Convergence of a modified Halpern-type iteration algorithm for quasi-ϕ-nonexpansive mappings

Abstract

a b s t r a c tThe purpose of this work is to modify the Halpern-type iteration algorithm to have strong convergence under a limit condition only in the framework of Banach spaces. The results presented in this work improve on the corresponding ones announced by many others.

Strong convergence theorems of quasi-ϕ-asymptotically nonexpansive semi-groups in Banach spaces

“…Remark 3.3. Theorems 3.1 and 3.2 improve and extend the corresponding results of Suzuki [5], Xu [6], Chang et al Chang et al [7], Zhang [8], Chang et al [9], Cho et al [11], Thong [12], Buong [13], Mann [14], Halpern [15], Qin et al [16], Nakajo and Takahashi [19], Kang et al [23], Chang et al [24], and others.…”

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

“…In view of the mappings, and the framework of the spaces, we see that Theorem 3.8 can be viewed as a generalization of the corresponding results announced in Cho et al [27], Qin et al [28], and Qin and Su [29].…”

“…Due to the restriction of (C2), Algorithm (1.1) is widely believed to have slow convergence though the rate of convergence has not be determined. Thus to improve the rate of convergence of algorithm (1.1), one can not rely only on the process itself; instead, some additional step of iteration should be taken; see [27][28][29][30] and the references therein. One of the purposes of this article is to show algorithm (1.1) is strong convergence under (C1) only with the help of projections.…”

Fixed Point of a Countable Family of Uniformly Totally Quasi- <i>&Oslash</i> -Asymptotically Nonexpansive Multi-Valued Mappings in Reflexive Banach Spaces with Applications

“…is a real sequence in   0 1  and K P denotes the metric projection from a Hilbert space H onto a closed convex subset K of H. It should be noted here that the iteration above works only in Hilbert space setting. To extend this iteration to a Banach space, the concept of relatively nonexpansive mappings and quasi- -nonexpansive mappings are introduced by Aoyama et al [6], Chang et al [7,8], Chidume et al [9], Matsushita et al [10][11][12], Qin et al [13], Song et al [14], Wang et al [15] and others.…”