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SHRINKING PROJECTION ALGORITHMS FOR THE SPLIT COMMON NULL POINT PROBLEM

Part of: Nonlinear operators and their properties

Published online by Cambridge University Press:  29 March 2017

VAHID DADASHI Open the ORCID record for VAHID DADASHI [Opens in a new window]
VAHID DADASHI*
Affiliation:
Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran email [email protected]
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Abstract

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We consider the split common null point problem in Hilbert space. We introduce and study a shrinking projection method for finding a solution using the resolvent of a maximal monotone operator and prove a strong convergence theorem for the algorithm.

Keywords

maximal monotone operatorresolvent operatorprojectionsplit common null point problemshrinking projection algorithm

MSC classification

Primary: 47H05: Monotone operators and generalizations
Secondary: 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 2 , October 2017 , pp. 299 - 306
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

Alsulami, S. M., Latif, A. and Takahashi, W., ‘Strong convergence theorems by hybrid methods for the split feasibility problem in Banach spaces’, Linear Nonlinear Anal. 1(1) (2015), 1–11.Google Scholar
Alsulami, S. M. and Takahashi, W., ‘The split common null point problem for maximal monotone mappings in Hilbert spaces and applications’, J. Nonlinear Convex Anal. 15 (2014), 793–808.Google Scholar
Byrne, C., ‘Iterative oblique projection onto convex subsets and the split feasibility problem’, Inverse Probl. 18 (2002), 441–453.CrossRefGoogle Scholar
Byrne, C., ‘A unified treatment of some iterative algorithms in signal processing and image reconstruction’, Inverse Probl. 20 (2004), 103–120.CrossRefGoogle Scholar
Byrne, C., Censor, Y., Gibali, A. and Reich, S., ‘The split common null point problem’, J. Nonlinear Convex Anal. 13 (2012), 759–775.Google Scholar
Censor, Y., Bortfeld, T., Martin, B. and Trofimov, A., ‘A unified approach for inversion problems in intensity-modulated radiation therapy’, Phys. Med. Biol. 51 (2006), 2353–2365.CrossRefGoogle ScholarPubMed
Censor, Y. and Elfving, T., ‘A multiprojection algorithm using Bregman projections in a product space’, Numer. Algorithms 8 (1994), 221–239.CrossRefGoogle Scholar
Censor, Y., Elfving, T., Kopf, N. and Bortfeld, T., ‘The multiple-sets split feasibility problem and its applications for inverse problems’, Inverse Problems 21 (2005), 2071–2084.CrossRefGoogle Scholar
Cioranescu, I., Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and Its Applications, 62 (Kluwer Academic Publishers, Dordrecht, 1990).CrossRefGoogle Scholar
Combettes, P. L., ‘The convex feasibility problem in image recovery’, in: Advances in Imaging and Electron Physics (ed. Hawkes, P.) (Academic Press, New York, 1996), 155–270.Google Scholar
Mosco, U., ‘Convergence of convex sets and of solutions of variational inequalities’, Adv. Math. 3 (1969), 510–585.CrossRefGoogle Scholar
Takahashi, W., Nonlinear Functional Analysis (Yokohama Publishers, Yokohama, 2000).Google Scholar
Takahashi, W., ‘The split feasibility problem and the shrinking projection method in Banach spaces’, J. Nonlinear Convex Anal. 16(7) (2015), 1449–1450.Google Scholar
Takahashi, W. and Yao, J.-C., ‘Strong convergence theorems by hybrid methods for the split common null point problem in Banach spaces’, Fixed Point Theory Appl. 2015 (2015), article ID 87, 13 pages.CrossRefGoogle Scholar
Tsukada, M., ‘Convergence of best approximations in a smooth Banach space’, J. Approx. Theory 40 (1984), 301–309.CrossRefGoogle Scholar
Xu, H. K., ‘Iterative methods for the split feasibility problem in infinite dimensional Hilbert spaces’, Inverse Problems 26 (2010), article ID 105018.CrossRefGoogle Scholar