Some Krasnonsel'skiĭ-Mann Algorithms and the Multiple-Set Split Feasibility Problem
oleh: Noor MuhammadAslam, He Huimin, Liu Sanyang
| Format: | Article |
|---|---|
| Diterbitkan: | SpringerOpen 2010-01-01 |
Deskripsi
<p/> <p>Some variable Krasnonsel'skiĭ-Mann iteration algorithms generate some sequences <inline-formula> <graphic file="1687-1812-2010-513956-i1.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1812-2010-513956-i2.gif"/></inline-formula>, and<inline-formula> <graphic file="1687-1812-2010-513956-i3.gif"/></inline-formula>, respectively, via the formula <inline-formula> <graphic file="1687-1812-2010-513956-i4.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1812-2010-513956-i5.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1812-2010-513956-i6.gif"/></inline-formula>, where <inline-formula> <graphic file="1687-1812-2010-513956-i7.gif"/></inline-formula> and the mod function takes values in <inline-formula> <graphic file="1687-1812-2010-513956-i8.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1812-2010-513956-i9.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1812-2010-513956-i10.gif"/></inline-formula>, and<inline-formula> <graphic file="1687-1812-2010-513956-i11.gif"/></inline-formula> are sequences in <inline-formula> <graphic file="1687-1812-2010-513956-i12.gif"/></inline-formula> and <inline-formula> <graphic file="1687-1812-2010-513956-i13.gif"/></inline-formula> are sequences of nonexpansive mappings. We will show, in a fairly general Banach space, that the sequence <inline-formula> <graphic file="1687-1812-2010-513956-i14.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1812-2010-513956-i15.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1812-2010-513956-i16.gif"/></inline-formula> generated by the above formulas converge weakly to the common fixed point of <inline-formula> <graphic file="1687-1812-2010-513956-i17.gif"/></inline-formula>, respectively. These results are used to solve the multiple-set split feasibility problem recently introduced by Censor et al. (2005). The purpose of this paper is to introduce convergence theorems of some variable Krasnonsel'skiĭ-Mann iteration algorithms in Banach space and their applications which solve the multiple-set split feasibility problem.</p>