Convergence of algorithms for fixed points of generalized asymptotically quasi-ϕ-nonexpansive mappings with applications

Abstract

In this article, strong convergence of Krasnoselski-Mann iterative sequences and Halpern iterative sequences are investigated based on hybrid projection methods. Strong convergence theorems for common fixed points of a family of generalized asymptotically quasi-j-nonexpansive mappings are established in the framework of Banach spaces. Mathematics Subject Classification 2000: 47H09; 47J05; 47J25 Keywords: asymptotically quasi-j-nonexpansive mapping, asymptotically nonexpansive mapping, fixed point, generalized asymptotically quasi-j-nonexpansive mapping, generalized asymptotically quasi-nonexpansive mapping 1.IntroductionFixed point theory as an important branch of nonlinear analysis theory has been applied in the study of nonlinear phenomena. During the four decades, many famous existence theorems of fixed points were established; see, for example, [1][2][3][4][5]. However, from the standpoint of real world applications it is not only to know the existence of fixed points of nonlinear mappings, but also to be able to construct an iterative process to approximate their fixed points. The computation of fixed points is important in the study of many real world problems, including inverse problems; for instance, it is not hard to show that the split feasibility problem and the convex feasibility problem in signal processing and image reconstruction can both be formulated as a problem of finding fixed points of certain operators, respectively (see [6,7] for more details and the references therein).Recently, the study of the convergence of various iterative processes for solving various nonlinear mathematical models forms the major part of numerical mathematics. Among these iterative processes, Krasnoselski-Mann iterative process and Halpern iterative process are popular and hot. Let C be a nonempty, closed, and convex subset of a underlying space X, and T : C C a mapping. Halpern iterative process generates a sequence {x n } in the following manner:x 0 ∈ C, x n+1 = α n u + (1 − α n )Tx n , ∀n ≥ 0,(1:1)

Hybrid projection algorithms for total asymptotically strict quasi-ϕ -pseudo-contractions

“…The class of generalized asymptotically quasi-φ-nonexpansive mappings was first considered in Qin, Agarwal, Cho and Kang [19].…”

“…In 2011, Qin, Wang, and Cho [22] introduced a new nonlinear mapping, which was called asymptotically strict quasi-φ-pseudocontraction, and proved a strong convergence theorem for fixed points of an asymptotically strict quasi-φ-pseudocontraction in some Banach space. In 2012, Qin, Agarwal, Cho, and Kang [19] established strong convergence theorems for common fixed points of a family of generalized asymptotically quasi-φ-nonexpansive mappings in the framework of Banach spaces. In the same year, Qin, Wang, Kang [23] proved strong convergence theorems for fixed points of asymptotically strict quasi-φ-pseudo-contractions, in the intermediate, sense in a real Banach space.…”

Block methods for a convex feasibility problem in a Banach space

“…The class of generalized asymptotically quasi-φ-nonexpansive mappings, which was introduced in [21] is a generalization of the class of generalized asymptotically quasi-nonexpansive mappings [2] in the framework of Banach spaces. Common fixed points of generalized asymptotically quasi-nonexpansive mappings were investigated via the implicit iterations in [2].…”

Strong convergence analysis of a monotone projection algorithm in a Banach space

“…In these problems, strong convergence is often much more desirable than the weak convergence [18]. To obtain the strong convergence of mean-valued algorithms, different regularization techniques have been considered; see [4,5,8,9,10,11,15,21,22,23,24,25,26,31,32,33] and the references therein. The projection method which was first introduced by Haugazeau [19] has been investigated for the approximation of fixed points of nonlinear operators.…”