Strong convergence of Halpern iterations for quasi-nonexpansive mappings and accretive operators in Banach spaces
Chin-Tzong Pang, Eskandar Naraghirad · Fixed Point Theory and Applications · 2013
In this paper, we first introduce a new Halpern-type iterative scheme to approximate common fixed points of an infinite family of quasi-nonexpansive mappings and obtain a strongly convergent iterative sequence to the common fixed points of these mappings in a uniformly convex Banach space. We then apply our method to approximate zeros of an infinite family of accretive operators and derive a strong convergence theorem for these operators. It is important to state clearly that the contribution of this paper in relation with the previous works (see, for example, Yao et al. (Nonlinear Anal. 70:2332-2336, 2009)) is a technical method to establish a strong convergence theorem of Halpern type for a wide class of quasi-nonexpansive mappings. The method provides a positive answer to an old problem in fixed point theory and applications. Our results improve and generalize many known results in the current literature. MSC: 47H10, 37C25.