Strong convergence theorems for a nonexpansive mapping and its applications for solving the split feasibility problem

Volume 10, Issue 4, pp 1470--1477 http://dx.doi.org/10.22436/jnsa.010.04.16

Publication Date: April 20, 2017 Submission Date: February 18, 2017

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Authors

Qinwei Fan - School of Science, Xi’an Polytechnic University, Xi’an 710048, P. R. China. Zhangsong Yao - School of Information Engineering, Nanjing Xiaozhuang University, Nanjing 211171, P. R. China.

Abstract

The aim of this paper is to propose some novel algorithms and their strong convergence theorems for solving the split feasibility problem, and we obtain the corresponding strong convergence results under mild conditions. The split feasibility problem was proposed by [Y. Censor, Y. Elfving, Numer. Algorithms, 8 (1994), 221–239]. So far a lot of algorithms have been given for solving this problem due to its applications in intensity-modulated radiation therapy, signal processing, and image reconstruction. But most of these algorithms are of weak convergence. In this paper, we propose the new algorithms which can provide useful guidelines for solving the relevant problem, such as the split common fixed point problem (SCFP), multi-set split feasibility problem and so on.

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Keywords

• Split feasibility problem
• strong convergence
• nonexpansive mapping
• Hilbert space.

MSC

•  47J25
•  47H45

References

• [1] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441–453.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103–120.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] L.-C. Ceng, Q. H. Ansari, J.-C. Yao, Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem, Nonlinear Anal., 75 (2012), 2116–2125.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] Y. Censor, T. Borfeid, B. Martin, A. Troimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2005), 2353–2365.
  • View Article
  • Google Scholar


• [5] Y. Censor, Y. Elfving, A multiprojection algorithm using Bregman projection in a product space, Numer. Algorithms, 8 (1994), 221–239.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] Y. Censor, Y. Elfving, N. Kopf, T. Bottfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071–2084.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] Q.-W. Fan, W. Wu, J. M. Zurada, Convergence of batch gradient learning with smoothing regularization and adaptive momentum for neural, SpringerPlus, 5 (2016), 17 pages.
  • View Article
  • Google Scholar


• [8] T.-H. Kim, H.-K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal., 61 (2005), 51–60.
  • View Article
  • Google Scholar


• [9] R. Kraikaew, S. Saejung, On split common fixed point problems, J. Math. Anal. Appl., 415 (2014), 513–524.
  • View Article
  • Google Scholar


• [10] A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems, Nonlinear Anal., 79 (2013), 117–121.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] X.-L. Qin, J.-C. Yao, Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators, J. Inequal. Appl., 2016 (2016), 9 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] B. Qu, B.-H. Liu, N. Zheng, On the computation of the step-size for the CQ-like algorithms for the split feasibility problem, Appl. Math. Comput., 262 (2015), 218–223.
  • View Article
  • MathSciNet
  • Google Scholar


• [13] B. Qu, N.-H. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems, 21 (2005), 1655–1665.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] B. Qu, N.-H. Xiu, A new halfspace-relaxation projection method for the split feasibility problem, Linear Algebra Appl., 428 (2008), 1218–1229.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] W. Takahashi, Nonlinear functional analysis, Fixed point theory and its applications, Yokohama Publishers, Yokohama (2000)
  • Google Scholar


• [16] Z.-W. Wang, Q.-Z. Yang, Y.-N. Yang, The relaxed inexact projection methods for the split feasibility problem, Appl. Math. Comput., 217 (2011), 5347–5359.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256.
  • View Article
  • Google Scholar


• [18] H.-K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116 (2003), 659–678.
  • View Article
  • MathSciNet
  • Google Scholar


• [19] H.-K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279–291.
  • View Article
  • Google Scholar


• [20] H.-K. Xu, A variable Krasonsel’skiî-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021–2034.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] H.-K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Problems, 26 (2010), 17 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [22] Q.-Z. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20 (2004), 1261–1266.
  • View Article
  • MathSciNet
  • Google Scholar


• [23] Y.-H. Yao, R. P. Agarwal, M. Postolache, Y.-C. Liou, Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem, Fixed Point Theory Appl., 2014 (2014 ), 14 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [24] Y.-H. Yao, Y.-C. Liou, J.-C. Yao, Split common fixed point problem for two quasi-pseudo-contractive operators and its algorithm construction, Fixed Point Theory Appl., 2015 (2015), 19 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] Y.-H. Yao, G. Marino, H.-K. Xu, Y.-C. Liou, Construction of minimum-norm fixed points of pseudocontractions in Hilbert spaces, J. Inequal. Appl., 2014 (2014), 14 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] Y.-H. Yao, M. Postolache, Y.-C. Liou, Strong convergence of a self-adaptive method for the split feasibility problem, Fixed Point Theory Appl., 2013 (2013), 12 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] J.-L. Zhao, Q.-Z. Yang, Several solution methods for the split feasibility problem, Inverse Problems, 21 (2005), 1791–1799.
  • View Article
  • MathSciNet
  • Google Scholar


• [28] J.-L. Zhao, Y.-J. Zhang, Q.-Z. Yang, Modified projection methods for the split feasibility problem and the multiple-sets split feasibility problem, Appl. Math. Comput., 219 (2012), 1644–1653.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH