A variable Krasnosel'skii–Mann algorithm and the multiple-set split feasibility problem

Abstract

A variable Krasnosel'skii–Mann algorithm generates a sequence {xn} via the formula xn+1 = (1 − αn)xn + αnTnxn, where {αn} is a sequence in [0, 1] and {Tn} is a sequence of nonexpansive mappings. We will show, in a fairly general Banach space, that the sequence {xn} generated converges weakly. This result is used to solve the split feasibility problem which is to find a point x with the property that x ∊ C and Ax ∊ Q, where C and Q are closed convex subsets of Hilbert spaces H1 and H2, respectively, and A is a bounded linear operator from H1 to H2. The multiple-set split feasibility problem recently introduced by Censor et al is stated as finding a point x ∊ ∩Ni=1Ci such that Ax ∊ ∩Mj=1Qj, where N and M are positive integers, {C1, …, CN} and {Q1, …, QM} are closed convex subsets of H1 and H2, respectively, and A is again a linear bounded operator from H1 to H2. One of the purposes of this paper is to introduce more iterative algorithms that solve this problem in the framework of infinite-dimensional Hilbert spaces.