete convergence
• fixed point
• Mann iteration
• negative association
• quasi-associated random errors
• rate of convergence

Acknowledgments

The authors are thankful to the Editor, Associate Editor, and referees for their time, effort, and comments in reviewing this article.