Strong Convergence Theorems for Mixed Equilibrium Problems and Uniformly Bregman Totally Quasi-Asymptotically Nonexpansive Multi-Valued Mappings in Reflexive Banach Spaces
Abstract
In this paper, we propose a new iterative method for finding common solutions of mixed equilibrium problems and common fixed points of uniformly Bregman totally quasi-asymptotically nonexpansive multi-valued mappings in reflexive Banach spaces and prove the strong convergence theorems under some suitable control conditions. Keywords : mixed equilibrium problems, Bregman totally quasi-asymptotically nonexpansive multi-valued mappings, reflexive Banach spaces.References
Blum, E., & Oettli, W. (1994). From optimization and variational inequalities to equilibrium problems. Math Stud, 63, 123-145.
Bregman, L.M. (1967). 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(3), 200-217.
Butnariu, D., & Iusem, A.N. (2000). Totally convex functions for fixed points computation and infinite dimensional optimization. Dordrecht: Kluwer Academic.
Butnariu, D., & Resmerita, E. (2006). Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces. Abstr Appl Anal, Art ID: 84919.
Ceng, L.C., & Yao, J.C. (2008). A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J Comp Appl Math, 214(1), 186-201.
Censor, Y., & Lent, A. (1981). An iterative row-action method for interval convex programing. J Optim Theory Appl, 34(3), 321-353.
Chang, S.S., Wang, L., Wang, X.R., & Chan, C.K. (2013). Strong convergence theorems for Bregman totally quasi-asymptotically nonexpansive mappings in reflexive Banach spaces. Appl Math Comput, doi:10.1016/j.amc.2013.11.074.
Darvish, V. (2015). A new algorithm for mixed equilibrium problem and Bregman strongly nonexpansive mapping in Banach spaces. Math FA, 2015.
Kassay, G., Reich, S., & Sabach, S. (2011). Iterative methods for solving systems of variatio nal inequalities in reflexive Banach spaces. SIAM J Optim, 21(4), 1319-1344.
Kirk, W.A, & Massa, S. (1990). Remarks on asymptotic and Chebyshev centers. Houston J. Math, 16, 357–364.
Li, Y., & Liu, H. (2014). Strong convergence of hybrid Halpern iteration for Bregman totally quasi-asymptotically nonexpansive multi-valued mappings in reflexive Banach spaces with application. Fixed Point Theory Appl, 186.
Li, Y., Liu, H., & Zheng, K. (2013). Halpern's iteration for Bregman strongly nonexpansive multi-valued mappings in reflexive Banach spaces with application. Fixed Point Theory Appl, 197.
Nilsrakoo, W., & Saejung, S. (2011). Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spaces. Appl Math Comput, 217(14), 6577-6586.
Reich, S., & Sabach, S. (2010). Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer Funct Anal Optim, 31(1), 22-44.
Reich, S., & Sabach, S. (2010). Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces. Nonlinear Anal, 73(1), 122-135.
Zhu, S., & Huang, J. (2016). Strong convergence theorems for equilibrium problem and Bregman totally quasi-asymptotically nonexpansive mapping in Banach spaces. Acta Mathematica Scientia, 36B(5),
1433-1444.
Bregman, L.M. (1967). 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(3), 200-217.
Butnariu, D., & Iusem, A.N. (2000). Totally convex functions for fixed points computation and infinite dimensional optimization. Dordrecht: Kluwer Academic.
Butnariu, D., & Resmerita, E. (2006). Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces. Abstr Appl Anal, Art ID: 84919.
Ceng, L.C., & Yao, J.C. (2008). A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J Comp Appl Math, 214(1), 186-201.
Censor, Y., & Lent, A. (1981). An iterative row-action method for interval convex programing. J Optim Theory Appl, 34(3), 321-353.
Chang, S.S., Wang, L., Wang, X.R., & Chan, C.K. (2013). Strong convergence theorems for Bregman totally quasi-asymptotically nonexpansive mappings in reflexive Banach spaces. Appl Math Comput, doi:10.1016/j.amc.2013.11.074.
Darvish, V. (2015). A new algorithm for mixed equilibrium problem and Bregman strongly nonexpansive mapping in Banach spaces. Math FA, 2015.
Kassay, G., Reich, S., & Sabach, S. (2011). Iterative methods for solving systems of variatio nal inequalities in reflexive Banach spaces. SIAM J Optim, 21(4), 1319-1344.
Kirk, W.A, & Massa, S. (1990). Remarks on asymptotic and Chebyshev centers. Houston J. Math, 16, 357–364.
Li, Y., & Liu, H. (2014). Strong convergence of hybrid Halpern iteration for Bregman totally quasi-asymptotically nonexpansive multi-valued mappings in reflexive Banach spaces with application. Fixed Point Theory Appl, 186.
Li, Y., Liu, H., & Zheng, K. (2013). Halpern's iteration for Bregman strongly nonexpansive multi-valued mappings in reflexive Banach spaces with application. Fixed Point Theory Appl, 197.
Nilsrakoo, W., & Saejung, S. (2011). Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spaces. Appl Math Comput, 217(14), 6577-6586.
Reich, S., & Sabach, S. (2010). Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer Funct Anal Optim, 31(1), 22-44.
Reich, S., & Sabach, S. (2010). Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces. Nonlinear Anal, 73(1), 122-135.
Zhu, S., & Huang, J. (2016). Strong convergence theorems for equilibrium problem and Bregman totally quasi-asymptotically nonexpansive mapping in Banach spaces. Acta Mathematica Scientia, 36B(5),
1433-1444.
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2018-02-09
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