Hybrid iterative algorithms for the split common fixed point problems

Volume 10, Issue 4, pp 2214--2228 http://dx.doi.org/10.22436/jnsa.010.04.72

Publication Date: April 20, 2017 Submission Date: October 28, 2016

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1820 Downloads
• 2876 Views

Authors

Jong Soo Jung - Department of Mathematics, Dong-A University, Busan 49315, Korea.

Abstract

In this paper, we introduce two iterative algorithms (one implicit algorithm and one explicit algorithm) based on the hybrid steepest descent method for solving the split common fixed point problems. We establish the strong convergence of the sequences generated by the proposed algorithms to a solution of the split common fixed point problems, which is also a solution of a certain variational inequality. In particular, the minimum norm solution of the split common fixed point problems is obtained. As applications, variational problems and equilibrium problems are considered.

Share and Cite

Keywords

• Split common fixed point problem
• firmly nonexpansive mapping
• nonexpansive mapping
• variational inequality
• minimum-norm
• bounded linear operator
• variational problems
• equilibrium problems
• iterative algorithms.

MSC

•  47J05
•  49J40
•  49J52
•  47J20
•  47H10

References

• [1] R. P. Agarwal, D. O’Regan, D. R. Sahu, Fixed point theory for Lipschitzian-type mappings with applications, Topological Fixed Point Theory and Its Applications, Springer, New York (2009)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] Q. H. Ansari, A. Rehan, C.-F. Wen, Implicit and explicit algorithms for split common fixed point problems, J. Nonlinear Convex Anal., 17 (2016), 1381–1397.
  • MathSciNet
  • Google Scholar
  • MATH


• [3] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123–146.
  • MathSciNet
  • Google Scholar


• [4] 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


• [5] 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


• [6] A. Cegielski, Iterative methods for fixed point problems in Hilbert spaces, Lecture Notes in Mathematics, Springer, Heidelberg (2012)
  • Google Scholar


• [7] Y. Censor, T. Bortfeld, B. Martin, A. Tronfimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353–2365.
  • View Article
  • Google Scholar


• [8] 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


• [9] 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


• [10] Y. Censor, A. Motova, A. Segal, Perturbed projections and subgradient projections for the multiple-sets split feasibility problem, J. Math. Anal. Appl., 327 (2007), 1244–1256.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


• [12] P. L. Combettes, S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117–136.
  • MathSciNet
  • Google Scholar


• [13] 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
  • MathSciNet
  • Google Scholar


• [14] Y.-Z. Dang, Y. Gao, The strong convergence of a KM-CQ-like algorithm for a split feasibility problem, Inverse Problems, 27 (2011), 9 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1990)
  • View Article
  • MathSciNet
  • Google Scholar


• [16] J. S. Jung, Iterative algorithms based on the hybrid steepest descent method for the split feasibility problem, J. Nonlinear Sci. Appl., 9 (2016), 4214–4225.
  • MathSciNet
  • Google Scholar
  • MATH


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


• [18] A. Latif, D. Y. Sahu, Q. H. Ansari, Variable KM-like algorithms for fixed point problems and split feasibility problems, Fixed Point Theory Appl., 2014 (2014), 20 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] G. J. Minty, On the generalization of a direct method of the calculus of variations, Bull. Amer. Math. Soc., 73 (1967), 315–321.
  • View Article
  • MathSciNet
  • Google Scholar


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


• [21] 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


• [22] 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


• [23] T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227–239.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


• [25] F.-H. Wang, H.-K. Xu, Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem, J. Inequal. Appl., 2010 (2010), 13 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


• [27] 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


• [28] I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, Inherently parallel algorithms in feasibility and optimization and their applications, Haifa, (2000), Stud. Comput. Math., North-Holland, Amsterdam, 8 (2001), 473–504.
  • View Article
  • MathSciNet
  • Google Scholar


• [29] I. Yamada, N. Ogura, Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings, Numer. Funct. Anal. Optim., 25 (2004), 619–655.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


• [31] Y.-H. Yao, P.-X. Yang, S. M. Kang, Composite projection algorithms for the split feasibility problem, Math. Comput. Modelling, 57 (2013), 693–700.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] 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