**Volume 11, Issue 1, pp 108--130**

**Publication Date**: 2017-12-31

http://dx.doi.org/10.22436/jnsa.011.01.09

Jingling Zhang - Department of Mathematics, Tianjin University, Tianjin 300350, P. R. China

Ravi P. Agarwal - Department of Mathematics, Texas A \(\&\) M University-Kingsville, Texas 78363, USA.

Nan Jiang - Department of Mathematics, Tianjin University, Tianjin 300350, P. R. China

The purpose of this paper is to introduce and consider a new accelerated hybrid shrinking projection method for finding a common element of the set \(EP \cap F\) in reflexive Banach spaces, where \(EP\) is the set of all solutions of a generalized equilibrium problem, and \(F\) is the common fixed point set of finite uniformly closed families of countable Bregman quasi-Lipschitz mappings. It is proved that the sequence generated by the accelerated hybrid shrinking projection iteration, converges strongly to the point in \(EP \cap F,\) under some conditions. This result is also applied to find the fixed point of Bregman asymptotically quasi-nonexpansive mappings. It is worth mentioning that, there are multiple projection points from the multiple points in the projection algorithm. Therefore the new projection method in this paper can accelerate the convergence speed of iterative sequence. The new results improve and extend the previously known ones in the literature.

Bregman distance, Bregman quasi-Lipschitz mapping, accelerated hybrid algorithm, Bregman asymptotically quasi-nonexpansive mappings, equilibrium problem

[1] Y. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, AG (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes Pure Appl. Math., 178 (1996), 15–50.

[2] Y. Alber, 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.

[3] V. Barbu, T. Precupanu, Convexity and Optimization in Banach Spaces, Springer, Dordrecht, (2012).

[4] H. H. Bauschke, J. M. Borwein, P. L. Combettes, Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces, Commun. Contemp. Math., 3 (2001), 615–647.

[5] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123–145.

[6] L. M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math. Phys., 7 (1967), 200–217.

[7] R. Bruck, T. Kuczumow, S. Reich, Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property, Colloq. Math., 65 (1993), 169–179.

[8] D. Butnariu, A. N. Iusem, C. Zălinescu, On uniform convexity, total convexity and convergence of the proximal points and outer Bregman projection algorithms in Banach spaces, J. Convex Anal., 10 (2003), 35–61.

[9] D. Butnariu, E. Resmerita, Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal., 2006 (2006), 39 pages.

[10] Y. Censor, A. Lent, An iterative row-action method for interval convex programming, J. Optim. Theory Appl., 34 (1981), 321–353.

[11] M. Chen, J. Bi, Y. Su, Hybrid iterative algorithm for finite families of countable Bregman quasi-Lipschitz mappings with applications in Banach spaces, J. Inequal. Appl., 2015 (2015), 19 pages.

[12] G. Chen, M. Teboulle, Convergence analysis of a proximal-like minimization algorithm using Bregman functions, SIAM J. Optim., 3 (1993), 538–543.

[13] K. Goebel, W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171–174.

[14] Y. Hao, Some results on a modified Mann iterative scheme in a reflexive Banach space, Fixed Point Theory Appl., 2013 (2013), 14 pages.

[15] Y. Hecai, L. Aichao, Projection algorithms for treating asymptotically quasi-\(\phi\)-nonexpansive mappings in the intermediate sense, J. Inequal. Appl., 2013 (2013), 15 pages.

[16] I. Inchan, Strong convergence theorems of modified Mann iteration methods for asymptotically nonexpansive mappings in Hilbert spaces, Int. J. Math. Anal., 2 (2008), 1135–1145.

[17] F. Kohsaka, W. Takahashi, Strong convergence of an iterative sequence for maximal monotone operators in a Banach space, Abstr. Appl. Anal., 3 (2004), 239–249.

[18] V. Martín-Márquez, S. Reich, S. Sabach, Iterative methods for approximating fixed points of Bregman nonexpansive operators, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1043–1063.

[19] B. Martinet, Regularisation d’inequations variationnelles par approximations successives, Rev. Franaise Informat. Recherche Oprationnelle, 4 (1970), 154–158.

[20] E. Naraghirad, J.-C. Yao, Bregman weak relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 2013 (2013), 43 pages.

[21] X. Qin, S. Huang, T. Wang, On the convergence of hybrid projection algorithms for asymptotically quasi-\(\phi\)-nonexpansive mappings, Comput. Math. Appl., 61 (2011), 851–859.

[22] X. Qin, L. Wang, On asymptotically quasi-\(\phi\)-nonexpansive mappings in the intermediate sense, Abstr. Appl. Anal., 2012 (2012), 13 pages.

[23] S. Reich, S. Sabach, A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces, J. Nonlinear Convex Anal., 10 (2009), 471–485.

[24] S. Reich, S. Sabach, Two strong convergence theorems for a proximal method in reflexive Banach spaces, Numer. Funct. Anal. Optim., 31 (2010), 22–44.

[25] S. Reich, S. Sabach, Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces, Nonlinear Anal., 73 (2010), 122–135.

[26] S. Reich, 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 Optim. Appl., 49 (2011), 301–316.

[27] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43 (1991), 153–159.

[28] S. Suantai, Y. J. Cho, P. Cholamjiak, Halpern’s iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces, Comput. Math. Appl., 64 (2012), 489–499.

[29] S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal., 69 (2008), 1025–1033.

[30] W. Takahashi, Y. Takeuchi, R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 341 (2008), 276–286.

[31] Y. Tomizawa, A strong convergence theorem for Bregman asymptotically quasi-nonexpansive mappings in the intermediate sense, Fixed Point Theory Appl., 2014 (2014), 14 pages.

[32] Y.-H. Yao, M. Postolache, S. M. Kang, Strong convergence of approximated iterations for asymptotically pseudocontractive mappings, Fixed Point Theory Appl., 2014 (2014), 13 pages.

[33] Y.-H. Yao, N. Shahzad, Y.-C. Liou, Modified semi- implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 2015 (2015), 15 pages.

[34] Q. Yuan, Some results on asymptotically quasi-\(\phi\)-nonexpansive mappings in the intermediate sense, J. Fixed Point Theory, 2012 (2012), Article ID 1.

[35] C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific, River Edge, (2002).