- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 232170, 7 pages
Proximal Point Algorithms for Finding a Zero of a Finite Sum of Monotone Mappings in Banach Spaces
1Departement of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Received 15 February 2013; Revised 30 March 2013; Accepted 31 March 2013
Academic Editor: Yisheng Song
Copyright © 2013 H. Zegeye and N. Shahzad. 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.
- W. Takahashi, Nonlinear Functional Analysis, Kindikagaku, Tokyo, Japan, 1988.
- E. H. Zarantonello, “Solving functional equations by contractive averaging,” Tech. Rep. 160, Mathematics Research Centre, Univesity of Wisconsin, Madison, Wis, USA, 1960.
- G. J. Minty, “Monotone (nonlinear) operators in Hilbert space,” Duke Mathematical Journal, vol. 29, pp. 341–346, 1962.
- R. I. Kačurovskiĭ, “On monotone operators and convex functionals,” Uspekhi Mathematicheskikh Nauk, vol. 15, no. 4, pp. 213–215, 1960.
- M. M. Vaĭnberg and R. I. Kačurovskiĭ, “On the variational theory of non-linear operators and equations,” Doklady Akademii Nauk SSSR, vol. 129, pp. 1199–1202, 1959.
- P.-L. Lions and B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,” SIAM Journal on Numerical Analysis, vol. 16, no. 6, pp. 964–979, 1979.
- P. L. Combettes and V. R. Wajs, “Signal recovery by proximal forward-backward splitting,” Multiscale Modeling & Simulation, vol. 4, no. 4, pp. 1168–1200, 2005.
- H. Brézis and P.-L. Lions, “Produits infinis de résolvantes,” Israel Journal of Mathematics, vol. 29, no. 4, pp. 329–345, 1978.
- S. Kamimura and W. Takahashi, “Approximating solutions of maximal monotone operators in Hilbert spaces,” Journal of Approximation Theory, vol. 106, no. 2, pp. 226–240, 2000.
- B. Martinet, “Régularisation d'inéquations variationnelles par approximations successives,” Revue Française d'Informatique et de Recherche Opérationnelle, vol. 4, pp. 154–158, 1970.
- R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM Journal on Control and Optimization, vol. 14, no. 5, pp. 877–898, 1976.
- M. V. Solodov and B. F. Svaiter, “Forcing strong convergence of proximal point iterations in a Hilbert space,” Mathematical Programming, vol. 87, no. 1, pp. 189–202, 2000.
- H. H. Bauschke, E. Matoušková, and S. Reich, “Projection and proximal point methods: convergence results and counterexamples,” Nonlinear Analysis: Theory, Methods &Applications, vol. 56, no. 5, pp. 715–738, 2004.
- R. E. Bruck and S. Reich, “Nonexpansive projections and resolvents of accretive operators in Banach spaces,” Houston Journal of Mathematics, vol. 3, no. 4, pp. 459–470, 1977.
- O. Güler, “On the convergence of the proximal point algorithm for convex minimization,” SIAM Journal on Control and Optimization, vol. 29, no. 2, pp. 403–419, 1991.
- P.-L. Lions, “Une méthode itérative de résolution d'une inéquation variationnelle,” Israel Journal of Mathematics, vol. 31, no. 2, pp. 204–208, 1978.
- O. Nevanlinna and S. Reich, “Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces,” Israel Journal of Mathematics, vol. 32, no. 1, pp. 44–58, 1979.
- G. B. Passty, “Ergodic convergence to a zero of the sum of monotone operators in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 72, no. 2, pp. 383–390, 1979.
- S. Reich and S. Sabach, “A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 10, no. 3, pp. 471–485, 2009.
- Y. Censor and S. Reich, “Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization,” Optimization, vol. 37, no. 4, pp. 323–339, 1996.
- L. Hu and L. Liu, “A new iterative algorithm for common solutions of a finite family of accretive operators,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 6, pp. 2344–2351, 2009.
- S. Reich and S. Sabach, “Two strong convergence theorems for a proximal method in reflexive Banach spaces,” Numerical Functional Analysis and Optimization, vol. 31, no. 1–3, pp. 22–44, 2010.
- S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002.
- H. Zegeye and N. Shahzad, “Strong convergence theorems for a common zero for a finite family of -accretive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 5, pp. 1161–1169, 2007.
- H. Zegeye and N. Shahzad, “Approximating common solution of variational inequality problems for two monotone mappings in Banach spaces,” Optimization Letters, vol. 5, no. 4, pp. 691–704, 2011.
- K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984.
- R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 75–88, 1970.
- F. E. Browder, “Nonlinear maximal monotone operators in Banach space,” Mathematische Annalen, vol. 175, pp. 89–113, 1968.
- Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos, Ed., vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Dekker, New York, NY, USA, 1996.
- K. Aoyama, F. Kohsaka, and W. Takahashi, “Proximal point methods for monotone operators in Banach spaces,” Taiwanese Journal of Mathematics, vol. 15, no. 1, pp. 259–281, 2011.
- S. Kamimura, F. Kohsaka, and W. Takahashi, “Weak and strong convergence theorems for maximal monotone operators in a Banach space,” Set-Valued Analysis, vol. 12, no. 4, pp. 417–429, 2004.
- H.-K. Xu, “Another control condition in an iterative method for nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 65, no. 1, pp. 109–113, 2002.
- P.-E. Maingé, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,” Set-Valued Analysis, vol. 16, no. 7-8, pp. 899–912, 2008.