Some Krasnonsel’skiı̆-Mann Algorithms and the Multiple-Set Split Feasibility Problem

Some variable Krasnonsel’skiı̆-Mann iteration algorithms generate some sequences {xn}, {yn}, and {zn}, respectively, via the formula xn 1 1 − αn xn αnTN · · · T2T1xn, yn 1 1 − βn yn βn ∑N i 1 λiTiyn, zn 1 1 − γn 1 zn γn 1T n 1 zn, where T n TnmodN and the mod function takes values in {1, 2, . . . ,N}, {αn}, {βn}, and {γn} are sequences in 0, 1 , and {T1, T2, . . . , TN} are sequences of nonexpansive mappings. We will show, in a fairly general Banach space, that the sequence {xn}, {yn}, {zn} generated by the above formulas converge weakly to the common fixed point of {T1, T2, . . . , TN}, 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’skiı̆-Mann iteration algorithms in Banach space and their applications which solve the multiple-set split feasibility problem.