Approximation of common fixed points of left Bregman strongly nonexpansive mappings and solutions of equilibrium problems
In this paper we propose an iterative algorithm based on the hybrid method in mathematical programming for approximating a common fixed point of an infinite family of left Bregman strongly nonexpansive mappings which also solves a finite system of equilibrium problems in a reflexive real Banach space.We further prove that our iterative sequence converges strongly to a common fixed point of an infinite family of left Bregman strongly nonexpansive mappings which is also a common solution to a finite system of equilibrium problems. Our result extends many recent and important results in the literature.
Bibliogr. 48 poz.
-  R. P. Agarwal, J. W. Chen, Y. J. Cho and Z. Wan, Stability analysis for parametric generalized vector quasivariational-like inequality problems, J. Inequal. Appl. 2012 (2012), Article No. 57.
-  Y. I. Alber, Generalized projection operators in Banach spaces: Properties and applications, Funct. Differ. Equ. 1 (1993), 1–21.
-  Y. I. Alber, Metric and generalized Projection operators in Banach spaces: properties and applications, in: Theory and Applications of Nonlinear Operators of Monotonic and Accretive Type, Dekker New York (1996), 15–50.
-  Y. I. Alber and D. Butnariu, Convergence of Bregman projection methods for solving consistent convex feasibility problems in reflexive Banach spaces, J. Optim. Theory Appl. 92 (1997), 33–61.
-  H. H. Bauschke and J. M. Borwein, Legendre functions and the method of random Bregman projections, J. Convex Anal. 4 (1997), 27–67.
-  H. H. Bauschke, J. M. Borwein and P. L. Combettes, Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces, Commun. Contemp. Math. 3 (2001), 615–647.
-  H. H. Bauschke, J. M. Borwein and P. L. Combettes, Bregman monotone optimization algorithms, SIAM J. Control Optim. 42 (2003), 596–636.
-  H. H. Bauschke and A. S. Lewis, Dykstra’s algorithm with Bregman projections: A convergence proof, Optimization 48 (2000), 409–427.
-  E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), 123–145.
-  J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 2000.
-  J. M. Borwein, S. Reich and S. Sabach, A characterization of Bregman firmly nonexpansive operators using a new monotonicity concept, J. Nonlinear Convex Anal. 12 (2011), 161–184.
-  L. M. Bregman, The relaxation method for finding common points of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math. Phys. 7 (1967), 200–217.
-  R. S. Burachik, Generalized proximal point methods for the variational inequality problem, Ph.D. thesis, Instituto de Mathematica Pura e Aplicada (IMPA), Rio de Janeiro, 1995.
-  R. S. Burachik and S. Scheimberg, A proximal point method for the variational inequality problem in Banach spaces, SIAM J. Control Optim. 39 (2000), 1633–1649.
-  D. Butnariu, Y. Censor and S. Reich, Iterative averaging of entropic projections for solving stochastic convex feasibility problems, Comput. Optim. Appl. 8 (1997), 21–39.
-  D. Butnariu and A. N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, Appl. Optim. 40, Kluwer Academic, Dordrecht, 2000.
-  D. Butnariu, A. N. Iusem and C. Zalinescu, On uniform convexity, total convexity and con-vergence of the proximal point and outer Bregman projection algorithms in Banach spaces, J.Convex Anal. 10 (2003), 35–61.
-  D. Butnariu and G. Kassay, A proximal-projection method for finding zeroes of set valued operators, SIAM J.Control Optim. 47 (2008), 2096–2136.
-  D. Butnariu and E. Resmerita, Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal. 2006 (2006), Article ID 84919.
-  L.-C. Ceng, Q. H. Ansari and J.-C. Yao, Viscosity approximation methods for generalized equilibrium problems and fixed point problems, J. Global Optim. 43 (2009), 487–502.
-  Y. Censor and A. Lent, An iterative row-action method for interval convex programming, J. Optim. Theory Appl. 34 (1981), 321–353.
-  J. W. Chen, Z. Wan and L. Yuan, Approximation of fixed points of weak Bregman relatively nonexpansive mappings in Banach spaces, Int. J. Math. Math. Sci. 2011 (2011), Article ID 420192.
