Abstract
The purpose of this article is to discuss a modified Halpern-type iteration algorithm for a countable family of uniformly totally quasi- ? -asymptotically nonexpansive multi-valued mappings and establish some strong convergence theorems under certain conditions. We utilize the theorems to study a modified Halpern-type iterative algorithm for a system of equilibrium problems. The results improve and extend the corresponding results of Chang et al. (Applied Mathematics and Computation, 218, 6489-6497).
Highlights
Let X be a real Banach space with dual X
Inspired by the work of Matsushita and Takahashi, in this paper, we introduce modifying Halpern-Mann iterations sequence for finding a fixed point of a countable family of uniformly totally quasi- -asymptotically nonexpansive multi-valued mappings in reflexive Banach spaces Ti D D i 1 2 3, and some strong convergence theorems are proved
We present an example of relatively nonexpansive multi-valued mapping
Summary
Let X be a real Banach space with dual X. (7) Each uniformly convex Banach space X has the. Where n is a real sequence in 0 1 and PK 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
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