Fixed Point Theory and Applications
Volume 2009 (2009), Article ID 957407, 47 pages
doi:10.1155/2009/957407
Review Article

Super-Relaxed (η)-Proximal Point Algorithms, Relaxed (η)-Proximal Point Algorithms, Linear Convergence Analysis, and Nonlinear Variational Inclusions

Ravi P. Agarwal1,2 and Ram U. Verma1,3

1Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA
2Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
3International Publications (USA), 12085 Lake Cypress Circle, Suite I109, Orlando, FL 32828, USA

Received 26 June 2009; Accepted 30 August 2009

Academic Editor: Lai Jiu Lin

Copyright © 2009 Ravi P. Agarwal and Ram U. Verma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We glance at recent advances to the general theory of maximal (set-valued) monotone mappings and their role demonstrated to examine the convex programming and closely related field of nonlinear variational inequalities. We focus mostly on applications of the super-relaxed (η)-proximal point algorithm to the context of solving a class of nonlinear variational inclusion problems, based on the notion of maximal (η)-monotonicity. Investigations highlighted in this communication are greatly influenced by the celebrated work of Rockafellar (1976), while others have played a significant part as well in generalizing the proximal point algorithm considered by Rockafellar (1976) to the case of the relaxed proximal point algorithm by Eckstein and Bertsekas (1992). Even for the linear convergence analysis for the overrelaxed (or super-relaxed) (η)-proximal point algorithm, the fundamental model for Rockafellar's case does the job. Furthermore, we attempt to explore possibilities of generalizing the Yosida regularization/approximation in light of maximal (η)-monotonicity, and then applying to first-order evolution equations/inclusions.