Proximal forward-backward splitting method for zeros of sum accretive operators for a fixed point set and inverse problems

Volume 17, Issue 4, pp 506-526 http://dx.doi.org/10.22436/jmcs.017.04.06

Publication Date: October 27, 2017 Submission Date: June 22, 2017

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)

• 2243 Downloads
• 5083 Views

Authors

K. Sitthithakerngkiet - Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut K. Promluang - KMUTTFixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut P. Thounthong - Renewable Energy Research Centre \(\&\) Department of Teacher Training in Electrical Engineering, Faculty of Technical Education, King Mongkut P. Kumam - KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut

Abstract

In this paper, we investigate regularization method via a proximal point algorithm for solving treating sum of two accretive operators and fixed point problems. Strong convergence theorems are established in the framework of Banach spaces. Also we apply our result to variational inequalities and equilibrium problems. Furthermore, an illustrative numerical example is presented.

Share and Cite

Keywords

• Regularization method
• proximal point algorithm
• zero points
• accretive operators
• inverse problems

MSC

•  47H09
•  47H17
•  47J25
•  49J40

References

• [1] K. Aoyama, H. Iiduka, W. Takahashi, Weak convergence of an iterative sequence for accretive operators in Banach spaces, Fixed Point Theory Appl., 2006 (2006 ), 13 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [2] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Translated from the Romanian, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden (1976)
  • Google Scholar


• [3] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123–145.
  • MathSciNet


• [4] F. E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1272–1276.
  • MathSciNet
  • Google Scholar


• [5] S. Y. Cho, X.-L. Qin, L. Wang, Strong convergence of a splitting algorithm for treating monotone operators , Fixed Point Theory Appl., 2014 (2014 ), 15 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


• [7] J.-P. Gossez, E. Lami Dazo, Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math., 40 (1972), 565–573.
  • MathSciNet
  • Google Scholar


• [8] S. Kitahara, W. Takahashi, Image recovery by convex combinations of sunny nonexpansive retractions, Topol. Methods Nonlinear Anal., 2 (1993), 333–342.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] N. Lehdili, A. Moudafi, Combining the proximal algorithm and Tikhonov regularization, Optimization, 37 (1996), 239– 252.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] L.-S. Liu, Ishikawa-type and Mann-type iterative processes with errors for constructing solutions of nonlinear equations involving m-accretive operators in Banach spaces , Nonlinear Anal. , 34 (1998), 307–317.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] G. López, V. Martín-Márquez, F.-H. Wang, H.-K. Xu, Forward-backward splitting methods for accretive operators in Banach spaces, Abstr. Appl. Anal., 2012 (2012 ), 25 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [12] H. Manaka, W. Takahashi, Weak convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space, Cubo, 13 (2011), 11–24.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] B. Martinet, Régularisation d’inéquations variationnelles par approximations successives, (French) Rev. Française Informat. Recherche Opérationnelle, 4 (1970), 154–159.
  • MathSciNet
  • MATH


• [14] Y. Qing, S. Y. Cho, A regularization algorithm for zero points of accretive operators , Fixed Point Theory Appl., 2013 (2013 ), 9 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces , J. Math. Anal. Appl., 67 (1979), 274–276.
  • View Article
  • Google Scholar


• [16] R. T. Rockafellar , Monotone operators and the proximal point algorithm, SIAM J. Control Optimization, 14 (1976), 877–898.
  • View Article
  • Google Scholar


• [17] D. R. Sahu, J. C. Yao, The prox-Tikhonov regularization method for the proximal point algorithm in Banach spaces, J. Global Optim., 51 (2011), 641–655.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] Y.-L. Song, L.-C. Ceng, Weak and strong convergence theorems for zeroes of accretive operators in Banach spaces, J. Appl. Math., 2014 (2014 ), 11 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [19] 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


• [20] W. Takahashi, Viscosity approximation methods for resolvents of accretive operators in Banach spaces , J. Fixed Point Theory Appl., 1 (2007), 135–147.
  • View Article
  • MathSciNet
  • Google Scholar


• [21] W. Takahashi, Introduction to nonlinear and convex analysis, Yokohama Publishers, Yokohama (2009)
  • MathSciNet
  • Google Scholar


• [22] S. Takahashi, W. Takahashi, M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces , J. Optim. Theory Appl., 147 (2010), 27–41.
  • View Article
  • MathSciNet
  • Google Scholar


• [23] A. N. Tikhonov, Solution of incorrectly formulated problems and the regularization method, Soviet Meth. Dokl., 4 (1963), 1035–1038.
  • Google Scholar
  • MATH


• [24] A. N. Tikhonov, Improper problems of optimal planning and stable methods of their solution, Soviet Meth. Dokl., 6 (1965), 1264–1267.
  • MATH


• [25] A. N. Tikhonov, V. Y. Arsenin , Solutions of ill-posed problems, Translated from the Russian, Preface by translation editor Fritz John, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York-Toronto, Ont.-London (1977)