Strong convergence of hybrid Halpern iteration for Bregman totally quasi-asymptotically nonexpansive multi-valued mappings in reflexive Banach spaces with application

In this paper, Bregman totally quasi-asymptotically nonexpansive multi-valued mappings in the framework of reflexive Banach spaces are established. Under suitable limit conditions, by using the shrinking projection method introduced by Takahashi, Kubota and Takeuchi, some strong convergence theorems for hybrid Halpern's iteration for a countable family of Bregman totally quasi-asymptotically nonexpansive multi-valued mappings are proved. We apply our main results to solve classical equilibrium problems in the framework of reflexive Banach spaces. The main result presented in the paper improves and extends the corresponding result in the work by Chang (Appl. Math. Comput. 2013, doi: 10.1016/j.amc.2013.11.074; Appl. Math. Comput. 228:38-48, 2014), Suthep (Comput. Math. Appl., 64:489-499, 2012), Yi Li (Fixed Point Theory Appl. 2013: 197, 2013), Reich and Sabach (Nonlinear Anal. 73:122-135, 2010), Nilsrakoo and Saejung (Appl. Math. Comput. 217(14):6577-6586, 2011), Qin et al. (Appl. Math. Lett. 22:1051-1055, 2009), Wang et al. (J. Comput. Appl. Math. 235:2364-2371, 2011), Su et al. (Nonlinear Anal. 73:3890-3906, 2010) and others.