Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Volume 67, 2018 - Issue 10
115
Views
2
CrossRef citations to date
0
Altmetric
Articles

A cyclic iterative method for solving a class of variational inequalities in Hilbert spaces

Pages 1769-1796 | Received 05 Sep 2017, Accepted 05 Jul 2018, Published online: 31 Jul 2018

References

  • Stampacchia G. Formes bilineaires coercitives sur les ensembles convexes. C R Acad Sci Paris. 1964;258:4413–4416.
  • Moudafi A. Vicosity approximation methods for fixed point problems. J Math Anal Appl. 2000;241:46–55. doi: 10.1006/jmaa.1999.6615
  • Marino G, Xu H-K. Weak and strong convergence theorems for strict pseudo contractions in Hilbert spaces. J Math Anal Appl. 2007;329:336–346. doi: 10.1016/j.jmaa.2006.06.055
  • Bauschke HH. The approximation of fixed points of compositions of nonexpansive mappings in Hilbert spaces. J Math Anal Appl. 1996;202:150–159. doi: 10.1006/jmaa.1996.0308
  • Chang S-S. Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces. J Math Anal Appl. 2006;323:1402–1416. doi: 10.1016/j.jmaa.2005.11.057
  • Chang S-S, Yao J- C, Kim JK, et al. Iterative approximation to convex feasibility problems in Banach space. Fixed Point Theory Appl. 2007;2007:046797.
  • Ceng LC, Cubiotti P, Yao J-C. Strong convergence theorems for finitely many nonexpansive mappings and applications. Nonlinear Anal. 2007;67:1464–1473. doi: 10.1016/j.na.2006.06.055
  • Chidume CE, Ali B. Convergence theorems for common fixed points for infinite families of nonexpansive mappings in reflexive Banach spaces. Nonlinear Anal. 2008;68:3410–3418. doi: 10.1016/j.na.2007.03.032
  • Chidume CE, Zegeye H, Shahzad N. Convergence theorems for a common fixed point of finite family of nonself nonexpansive mappings. Fixed Point Theory Appl. 2005;2005:233–241.
  • Dye J, Khamsi MA, Reich S. Random products of contractions in Banach spaces. Trans Am Math Soc. 1991;325:87–99. doi: 10.1090/S0002-9947-1991-0989572-5
  • Dye JM, Reich S. On the unrestricted iteration of projections in Hilbert space. J Math Anal Appl. 1991;156:101–119. doi: 10.1016/0022-247X(91)90385-D
  • Dye JM, Reich S. Unrestricted iterations of nonexpansive mappings in Hilbert space. Nonlinear Anal. 1992;18:199–207. doi: 10.1016/0362-546X(92)90094-U
  • O'Hara JG, Pilla P, Xu H-K. Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces. Nonlinear Anal. 2003;54:1417–1426. doi: 10.1016/S0362-546X(03)00193-7
  • Jung JS. Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. J Math Anal Appl. 2005;302:509–520. doi: 10.1016/j.jmaa.2004.08.022
  • Takahashi W, Shimoji K. Convergence theorem for nonexpansive mappings and feasibility problems. Math Comp Model. 2000;32:1463–1471. doi: 10.1016/S0895-7177(00)00218-1
  • Yamada Y. The hybrid steepest-descent method for variational inequalities problems over the intersection of the fixed point sets of nonexpansive mappings. In: Butnariu D, Censor Y, Reich S, editors. Inherently parallel algorithms in feasibility and optimization and their applications. Amsterdam: North-Holland; 2001. p. 473–504.
  • Aoyama K, Kimura Y, Takahashi W, et al. Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal. 2007;67(8):2350–2360. doi: 10.1016/j.na.2006.08.032
  • Ceng L-C, Ansari QH, Yao J-C. Some iterative methods for finding fixed points and for solving constrained convex minimization problems. Nonlinear Anal. 2011;74(16):5286–5302. doi: 10.1016/j.na.2011.05.005
  • C.Wong N, Sahu DR, Yao JC. A generalized hybrid steepest-descent method for variational inequalities in Banach spaces. Fixed Point Theory Appl. 2011;2011, Article ID 754702, 28 pages.
  • Sahu DR, Kang SM, Sagar V. Approximation of common fixed points of a sequence of nearly nonexpansive mappings and solutions of variational inequality problems. J Appl Math. 2012;2012, Article ID 902437, 12 pages. doi: 10.1155/2012/902437
  • Combettes PL. On the numerical robustness of the parallel projection method in signal synthesis. IEEE Signal Process Lett. 2001;8:45–47. doi: 10.1109/97.895371
  • Kim T-H, Xu H-K. Robustness of Mann's algorithm for nonexpansive mappings. J Math Anal Appl. 2007;327:1105–1115. doi: 10.1016/j.jmaa.2006.05.009
  • Reich S. Extension problems for accretive sets in Banach spaces. J Funct Anal. 1977;26:378–395. doi: 10.1016/0022-1236(77)90022-2
  • Goebel K, Kirk WA. Topics in metric fixed point theory. Cambridge: Cambridge Univ. Press; 1990. (Cambridge Stud. Adv. Math.; 28).
