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Abstract: The purpose of this article is by using the shrinking projection method introduced by Takahashi, Kubota and Takeuchi to propose an iteration algorithm for Bregman total quasi- -asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of reflexive Banach spaces. As applications, we apply our results to a system of equilibrium problems and zero point problem of maximal monotone mappings in reflexive Banach spaces. The results presented in the paper improve and extend the corresponding results of Reich and Sabach (2010) [12], Suantai et al. (2012) [13], Nilsrakoo and Saejung (2011) [11], Qin et al. (2009) [5], Wang et al. (2011) [6], Su et al. (2010) [7], Martinez-Yanes and Xu (2006) [3] and others. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to introduce the concept of total asymptotically nonexpansive mappings and to prove the demiclosed principle for this kind of mappings in CAT(0) spaces. As a consequence, we obtain a -convergence theorem of the Krasnoselskii–Mann type iteration for total asymptotically nonexpansive mappings in this setting. Our results extend and improve the corresponding recent results announced by many authors. [Copyright &y& Elsevier]

Abstract: The purpose of this article is to modify the Halpern-type iteration algorithm for total quasi-ϕ-asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding results of [X.L. Qin, Y.J. Cho, S.M. Kang, H. Y. Zhou, Convergence of a modified Halpern-type iterative algorithm for quasi-ϕ-nonexpansive mappings, Appl. Math. Lett. 22 (2009) 1051–1055], [Z.M. Wang, Y.F. Su, D.X. Wang, Y.C. Dong, A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces, J. Comput. Appl. Math. 235 (2011) 2364–2371], [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890–3906], [C. Martinez-Yanes, H.K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006) 2400–2411] and others. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study Reich’s strongly convergence theorems for asymptotically nonexpansive mappings in Banach spaces. Under some general conditions an affirmative partial answer to Reich’s open question is given and some recent results are improved and generalized. [Copyright &y& Elsevier]

Some equivalence conditions between the convergence of modified Picard, modified Mann, and modified Ishikawa iterations for some kinds of nonlinear mappings in Banach spaces are obtained. [Copyright &y& Elsevier]