Abstract. A variable Krasnosel'skii–Mann algorithm generates a sequence {x_n} via the formula x_n+1 = (1 - α_n)x_n + α_nT_nx_n, where {α_n} is a sequence in [0, 1] and {T_n} is a sequence of nonexpansive mappings. We will show, in a fairly general Banach space, that the sequence {x_n} 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 ∈ ∩^N_i=1C_i such that Ax ∈ ∩^M_j=1Q_j, where N and M are positive integers, {C₁, ..., C_N} and {Q₁, ..., Q_M} 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.

Print publication: Issue 6 (December 2006)
Received 25 December 2005, in final form 19 September 2006
Published 13 October 2006