Convergence analysis of an iterative method for solving multiple-set split feasibility problems in certain Banach spaces

References

• 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, Lecture Notes in Pure and Applied Mathematics, Vol. 178, pp. 15–50, Marcel Dekker, New York, 1996.
   Google Scholar
• C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Prob. 18 (2002), 441–453. doi: 10.1088/0266-5611/18/2/310
   Web of Science ®Google Scholar
• C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Prob. 20 (2004), 103–120. doi: 10.1088/0266-5611/20/1/006
   Web of Science ®Google Scholar
• Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol. 51 (2003), 2353–2365. doi: 10.1088/0031-9155/51/10/001
   Web of Science ®Google Scholar
• Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, The split feasibility model leading to a unified approach for inversion problems in intensity-modulated radiation therapy, Technical Report 20, Department of Mathematics, University of Haifa, Israel, April 2005.
   Google Scholar
• Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projection in product space, Numer. Algorithms 8 (1994), 221–239. doi: 10.1007/BF02142692
   Google Scholar
• Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Prob. 21 (2005), 2071–2084. doi: 10.1088/0266-5611/21/6/017
   Web of Science ®Google Scholar
• Y. Censor and A. Lent, An iterative row-action method for interval convex programming, J. Optim. Theory Appl. 34 (1981), 321–353. doi: 10.1007/BF00934676
   Web of Science ®Google Scholar
• Y. Censor and S. Reich, Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization 37 (1996), 323–339. doi: 10.1080/02331939608844225
   Google Scholar
• G. Lopez, V. Martin-Marquez and H.-K. Xu, Iterative algorithms for the multiple-sets split feasibility problem, In: Biomedical mathematics: promising directions in imaging therapy planning and inverse problems, Y. Censor, M. Jiang, G. Wang, eds., pp. 243–279, Medical Physics Publishing, Madison (WI), 2010.
   Google Scholar
• I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht, 1990.
   Google Scholar
• Y. Dang and Y. Gao, The strong convergence of a KM-CQ-like algorithm for a split feasibility problem, Inverse Prob. 27 (2011), 015007. doi: 10.1088/0266-5611/27/1/015007
   Web of Science ®Google Scholar
• S. He, Z. Zhao and B. Luo, A relaxed self adaptive CQ algorithm for the multiple-sets split feasibility problem, Optimization doi:10.1080/02331934.2014-895898.
   Web of Science ®Google Scholar
• B. He, X. He, H. Liu and T. Wu, Self-adaptive projection method for co-coercive variational inequalities, Eur. J. Oper. Res. 196 (2009), 43–48. doi: 10.1016/j.ejor.2008.03.004
   Web of Science ®Google Scholar
• G. López, V. Martín-Márquez, F.H. Wang and H.K. Xu, Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Prob. (2012), doi:10.1088-0266-5611/28/8/085004.
   Web of Science ®Google Scholar
• G. López, V. Martín-Márquez and H.K. Xu, Perturbation techniques for non-expansive mappings with applications, Nonlinear Anal. Real World Appl. 10 (2009), 2369–2383. doi: 10.1016/j.nonrwa.2008.04.020
   Web of Science ®Google Scholar
• P.E. Maingé, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim. 47 (2008), 1499–1515. doi: 10.1137/060675319
   Web of Science ®Google Scholar
• P.E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal. 16 (2008), 899–912. doi: 10.1007/s11228-008-0102-z
   Google Scholar
• V. Martín-Márquez, S. Reich and S. Sabach, Bregman strongly nonexpansive operators in reflexive Banach spaces, J. Math. Anal. Appl. 400 (2013), 597–614. doi: 10.1016/j.jmaa.2012.11.059
   Web of Science ®Google Scholar
• V. Martín-Márquez, S. Reich and S. Sabach, Right Bregman nonexpansive operators in Banach spaces, Nonlinear Anal. 75 (2012), 5448–5465. doi: 10.1016/j.na.2012.04.048
   Web of Science ®Google Scholar
• V. Martín-Márquez, S. Reich and S. Sabach, Existence and approximation of fixed points of right Bregman nonexpansive operators, In: Computational and Analytical Mathematics, pp. 501–520, Springer, New York, 2013. doi: 10.1007/978-1-4614-7621-4_24
   Google Scholar
• E. Masad and S. Reich, A note on the multiple-set split convex feasibility problem in Hilbert space, J. Nonlinear Convex Anal. 8 (2007), 367–371.