  • Goebel K, Reich S. Uniform convexity, hyperbolic geometry, and nonexpansive mappings. New York: Marcel Dekker; 1984.
  • Takahashi W. Nonlinear functional analysis, fixed point theory and its applications. Yokohama: Yokohama Publishers; 2000.
  • Xu H-K. Strong convergence of an iterative method for nonexpansive and accretive operators. J Math Anal Appl. 2006;314(2):631–643. doi: 10.1016/j.jmaa.2005.04.082
  • Nakajo K, Takahashi W. Strong convergence theorem for nonexpansive mappings and nonexpansive semigroup. J Math Anal Appl. 2003;279:372–379. doi: 10.1016/S0022-247X(02)00458-4
  • Takahashi W, Takeuchi Y, Kubota R. Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J Math Anal Appl. 2008;341:276–286. doi: 10.1016/j.jmaa.2007.09.062
  • Reich S. Product formulas, nonlinear semigroups, and accretive operators. J Funct Anal. 1980;36:147–168. doi: 10.1016/0022-1236(80)90097-X
  • Nakajo K, Shimoji K, Takahashi W. Strong convergence to common fixed points of families of nonexpansive mappings in Banach spaces. J Nonlinear Convex Anal. 2007;8:11–34.
  • Martinet B. Régularisation d'inéquations variationnelles par approximations successives. Revue française d'informatique et de recherche opérationnelle. 1970;4:154–158.
  • Rockaffelar RT. Monotone operators and proximal point algorithm. SIAM J Control Optim. 1976;14:887–897.
  • Bruck RE, Reich S. Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J Math. 1977;3:459–470.
  • Nevanlinna O, Reich S. Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces. Israel J Math. 1979;32:44–58. doi: 10.1007/BF02761184
  • Güler O. On the convergence of the proximal point algorithm for convex minimization. SIAM J Control Optim. 1991;29(2):403–419. doi: 10.1137/0329022
  • Bauschke HH, Matoušková E, Reich S. Projection and proximal point methods: convergence results and counterexamples. Nonlinear Anal. 2004;56:715–738. doi: 10.1016/j.na.2003.10.010
  • Xu H-K. Iterative algorithms for nonlinear operators. J Lond Math Soc. 2002;66:240–256. doi: 10.1112/S0024610702003332
  • Kamimura S, Takahashi W. Approximating solutions of maximal monotone operators in Hilbert spaces. J Approx Theory. 2000;106:226–240. doi: 10.1006/jath.2000.3493
  • Halpern B. Fixed points of nonexpanding maps. Bull Am Math Soc. 1967;73:957–962. doi: 10.1090/S0002-9904-1967-11864-0
  • Lehdili N, Moudafi A. Combining the proximal algorithm and Tikhonov regularization. Optimization. 1996;37(3):239–252. doi: 10.1080/02331939608844217
  • Xu H-K. A regularization method for the proximal point algorithm. J Glob Optim. 2006;36(1):115–125. doi: 10.1007/s10898-006-9002-7
  • Song Y, Yang C. A note on a paper: a regularization method for the proximal point algorithm. J Glob Optim. 2009;43(1):171–174. doi: 10.1007/s10898-008-9279-9
  • Cho YJ, Qin X. Viscosity approximation methods for a family of m-accretive mapping in reflexive Banach spaces. Positivity. 2008;12:483–494. doi: 10.1007/s11117-007-2181-8
  • Jung J-S. Strong convergence of an iterative method for finding common zeros of a finite family of accretive operators. Commun Korean Math Soc. 2009;24(3):381–393. doi: 10.4134/CKMS.2009.24.3.381
  • Kim JK, Tuyen TM. Viscosity approximation method with Meir-Keeler contractions for common zero of accretive operators in Banach spaces. Fixed Point Theory Appl. 2015;2015:9. doi: 10.1186/s13663-014-0256-3
  • Reem D, Reich S. Solutions to inexact resolvent inclusion problems with applications to nonlinear analysis and optimization. Rend Circ Mat Palermo. 2018;67(2):337–371.
  • Reich S, Sabach S. A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces. J Nonlinear Convex Anal. 2009;10:471–485.
  • Reich S, Sabach S. Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer Funct Anal Optim. 2010;31:22–44. doi: 10.1080/01630560903499852
  • Reich S, Zaslavski AJ. Infinite products of resolvents of accretive operators. Topological Methods Nonlinear Anal. 2000;15:153–168. doi: 10.12775/TMNA.2000.012
  • Tuyen TM. Strong convergence theorem for a common zero of m-accretive mappings in Banach spaces by viscosity approximation methods. Nonlinear Funct Anal Appl. 2012;17:187–197.
  • Zegeye H, Shahzad N. Strong convergence theorems for a common zero of a finite family of m-accretive mappings. Nonlinear Anal. 2007;66:1161–1169. doi: 10.1016/j.na.2006.01.012

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.