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

A variable Krasnosel'skii-Mann algorithm generates a sequence {x(n)} via the formula x(n+1) = (1 - alpha(n))x(n) + alpha(n)T(n)x(n), where {alpha(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 is an element of C and Ax is an element of Q, where C and Q are closed convex subsets of Hilbert spaces H-1 and H-2, respectively, and A is a bounded linear operator from H-1 to H-2. The multiple-set split feasibility problem recently introduced by Censor et al is stated as finding a point x is an element of boolean AND C-N(i=1)i such that Ax is an element of boolean AND(M)(j=1) Q(j), where N and M are positive integers, {C-1,..., C-N} and {Q(1),..., Q(M)} are closed convex subsets of H-1 and H-2, respectively, and A is again a linear bounded operator from H-1 to H-2. 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.