Iterative methods for solving the split common fixed point problem of demicontractive mappings in Hilbert spaces

Volume 11, Issue 8, pp 960--970 http://dx.doi.org/10.22436/jnsa.011.08.03

Publication Date: June 07, 2018 Submission Date: February 12, 2018 Revision Date: April 17, 2018 Accteptance Date: April 18, 2018

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Authors

Chunxiang Zong - Department of Mathematics, Nanchang University, Nanchang 330031, P. R. China. Yuchao Tang - Department of Mathematics, Nanchang University, Nanchang 330031, P. R. China.

Abstract

The split common fixed point problem was proposed in recent years which required to find a common fixed point of a family of mappings in one space whose image under a linear transformation is a common fixed point of another family of mappings in the image space. In this paper, we study two iterative algorithms for solving this split common fixed point problem for the class of demicontractive mappings in Hilbert spaces. Under mild assumptions on the parameters, we prove the convergence of both iterative algorithms. As a consequence, we obtain new convergence theorems for solving the split common fixed point problem for the class of directed mappings. We compare the performance of the proposed iterative algorithms with the existing iterative algorithms and conclude from the numerical experiments that our iterative algorithms converge faster than these existing iterative algorithms in terms of iteration numbers.

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Keywords

• Split common fixed point problem
• demicontractive mappings
• cyclic iteration method
• simultaneous iteration method

MSC

•  47H10
•  90C25

References

• [1] H. H. Bauschke, J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1996), 367–426.
  • View Article
  • MathSciNet
  • Google Scholar


• [2] H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York (2011)
  • View Article
  • MathSciNet
  • Google Scholar


• [3] 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


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


• [5] L.-C. Ceng, Q. H. Ansari, J.-C. Yao, Mann type iterative methods for finding a common solution of split feasibility and fixed point problems, Positivity, 16 (2012), 471–495.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


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


• [8] Y. Censor, A. Segal , The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587–600.
  • MathSciNet
  • Google Scholar


• [9] S. S. Chang, H. W. Joseph Lee, C. K. Chan, L. Wang, L. J. Qin, Split feasibility problem for quasi-nonexpansive multi-valued mappings and total asymptotically strict pseudo-contracive mapping, Appl. Math. Comput., 219 (2013), 10416–10424.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] S. S. Chang, L. Wang, Y. K. Tang, L. Yang , The split common fixed point problems for total asymptotically strictly pseudocontracive mappings, J. Appl. Math., 2012 (2012), 13 pages.
  • View Article
  • Google Scholar


• [11] H.-H. Cui, M.-L. Su, F.-H. Wang, Damped projection method for split common fixed point problems, J. Inequal. Appl., 2013 (2013), 10 pages.
  • View Article
  • Google Scholar
  • MATH


• [12] H.-H. Cui, F.-H. Wang, Iterative methods for the split common fixed point problem in Hilbert spaces, Fixed Point Theory Appl., 2014 (2014), 8 pages.
  • View Article
  • Google Scholar


• [13] Q.-L. Dong, Y.-H. Yao, S.-N. He, Weak convergence theorems of the modified relaxed projection algorithms for the split feasibility problem in Hilbert spaces, Optim. Lett., 8 (2014), 1031–1046.
  • View Article
  • Google Scholar
  • MATH


• [14] Y.-Y. Huang, C.-C. Hong , Approximating common fixed points of averaged self-mappings with application to the split feasibility problem and maximal monotone operators in Hilbert spaces , Fixed Point Theory Appl., 2013 (2013), 20 pages.
  • View Article
  • MathSciNet
  • Google Scholar


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


• [16] A. Moudafi, The split common fixed point problem for demicontractive mappings, Inverse Problems, 26 (2010), 6 pages.
  • View Article
  • Google Scholar


• [17] A. Moudafi , A note on the split common fixed-point problem for quasi-nonexpansive operators, Nonlinear Anal., 74 (2011), 4083–4087.
  • View Article
  • MathSciNet
  • Google Scholar


• [18] A. Moudafi, Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J. Nonlinear Convex Anal., 15 (2014), 809–818.
  • MathSciNet
  • Google Scholar
  • MATH


• [19] Y.-C. Tang, L.-W. Liu, Several iterative algorithms for solving the split common fixed point problem of directed operators with applications, Optimization, 65 (2016), 53–65.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] Y.-C. Tang, J.-G. Peng, L.-W. Liu , A cyclic algorithm for the split common fixed point problem of demicontractive mappings in Hilbert spaces, Math. Model. Anal., 17 (2012), 457–466.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] Y.-C. Tang, J.-G. Peng, L.-W. Liu, A cyclic and simultaneous iterative algorithm for the multiple split common fixed point problem of demicontractive mappings, Bull. Korean. Math. Soc., 51 (2014), 1527–1538.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] D. V. Thong, D. V. Hieu , An inertial method for solving split common fixed point problems, J. Fixed Point Theory Appl., 19 (2017), 3029–3051.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] F.-H. Wang, H.-H. Cui, Convergence of a cyclic algorithm for the split common fixed point problem without continuity assumption, Math. Model. Anal., 18 (2013), 537–542.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] F.-H. Wang, H.-K. Xu, Cyclic algorithms for split feasibility problems in Hilbert spaces, Nonlinear Anal., 74 (2011), 4105–4111.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


• [26] 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


• [27] L.-J. Zhu, Y.-C. Liou, S. M. Kang, Y.-H. Yao, Algorithmic and analytical approach to the split common fixed points problem, Fixed Point Theory Appl., 2014 (2014), 10 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] L.-J. Zhu, Y.-C. Liu, J.-C. Yao, Y.-H. Yao , New algorithms for designed for the split common fixed point problem of quasi-pseudocontractions, J. Inequal. Appl., 2014 (2014), 13 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH