Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications

Abstract

In this paper, we consider a generalized iterative process with errors to approximate the common fixed points of two asymptotically quasi-nonexpansive mappings. A convergence theorem has been obtained which generalizes a known result. This theorem has then been used to prove another convergence theorem which, in turn, generalizes a number of results.

New iterative schemes for a finite family of nonself uniformly quasi-Lipschitzian mappings in Banach spaces

“…As one will see, our results extend and generalize the corresponding results in [2][3][4][5][6][7][8][9][10] as follows: (i) the condition…”

“…Fixed points iterative technique for (self or nonself) asymptotically nonexpansive mappings in Banach spaces, including Mann type iteration, Ishikawa type iteration, and three-step type iteration, have been studied by many authors (see, e.g., [2][3][4][5][6][7][8]). Recently, Khan et al [9] introduced an iterative scheme (which generalizes Mann iteration, Ishikawa iteration, and three-step iteration) for a finite family of asymptotically quasi-nonexpansive self-mappings {T i : i I}: C C, where I = {1, 2, .…”

“…We notice that (ii) In Corollary 2.2, we remove the condition: " ∞ n=1 (k in − 1) < ∞", which is required in many other article (see, e.g., [2,[4][5][6][7][8][9]). (iii) When considering iterative schemes with errors, many authors need the conditions: " ∞ n=1 γ in < ∞, i ∈ I ", see for example [4][5][6]. But in Corollary 2.2, we only need the condition: " ∞ n=1 θ n < ∞ ", where θ n = b kn + γ kn .…”

Convergence of modified S-iteration process for two asymptotically quasi-nonexpansive type mappings in CAT(0) spaces

“…The iterative approximation problems for asymptotically quasi-nonexpansive mappings was studied by many authors in a Banach space and a CAT(0) space (see, e.g. [6,11,16,17,21,23]). …”

Weak and strong convergence theorems of implicit iteration process on Banach spaces

“…In fact, for the above implicit iterative processes, most researchers always considered weak convergence theorems, and few researchers considered strong convergence theorem under suitable conditions. For example, one can see [5][6][7]. However, some conditions are not natural.…”