-  P. Cholamjiak and S. Suantai, Convergence analysis for a system of equilibrium problems and a countable family of relatively quasi-nonexpansive mappings in Banach spaces, Abstr. Appl. Anal. 2010 (2010), Article ID 141376.
-  P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), 117–136.
-  I. Eckstein, Nonlinear proximal point algorithms using Bregman function, with applications to convex programming, Math. Oper. Res. 18 (1993), 202–226.
-  F. Giannessi, A. Maugeri and P. M. Pardalos, Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Nonconvex Optim. Appl. 58, Kluwer Academic Publishers, Dordrecht, 2002.
-  U. Kamraksa and R. Wangkeeree, Generalized equilibrium problems and fixed point problems for nonexpansive semigroups in Hilbert spaces, J. Global Optim. 51 (2011), 689–714.
-  A. Kangtunyakarn, A new iterative scheme for fixed point problems of infinite family of ki-pseudo contractive mappings, equilibrium problem, variational inequality problems, J. Global Optim. 56 (2013), 1543–1562.
-  K. Kassay, S. Reich and S. Sabach, Iterative methods for solving systems of variational inequalities in reflexive Banach spaces, SIAM J. Optim. 21 (2011), 1319–1344.
-  K. C. Kiwiel, Proximal minimization methods with generalized Bregman functions, SIAM J. Control Optim. 35 (1997), 1142–1168.
-  V. Martín-Márquez, S. Reich and S. Sabach, Right Bregman nonexpansive operators in Banach spaces, Nonlinear Anal. 75 (2012), 5448–5465.
-  V. Martín-Márquez, S. Reich and S. Sabach, Iterative methods for approximating fixed points of Bregman nonexpansive operators, Discrete Contin. Dyn. Syst. Ser. S 6 (2013), 1043–1063.
-  P. M. Pardalos, T. M. Rassias and A. A. Khan, Nonlinear Analysis and Variational Problems, Springer Optim. Appl. 35 Springer, Berlin, 2010.
-  R. P. Phelps, Convex Functions, Monotone Operators, and Differentiability, 2nd ed., Lecture Notes in Mathematics 1364, Springer, Berlin, 1993.
-  X. Qin, Y. J. Cho and S. M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math. 225 (2009), 20–30.
-  X. Qin, S. M. Kang and Y. J. Cho, Convergence theorems on generalized equilibrium problems and fixed point problems with applications, Proc. Estonian Acad. Sci. 58 (2009), 170–318.
-  S. Reich, A weak convergence theorem for the alternating method with Bregman distances, in: Theory and Applications of Nonlinear Operators of Acretive and Monotone Type, Lecture Notes in Pure Appl. Math. 178, Dekker, New York (1996), 313–318.
-  S. Reich and S. Sabach, A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces, J. Nonlinear Convex Anal. 10 (2009), 471–485.
-  S. Reich and S. Sabach, Two strong convergence theorems for a proximal method in reflexive Banach spaces, Numer. Funct. Anal. Optim. 31 (2010), 22–44.
-  S. Reich and S. Sabach, Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces, Nonlinear Anal. 73 (2010), 122–135.
-  S. Reich and S. Sabach, A projection method for solving nonlinear problems in reflexive Banach spaces, J. Fixed Point Theory Appl. 9 (2011), 101–116.
-  S. Reich and S. Sabach, Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces, in: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer, New York (2011), 299–314.
-  E. Resmerita, On total convexity, Bregman ections and stability in Banach spaces, J. Convex Anal. 11 (2004), 1–16.
-  Y. Shehu, Iterative method for fixed point problem, variational inequality and generalized mixed equilibrium problems with applications, J. Global Optim. 52 (2012), 57–77.
-  Y. Shehu, Strong convergence theorem for nonexpansive semigroups and systems of equilibrium problems, J. Global Optim. 56 (2013), 1675–1688.
-  M. V. Solodov and B. F. Svaiter, An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions, Math. Oper. Res. 25 (2000), 214–230.
-  S. Suantai, Y. J. Cho and P. Cholamjiak, Halpern’s iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces, Comput. Math. Appl.64 (2012), 489–499.
-  Y. Yao, Y.-C. Liou and S. M. Kang, Minimization of equilibrium problems, variational inequality problems and fixed point problems, J. Global Optim. 48 (2010), 643–656.