Publication Cover
Applicable Analysis
An International Journal
Volume 100, 2021 - Issue 10
224
Views
15
CrossRef citations to date
0
Altmetric
Articles

A modified extragradient method for variational inclusion and fixed point problems in Banach spaces

ORCID Icon &
Pages 2049-2068 | Received 01 Jul 2019, Accepted 20 Sep 2019, Published online: 30 Sep 2019

References

  • Xiu NH, Zhang JZ. Some recent advances in projection-type methods for variational inequalities. J Comput Appl Math. 2003;152:559–587.
  • Korpelevich GM. The extragradient method for finding saddle points and other problems. Matecon. 1976;12:747–756.
  • Korpelevich GM. An extragradient method for finding saddle points and for other problems. Ekonom Matem Metody. 1976;12:747–756.
  • Ceng LC, Yao JC. Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwanese J Math. 2006;10:1293–1303.
  • Nadezhkina N, Takahashi W. Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J Optim Theory Appl. 2006;128:191–201.
  • Thong DV, Hieu DV. Some extragradient-viscosity algorithms for solving variational inequality problems and fixed point problems, Numer. Algor. (2018). Available from: https://doi.org/10.1007/s11075-018-0626-8.
  • Thong DV, Hieu DV. Modified Tseng's extragradient algorithms for variational inequality problems. J Fixed Point Theory Appl. 2018;20:152.
  • Thong DV, Vinh NT, Hieu DV. Accelerated hybrid and shrinking projection methods for variational inequality problems. Optimization. 2019;68:981–998.
  • Lopez G, Martin-Marquez V, Wang F, et al. Forward–backward splitting methods for accretive operators in Banach spaces, Abstr. Appl. Anal. Volume 2012, Article ID 109236, 25 pages.
  • Rockafellar RT. On the maximal monotonicity of subdifferential mappings. Pacific J Math. 1970;33:209–216.
  • Takahashi S, Takahashi W, Toyoda M. Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J Optim Theory Appl. 2010;147:27–41.
  • Combettes PL, Wajs VR. Signal recovery by proximal forward-backward splitting. Multiscale Model Simul. 2005;4:1168–1200.
  • Lions PL, Mercier B. Splitting algorithms for the sum of two nonlinear operators. SIAM J Numer Anal. 1979;16:964–979.
  • Brézis H, Lions PL. Produits infinis de resolvantes. Israel J Math. 1978;29:329–345.
  • Chen GHG, Rockafellar RT. Convergence rates in forward–backward splitting. SIAM J Optim. 1997;7:421–444.
  • Güler O. On the convergence of the proximal point algorithm for convex minimization. SIAM J Control Optim. 1991;29:403–419.
  • Bertsekas DP, Tsitsiklis JN. Parallel and distributed computation: numerical methods. Belmont (MA): Athena Scientific; 1997.
  • Dunn JC. Convexity, monotonicity, and gradient processes in Hilbert space. J Math Anal Appl. 1976;53:145–158.
  • Takahashi W, Wong NC, Yao JC. Two generalized strong convergence theorems of Halpern's type in Hilbert spaces and applications. Taiwanese J Math. 2012;16:1151–1172.
  • Pholasa N, Cholamjiak P, Cho YJ. Modified forward–backward splitting methods for accretive operators in Banach spaces. J Nonlinear Sci Appl. 2016;9:2766–2778.
  • Chang SS, Wen CF, Yao JC. Zero point problem of accretive operators in Banach spaces. Bull Malays Math Sci Soc. 2019;42:105. Available from: https://doi.org/10.1007/s40840-017-0470-3.
  • Chang SS, Wen CF, Yao JC. Common zero point for a finite family of inclusion problems of accretive mappings in Banach spaces. Optimization. 2018;67:1183–1196.
  • Chang SS, Wen CF, Yao JC. Generalized viscosity implicit rules for solving quasi-inclusion problems of accretive operators in Banach spaces. Optimization. 2017;66:1105–1117.
  • Chang SS, Wen CF, Yao JC. A generalized forward–backward splitting method for solving a system of quasi variational inclusions in Banach spaces. RACSAM. 2019;113:729–747.
  • Cholamjiak P. A generalized forward–backward splitting method for solving quasi inclusion problems in Banach spaces. Numer Algor. 2016;71:915–932.
  • Qin X, Cho SY, Wang L. Strong convergence of an iterative algorithm involving nonlinear mappings of nonexpansive and accretive type. Optimization. 2018;67:1377–1388.
  • Sunthrayuth P, Cholamjiak P. Iterative methods for solving quasi-variational inclusion and fixed point problem in q-uniformly smooth Banach spaces. Numer Algor. 2018;78:1019–1044.
  • Wang Y, Wang F. Strong convergence of the forward-backward splitting method with multiple parameters in Hilbert spaces. Optimization. 2018;67:493–505.
  • Manaka H, Takahashi W. Weak convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space. Cubo. 2011;13:11–24.
  • Alber Y, Ryazantseva I. Nonlinear ill-posed problems of monotone type. Netherlands: Springer; 2006.
  • Xu HK. Inequalities in Banach spaces with applications. Nonlinear Anal. 1991;16:1127–1138.
  • Cioranescu I. Geometry of Banach spaces, duality mappings and nonlinear problems. Dordrecht: Kluwer Academic; 1990.
  • Agarwal RP, O'Regan D, Sahu DR. Fixed point theory for Lipschitzian-type mappings with applications. New York: Springer; 2009.
  • Chidume C. Geometric properties of Banach spaces and nonlinear iterations. London: Springer; 2009.
  • Reich S. Strong convergence theorems for resolvents of accretive operators in Banach spaces. J Math Anal Appl. 1980;75:287–292.
  • Song Y, Ceng L. A general iteration scheme for variational inequality problem and common fixed point problems of nonexpansive mappings in q-uniformly smooth Banach spaces. J Glob Optim. 2013;57:1327–1348.
  • Takahashi W. Nonlinear functional analysis. Yokohama: Yokohama Publishers; 2000.
  • Browder FE. Fixed-point theorems for noncompact mappings in Hilbert space. Proc Nat Acad Sci USA. 1965;53:1272–1276.
  • Cai G, Bu S. An iterative algorithm for a general system of variational inequalities and fixed point problems in q-uniformly smooth Banach spaces. Optim Lett. 2013;7:267–287.
  • Kazarinoff ND. Analytic inequalities. New York: Holt, Rinehart and Winston; 1961.
  • Xu HK. Iterative algorithms for nonlinear operators. J London Math Soc. 2002;66:240–256.
  • Maingé PE. Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 2008;16:899–912.
  • Censor Y, Elfving T. A multiprojection algorithm using Bregman projections in product space. Numer Algor. 1994;8:221–239.
  • Byrne C. Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 2002;18:441–453.
  • Xu HK. Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 2010;26:105018. 17 pages.
  • Byrne C. A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 2004;20:103–120.
  • Tibshirani R. Regression shrinkage and selection via the lasso. J Roy Stat Soc Ser B. 1996;58:267–288.
  • Xu HK. Properties and iterative methods for the Lasso and its variants. Chin Ann Math. 2014;35:501–518.
  • Alghamdi MA, Alghamdi MA, Shahzad N, et al. Properties and iterative methods for the Q-Lasso, Abstr. Appl. Anal. Volume 2013, Article ID 250943, 8 pages.

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.