Convergence of an iteration to common fixed points of two Bregman totally quasi-asymptotically nonexpansive mappings in reflexive Banach spaces
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Abstract
In this paper, we introduce a hybrid iteration method and prove the convergence of this iteration process to common fixed points of two Bregman totally quasi-asymptotically nonexpansive mappings in reflexive Banach spaces. From this result, we gain some convergence results by such iterations for Bregman quasi-asymptotically nonexpansive mappings, totally quasi- -asymptotically nonexpansive mappings and quasi- -asymptotically nonexpansive mappings. In addition, we provide an example to illustrate the convergence of the proposed iteration.
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Keywords
Bregman totally quasi-asymptotically nonexpansive mapping, Bregman distance, reflexive Banach space
References
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Chang, S. S., Wang, L., Wang, X. R., & Chan, C. K. (2014). Strong convergence theorems for Bregman totally quasi-asymptotically nonexpansive mappings in reflexive Banach spaces. Appl. Math. Comput., 228, 38-48.
Kumam, W., Witthayarat, U., Kumam, P., Suantai, S., & Wattanawitoon, K. (2016). Convergence theorem for equilibrium problem and Bregman strongly nonexpansive mappings in Banach spaces. Optimization, 65(2), 265-280.
Kohsaka, F., & Takahashi, W. (2005). Proximal point algorithms with Bregman functions in Banach spaces. J. Nonlinear Convex Anal., 6(3), 505-523.
Naraghirad, X., & Yao, J. C. (2013). Bregman weak relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 2013(141), 1-43.
Ni, R., & Wen, C. (2018). Hybrid projection methods for Bregman totally quasi-D asymptotically nonexpansive mappings. Bull. Malays. Math. Sci. Soc., 41, 807-836.
Reich, S., & Sabach, S. (2010). Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer. Funct. Anal. Optim., 31, 22-44.
Reich, S., & Sabach, S. (2009). A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces. J. Nonlinear Convex Anal.,10, 471-485.
Resmerita, E. (2004). On total convexity, Bregman projections and stability in Banach spaces, J. Nonlinear Convex Anal., 11, 1-16.
Sabach, S. (2011). Products of finitely many resolvents of maximal monotone mappings in reflexive Banach spaces. SIAM J. Optim., 21, 1289-1308.
Zalinescu, Z. (2002). Convex Analysis In General Vector Spaces. River Xdge: World Scientific.
Zhu, S., & Huang, J. H. (2016). Strong convergence theorems for equilibrium problem and Bregman totally quasi-asymptotically nonexpansivemapping in Banach spaces. Acta Math. Sci. Ser. B. 36B(5),1433-1444.
Butnariu, D., & Iusem, A. N. (2000). Totally Convex Functions For Fixed Points Computation And Infinite Dimensional Optimization,Applied Optimization, Vol. 40. 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., 2006, 1-39.
Censor, Y., & Lent, A. (1981). An iterative row-action method for interval convex programming. J. Optim. Theory Appl., 34, 321-353.
Chang, S. S., Wang, L., Wang, X. R., & Chan, C. K. (2014). Strong convergence theorems for Bregman totally quasi-asymptotically nonexpansive mappings in reflexive Banach spaces. Appl. Math. Comput., 228, 38-48.
Kumam, W., Witthayarat, U., Kumam, P., Suantai, S., & Wattanawitoon, K. (2016). Convergence theorem for equilibrium problem and Bregman strongly nonexpansive mappings in Banach spaces. Optimization, 65(2), 265-280.
Kohsaka, F., & Takahashi, W. (2005). Proximal point algorithms with Bregman functions in Banach spaces. J. Nonlinear Convex Anal., 6(3), 505-523.
Naraghirad, X., & Yao, J. C. (2013). Bregman weak relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 2013(141), 1-43.
Ni, R., & Wen, C. (2018). Hybrid projection methods for Bregman totally quasi-D asymptotically nonexpansive mappings. Bull. Malays. Math. Sci. Soc., 41, 807-836.
Reich, S., & Sabach, S. (2010). Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer. Funct. Anal. Optim., 31, 22-44.
Reich, S., & Sabach, S. (2009). A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces. J. Nonlinear Convex Anal.,10, 471-485.
Resmerita, E. (2004). On total convexity, Bregman projections and stability in Banach spaces, J. Nonlinear Convex Anal., 11, 1-16.
Sabach, S. (2011). Products of finitely many resolvents of maximal monotone mappings in reflexive Banach spaces. SIAM J. Optim., 21, 1289-1308.
Zalinescu, Z. (2002). Convex Analysis In General Vector Spaces. River Xdge: World Scientific.
Zhu, S., & Huang, J. H. (2016). Strong convergence theorems for equilibrium problem and Bregman totally quasi-asymptotically nonexpansivemapping in Banach spaces. Acta Math. Sci. Ser. B. 36B(5),1433-1444.
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