An iterative approximation scheme for solving a split generalized equilibrium, variational inequalities and fixed point problems

References

• J.-P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer-Verlag, Berlin, 1998.
   Google Scholar
• C. Byrne, Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse Problems 18 (2002), pp. 441–453.
   Web of Science ®Google Scholar
• C. Byrne, Y. Censor, A. Gibali, and S. Reich, The split common null point problem, J. Nonlinear Convex Anal. 13(4) (2012), pp. 759–775.
   Web of Science ®Google Scholar
• L.C. Ceng and J.C. Yao, An extragradient like approximation method for variational inequality problems and fixed point problems, Appl. Math. Comput. 190 (2007), pp. 205–215.
   Web of Science ®Google Scholar
• L.C. Ceng and J.C. Yao, On the convergence analysis of inexact hybrid extragradient proximal point algorithms for maximal monotone operators, J. Comput. Appl. Math. 217 (2008), pp. 326–338.
   Web of Science ®Google Scholar
• L.C. Ceng and J.C. Yao, Approximate proximal algorithms for generalized variational inequalities with pseudomonotone multifunctions, J. Comput. Appl. Math. 213 (2008), pp. 423–438.
   Web of Science ®Google Scholar
• Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms 8(2–4) (1994), pp. 221–239.
   Google Scholar
• Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in product space, Numer. Algorithms 8 (1994), pp. 221–239.
   Google Scholar
• F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem, Fixed Point Theory Appl. 2010 (2010), pp. 383–740.
   Google Scholar
• B. Eicke, Iterative methods for convexly constrained ill-posed problem in Hilbert space, Numer. Funct. Anal. Optim. 13 (1992), pp. 413–429.
   Web of Science ®Google Scholar
• R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984.
   Google Scholar
• C. Jaiboon, P. Kumam, and H.W. Humphries, Weak convergence theorem by extragradient method for variational inequality, equilibrium problems and fixed point problems, Bull. Malays. Math. Sci. Soc. 2(32) (2009), pp. 173–185.
   Google Scholar
• P. Katchang and P. Kumam, A new iterative algorithm of solution for equilibrium problems, variational inequalities and fixed point problems in a Hilbert space, J. Appl. Math. Comput. 32(1) (2010), pp. 19–38.
   Google Scholar
• K.R. Kazmi and S.H. Rizvi, Iterative approximation of a common solution of a split generalized equilibrium problem and a fixed point problem for nonexpansive semigroup, Math. Sci. 7(1) (2013). doi: 10.1186/2251-7456-7-1.
   Google Scholar
• L. Landweber, An iterative formula for Fredholm integral equations of the first kind, Amer. J. Math. 73 (1951), pp. 615–625.
   Web of Science ®Google Scholar
• F. Liu and M.Z. Nasheed, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal. 6 (1998), pp. 313–344.
   Google Scholar
• H. Mahdioui and O. Chadli, On a system of generalized mixed equilibrium problems involving variational-like inequalities in Banach spaces: Existence and algorithmic aspects, Adv. Oper. Res. 2012 (2012), pp. 843–486.
   Google Scholar
• A. Moudafi, The split common fixed point problem for demicontractive mappings, Inverse Problems 26 (2010), p. 055007, 6pp.
   Web of Science ®Google Scholar
• A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000), pp. 46–55.
   Web of Science ®Google Scholar
• A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl. 150 (2011), pp. 275–283.
   Web of Science ®Google Scholar
• Z. Opial, Weak convergence of successive approximations for nonexpansive mappings, Bull. Am. Math. Soc. 73 (1967), pp. 591–597.
   Web of Science ®Google Scholar
• M.O. Osilike and D.I. Igbokwe, Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations, Comput. Math. Appl. 40 (2000), pp. 559–567.
   Web of Science ®Google Scholar
• K. Shimoji and W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math. 5(2) (2001), pp. 87–404.
   Web of Science ®Google Scholar
• G. Stampacchia, Formes bilineaires coercitivies sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), pp. 4413–4416.
   Google Scholar
• T. Suzuki, Strong convergence theorems for an infinite family of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl. 1 (2005), pp. 103–123.
   Google Scholar
• 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), pp. 227–239.
   Web of Science ®Google Scholar
• S. Wang, X. Gong, A.N. Abdou, and Y.J. Cho, Iterative algorithm for a family of split equilibrium problems and fixed point problems in Hilbert spaces with applications, Fixed Point Theory Appl. 2016(4) (2016). doi:doi: 10.1186/s13663-015-0475-2.
   Google Scholar
• R. Wangkeeree and R. Wangkeeree, A general iterative methods for variational inequality problems and mixed equilibrium problems and fixed point problems of strictly pseudocontractive mappings in Hilbert spaces, Fixed Point Theory Appl. (2007), p. 32 (Article ID 519065).
   Google Scholar
• H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004), pp. 279–291.
   Web of Science ®Google Scholar
• Y. Yao, Y.-C. Liou, and R. Chen, Convergence theorems for fixed point problems and variational inequality problems, J. Nonlinear Convex Anal. 9(2) (2008), pp. 239–248.
   Web of Science ®Google Scholar
• S.S. Zeng, N.C. Wong, and J.C. Yao, Convergence of hybrid steepest-descent methods for generalized variational inequalities, Acta Math. Sin. (Engl. Ser.) 22(1) (2006), pp. 1–12.
   Web of Science ®Google Scholar