Parallel iterative methods for solving systems of generalized mixed equilibrium problems in reflexive Banach spaces

References

• Ceng LC, Yao JC. A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J Comput Appl Math. 2008;214:186–201.
   Web of Science ®Google Scholar
• Takahashi W, Toyoda M. Weak convergence theorems for nonexpansive mappings and monotone mappings. J Optim Theory Appl. 2003;118:417–428.
   Web of Science ®Google Scholar
• Browder FE. Existence and approximation of solutions of nonlinear variational inequalities. Proc Natl Acad Sci USA. 1966;56:1080–1086.
   PubMed Web of Science ®Google Scholar
• Blum E, Oettli W. From optimization and variational inequalities to equilibrium problems. Math Student. 1994;63:123–145.
   Google Scholar
• Haugazeau Y. Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes [On the variational inequalities and the minimization of convex functions] [Thesis]. Paris: Université de Paris; 1968.
   Google Scholar
• Kim TH, Takahashi W. Strong convergence of modified iteration processes for relatively asymptotically nonexpansive mappings. Taiwan J Math. 2010;14(6):2163–2180.
   Web of Science ®Google Scholar
• Plubtieng S, Ungchittrakool K. Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space. J Approx Theory. 2002;149:103–115.
   Web of Science ®Google Scholar
• Qin X, Cho YJ, Kang SM. Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J Comput Appl Math. 2009;225:20–30.
   Web of Science ®Google Scholar
• Qin X, Cho YJ, Kang SM. On hybrid projection methods for asymptotically quasi-φ-nonexpansive mappings. Appl Math Comput. 2010;215:3874–3883.
   Web of Science ®Google Scholar
• Matsushita S, Takahashi W. Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2004;2004:37–47.
   Google Scholar
• Solodov MV, Svaiter BF. Forcing strong convergence of proximal point iterations in Hilbert space. Math Progr. 2000;87:189–202.
   Web of Science ®Google Scholar
• Takahashi W, Zembayashi K. Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 2009;70:45–57.
   Web of Science ®Google Scholar
• Kassay G, Reich S, Sabach S. Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J Optim. 2011;21:1319–1344.
   Web of Science ®Google Scholar
• Reich S, Sabach S. Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces. Contemp Math. 2012;568:225–240.
   Google Scholar
• 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.
   Web of Science ®Google Scholar
• Anh PK, Chung CV. Parallel hybrid methods for a finite family of relatively nonexpansive mappings. Numer Funct Anal Optim. 2014;35:649–664.
   Web of Science ®Google Scholar
• Butnariu D, Reich S, Zaslavski AJ. Asymptotic behavior of relatively nonexpansive operators in Banach spaces. J Appl Anal. 2001;7:151–174.
   Google Scholar
• Cioranescu I. Geometry of Banach spaces, duality mappings and nonlinear problems. Dordrecht: Kluwer Academic; 1990.
   Google Scholar
• Reich S. Review of geometry of Banach spaces, duality mappings and nonlinear problems by Ioana Cioranescu. Dordrecht: Kluwer Academic; 1990. Bull Amer Math Soc. 1992;26:367–370.
   Google Scholar
• Bonnans JF, Shapiro A. Perturbation analysis of optimization problem. New York (NY): Springer; 2000.
   Google Scholar
• Butnariu D, Resmerita E. Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces. Abstr Appl Anal. 2006;2006:1–39.
   Web of Science ®Google Scholar
• Bauschke HH, Borwein JM, Combettes PL. Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Commun Contemp Math. 2001;3:615–647.
   Web of Science ®Google Scholar
• Censor Y, Lent A. An iterative row-action method for interval convex programming. J Optim Theory Appl. 1981;34:321–353.
   Web of Science ®Google Scholar
• Alber YI. Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos AG, editor. Theory and applications of nonlinear operator of accretive and monotone type. New York (NY): Marcel Dekker; 1996. p. 15–50.
   Google Scholar
• Zegeye H. Convergence theorems for Bregman strongly nonexpansive mappings in reflexive Banach spaces. Filomat. 2014;7:1525–1536.
   Web of Science ®Google Scholar
• Darvish V. Strong convergence theorem for generalized mixed equilibrium problems and Bregman nonexpansive mapping in Banach spaces. Opsearch. 2016;53(3):584–603.
   Web of Science ®Google Scholar
• Censor Y, Reich S. Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization. 1996;37:323–339.
   Google Scholar
• Reich S. A weak convergence theorem for the alternating method with Bregman distances. In: Kartsatos AG, editor. Theory and applications of nonlinear operators of accretive and monotone type. New York (NY): Marcel Dekker; 1996. p. 313–318.
   Google Scholar
• Martin-Marquez V, Reich S, Sabach S. Bregman strongly nonexpansive operators in reflexive Banach spaces. J Math Anal Appl. 2013;400:597–614.
   Web of Science ®Google Scholar
• Reich S, Sabach S. Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces. In: Bauschke HH, Burachik RS, Combettes PL, et. al., editors. Fixed-point algorithms for inverse problems in science and engineering. New York (NY): Springer; 2011. p. 301–316.
   Google Scholar
• Ambrosetti A, Prodi G. A Primer of nonlinear analysis. Cambridge: Cambridge University Press; 1993.
   Google Scholar
• 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.
   Web of Science ®Google Scholar
• Resmerita E. On total convexity, Bregman projections and stability in Banach spaces. J Convex Anal. 2004;11:1–16.
   Web of Science ®Google Scholar
• Butnariu D, Iusem AN. Totally convex functions for fixed points computation and infinite dimensional optimization. Dordrecht: Kluwer Academic; 2000.
   Google Scholar
• Reich S, Sabach S. Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer Funct Anal Optim. 2010;31:22–44.
   Web of Science ®Google Scholar