   Web of Science ®Google Scholar
• R.P. Phelps, Convex Functions, Monotone Operators, and Differentiability, 2nd ed., Springer Verlag, Berlin, 1993.
   Google Scholar
• S. Reich, A review of Geometry of Banach spaces, duality mappings and nonlinear problems by Ioana Cioranescu, Kluwer Academic Publishers, Dordrecht, 1990; Bull. Amer. Math. Soc. 26(2) (1992), 367–370.
   Google Scholar
• S. Reich, A weak convergence theorem for the alternating method with Bregman distances, In: Theory and Applications of Nonlinear Operators, pp. 313–318, Marcel Dekker, New York, 1996.
   Google Scholar
• F. Schöpfer, Iterative regularisation method for the solution of the split feasibility problem in Banach spaces, Ph.D Thesis, Saarbrücken, 2007.
   Google Scholar
• F. Schöpfer, T. Schuster and A.K. Louis, An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse problems 24 (2008), 055008. doi: 10.1088/0266-5611/24/5/055008
   Web of Science ®Google Scholar
• T. Suzuki, A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings, Proc. Am. Math. Soc. 135 (2007), 99–106. doi: 10.1090/S0002-9939-06-08435-8
   Web of Science ®Google Scholar
• Y. Tang, C. Zhu and H. Yu, Iterative methods for solving the multiple-sets split feasibility problem with splitting self adaptive step size, Fixed point Theory and Appl. 2015 2015:178. doi:10.1186/s13663-015-0430-2. doi: 10.1186/s13663-015-0430-2
   Google Scholar
• F. Wang, A new algorithm for solving the multiple-sets split feasibility problem in Banach spaces, Numer. Funct. Anal. Opt. 35 (2014), 99–110. doi: 10.1080/01630563.2013.809360
   Web of Science ®Google Scholar
• H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66(2) (2002), 240–256. doi: 10.1112/S0024610702003332
   Web of Science ®Google Scholar
• H.K. Xu, A variable Krasnoselskii-Mann algorithm and the multiple-set split feasibility problem, Inverse Prob. 22 (2006), 2021–2034. doi: 10.1088/0266-5611/22/6/007
   Web of Science ®Google Scholar
• H.K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Prob. 26 (2010), 105018. doi: 10.1088/0266-5611/26/10/105018
   Web of Science ®Google Scholar
• Q. Yang, The relaxed CQ algorithm for solving the split feasibility problem, Inverse Prob. 20 (2004), 1261–1266. doi: 10.1088/0266-5611/20/4/014
   Web of Science ®Google Scholar
• Q. Yang, On variable-set relaxed projection algorithm for variational in-equalities, J. Math. Anal.Appl. 302 (2005), 166–179. doi: 10.1016/j.jmaa.2004.07.048
   Web of Science ®Google Scholar
• H. Zegeye and N. Shahzad, Convergence Theorems for Right Bregman Strongly Nonexpansive Mappings in Reflexive Banach Spaces, Abstr. Appl Anal. 2014, Article ID 584395.
   Google Scholar
• W. Zhang, D. Han and Z. Li, A self-adaptive projection method for solving the multiple-sets split feasibility problem, Inverse Prob. 25 (2009), 115001. doi: 10.1088/0266-5611/25/11/115001
   PubMed Web of Science ®Google Scholar
• J. Zhao and Q. Yang, Generalized KM theorems and their applications, Inverse Prob. 22 (2006), 833–844. doi: 10.1088/0266-5611/22/3/006
   Web of Science ®Google Scholar
• J. Zhao and Q. Yang, Self-adaptive projection methods for the multiple-sets split feasibility problem, Inverse Prob. 27 (2011), 035009. doi: 10.1088/0266-5611/27/3/035009
   Web of Science ®Google Scholar
• J. Zhao, Y. Zhang and Q. Yang, Modified projection methods for the split feasibility problem and multiple-sets feasibility problem, Appl. Math. Comput. 219 (2012), 1644–1653.
   Web of Science ®Google Scholar