Abstract
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.
Highlights
Throughout this paper, we denote the set of real numbers and the set of positive integers by R and N, respectively
We 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 is a technical method to establish a strong convergence theorem of Halpern type for a wide class of quasi-nonexpansive mappings
Summary
Throughout this paper, we denote the set of real numbers and the set of positive integers by R and N, respectively. The modulus δ of convexity of E is denoted by δ( ) = inf – x + y : x ≤ , y ≤ , x – y ≥. A Banach space E is said to be uniformly convex if δ( ) > for every >. The Banach space E is said to be strictly convex if x+y. It is well known that E is uniformly convex if and only if E∗ is. It is known that if E is reflexive, E is strictly convex if and only if E∗ is smooth; for more details, see [ ]. We define a mapping ρ : [ , ∞) → [ , ∞), the modulus of smoothness of E, as follows: ρ(t) = sup x+y + x–y.
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