Strong convergence results for variational inclusions, systems of variational inequalities and fixed point problems using composite viscosity implicit methods

References

• Ceng LC, Wang CY, Yao JC. Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math Methods Oper Res. 2008;67:375–390.
   Web of Science ®Google Scholar
• Ceng LC, Liou YC, Wen CF. Systems of variational inequalities with hierarchical variational inequality constraints in Banach spaces. J Nonlinear Sci Appl. 2017;10:3136–3154.
   Google Scholar
• Ceng LC, Wen CF. Three-step Mann iterations for a general system of variational inequalities and an infinite family of nonexpansive mappings in Banach spaces. J Inequal Appl. 2013;2013:Article ID 539, 27 pages.
   Web of Science ®Google Scholar
• Ceng LC, Cho SY, Qin X, et al. A general system of variational inequalities with nonlinear mappings in Banach spaces. J Nonlinear Convex Anal. 2019;20:395–410.
   Web of Science ®Google Scholar
• Ceng LC, Liou YC, Wen CF, et al. Convergence and some control conditions of hybrid steepest-descent methods for systems of variational inequalities and hierarchical variational inequalities. J Nonlinear Sci Appl. 2017;10:4574–4596.
   Google Scholar
• Ceng LC, Petrusel A, Yao JC, et al. Hybrid viscosity extragradient method for systems of variational inequalities, fixed points of nonexpansive mappings, zero points of accretive operators in Banach spaces. Fixed Point Theory. 2018;19:487–501.
   Web of Science ®Google Scholar
• Wang QW, Guan JL, Ceng LC, et al. General iterative methods for systems of variational inequalities with the constraints of generalized mixed equilibria and fixed point problem of pseudocontractions. J Inequal Appl. 2018;2018:Article ID 315, 35 pages.
   Web of Science ®Google Scholar
• Cai G, Bu SQ. Strong convergence theorems based on a new modified extragradient method for variational inequality problems and fixed point problems in Banach spaces. Comput Math Appl. 2011;62:2567–2579.
   Web of Science ®Google Scholar
• Ceng LC, Gupta H, Ansari QH. Implicit and explicit algorithms for a system of nonlinear variational inequalities in Banach spaces. J Nonlinear Convex Anal. 2015;16(5):965–984.
   Web of Science ®Google Scholar
• Yao Y, Liou YC, Kang SM, et al. Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces. Nonlinear Anal. 2011;74:6024–6034.
   Web of Science ®Google Scholar
• Ceng LC, Wen CF. Systems of variational inequalities with hierarchical variational inequality constraints for asymptotically nonexpansive and pseudocontractive mappings. Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM. 2019;113:2431–2447.
   Web of Science ®Google Scholar
• Ceng LC, Latif A, Yao JC. On solutions of a system of variational inequalities and fixed point problems in Banach spaces. Fixed Point Theory Appl. 2013;2013:Article ID 176, 34 pp.
   Google Scholar
• Song YL, Ceng LC. A general iteration scheme for variational inequality problem and common fixed point problems of nonexpansive mappings in q-uniformly smooth Banach spaces. J Global Optim. 2013;57:1327–1348.
   Web of Science ®Google Scholar
• Sunthrayuth P, Cholamjiak P. A modified extragradient method for variational inclusion and fixed point problems in Banach spaces. Appl Anal. 2019. Available from: https://doi.org/10.1080/00036811.2019.1673374.
   Web of Science ®Google Scholar
• Aoyama K, Iiduka H, Takahashi W. Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl. 2006;2006:Article ID 35390, 13 pages.
   Google Scholar
• 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.
   Web of Science ®Google Scholar
• Suantai S, Cholamjiak P, Sunthrayuth P. The generalized viscosity explicit rules for solving variational inclusion problem in Banach spaces. Optimization. To appear.
   Web of Science ®Google Scholar
• Kamimura S, Takahashi W. Strong convergence of a proximal-type algorithm in a Banach space. SIAM J Optim. 2002;13:938–945.
   Web of Science ®Google Scholar
• Takahashi Y, Hashimoto K, Kato M. On sharp uniform convexity, smoothness, and strong type, cotype inequalities. J Nonlinear Convex Anal. 2002;3:267–281.
   Google Scholar
• Xu HK. Inequalities in Banach spaces with applications. Nonlinear Anal. 1991;16:1127–1138.
   Web of Science ®Google Scholar
• Reich S. Weak convergence theorems for nonexpansive mappings in Banach spaces. J Math Anal Appl. 1979;67:274–276.
   Web of Science ®Google Scholar
• Takahashi W. Convex analysis and approximation of fixed points. Yokohama: Yokohama Publisher; 2000. (Japanese).
   Google Scholar
• López G, Martín-Márquez V, Wang F, et al. Forward-backward splitting methods for accretive operators in Banach spaces. Abstr Appl Anal. 2012;2012:Article ID 109236, 25 pages.
   Google Scholar
• Xu HK. Viscosity approximation methods for nonexpansive mappings. J Math Anal Appl. 2004;298:279–291.
   Web of Science ®Google Scholar
• Bruck RE. Properties of fixed point sets of nonexpansive mappings in Banach spaces. Trans Amer Math Soc. 1973;179:251–262.
   Web of Science ®Google Scholar
• Suzuki T. Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J Math Anal Appl. 2005;305:227–239.
   Web of Science ®Google Scholar
• Shimoji K, Takahashi W. Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese J Math. 2001;5(2):387–404.
   Web of Science ®Google Scholar
• Chang SS, Lee HWJ, Chan CK. A new method for solving equilibrium problem, fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 2009;70:3307–3319.
   Web of Science ®Google Scholar
• Xu HK, Kim TH. Convergence of hybrid steepest-descent methods for variational inequalities. J Optim Theory Appl. 2003;119(1):185–201.
   Web of Science ®Google Scholar
• Cho YJ, Zhou HY, Guo G. Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput Math Appl. 2004;47:707–717.
   Web of Science ®Google Scholar
• Baillon JB, Haddad G. Quelques proprietes des operateurs angle-bornes et N-cycliquement monotones. Isr J Math. 1977;26:137–150.
   Web of Science ®Google Scholar
• 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.
   Web of Science ®Google Scholar
• Ceng LC, Al-Homidan S, Ansari QH. Iterative algorithms with regularization for hierarchical variational inequality problems and convex minimization problems. Fixed Point Theory Appl. 2013;2013:Article ID 284, 24 pages.
   Google Scholar
• Xu HK. Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 2010;26(10):Article ID 105018, 17 pages.
   Web of Science ®Google Scholar
• Ceng LC, Ansari QH, Yao JC. Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem. Nonlinear Anal. 2012;75:2116–2125.
   Web of Science ®Google Scholar
• Ceng LC, Ansari QH, Yao JC. An extragradient method for solving split feasibility and fixed point problems. Comput Math Appl. 2012;64:633–642.
   Web of Science ®Google Scholar
• Ceng LC, Wong NC, Yao JC. Hybrid extragradient methods for finding minimum-norm solutions of split feasibility problems. J Nonlinear Convex Anal. 2015;16:1965–1983.
   Web of Science ®Google Scholar
• Byrne C. A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 2004;20:103–120.
   Web of Science ®Google Scholar
• Ceng LC, Coroian I, Qin X, et al. A general viscosity implicit iterative algorithm for split variational inclusions with hierarchical variational inequality constraints. Fixed Point Theory. 2019;20:469–482.
   Web of Science ®Google Scholar