- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 817392, 6 pages
The Problem of Image Recovery by the Metric Projections in Banach Spaces
1Department of Information Science, Toho University, Miyama, Funabashi, Chiba 274-8510, Japan
2Sundai Preparatory School, Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8313, Japan
Received 24 September 2012; Accepted 28 December 2012
Academic Editor: Jaan Janno
Copyright © 2013 Yasunori Kimura and Kazuhide Nakajo. 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.
- L. M. Bregman, “The method of successive projection for finding a common point of convexsets,” Soviet Mathematics, vol. 6, pp. 688–692, 1965.
- G. Crombez, “Image recovery by convex combinations of projections,” Journal of Mathematical Analysis and Applications, vol. 155, no. 2, pp. 413–419, 1991.
- S. Kitahara and W. Takahashi, “Image recovery by convex combinations of sunny nonexpansive retractions,” Topological Methods in Nonlinear Analysis, vol. 2, no. 2, pp. 333–342, 1993.
- W. Takahashi and T. Tamura, “Limit theorems of operators by convex combinations of nonexpansive retractions in Banach spaces,” Journal of Approximation Theory, vol. 91, no. 3, pp. 386–397, 1997.
- Y. I. Alber, “Metric and generalized projections operators in Banach spaces: properties andapplications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartasatos, Ed., vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.
- Y. I. Alber and S. Reich, “An iterative method for solving a class of nonlinear operator equations in Banach spaces,” Panamerican Mathematical Journal, vol. 4, no. 2, pp. 39–54, 1994.
- 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.
- Y. Haugazeau, Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes, [Thése], These, Universit é de Paris, Paris, France, 1968.
- H. H. Bauschke and P. L. Combettes, “A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces,” Mathematics of Operations Research, vol. 26, no. 2, pp. 248–264, 2001.
- K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–379, 2003.
- S. Ohsawa and W. Takahashi, “Strong convergence theorems for resolvents of maximal monotone operators in Banach spaces,” Archiv der Mathematik, vol. 81, no. 4, pp. 439–445, 2003.
- W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 276–286, 2008.
- Y. Kimura and W. Takahashi, “On a hybrid method for a family of relatively nonexpansive mappings in a Banach space,” Journal of Mathematical Analysis and Applications, vol. 357, no. 2, pp. 356–363, 2009.
- K. Nakajo, K. Shimoji, and W. Takahashi, “Strong convergence theorems by the hybrid method for families of mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 812–818, 2009.
- Y. Kimura, K. Nakajo, and W. Takahashi, “Strongly convergent iterative schemes for a sequence of nonlinear mappings,” Journal of Nonlinear and Convex Analysis, vol. 9, no. 3, pp. 407–416, 2008.
- F. Kohsaka and W. Takahashi, “Iterative scheme for finding a common point of infinitely many convex sets in a Banach space,” Journal of Nonlinear and Convex Analysis, vol. 5, no. 3, pp. 407–414, 2004.
- S.-y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005.
- S.-y. Matsushita, K. Nakajo, and W. Takahashi, “Strong convergence theorems obtained by a generalized projections hybrid method for families of mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 73, no. 6, pp. 1466–1480, 2010.
- X. Liu and L. Huang, “Total bounded variation-based Poissonian images recovery by split Bregman iteration,” Mathematical Methods in the Applied Sciences, vol. 35, no. 5, pp. 520–529, 2012.
- I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.
- J. Diestel, Geometry of Banach Spaces, Selected Topics, vol. 485 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975.
- H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 12, pp. 1127–1138, 1991.
- B. Beauzamy, Introduction to Banach Spaces and Their Geometry, vol. 68 of North-Holland Mathematics Studies, North-Holland Publishing, Amsterdam, The Netherlands, 2nd edition, 1985.
- K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984.
- W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, 2000.