A variable Krasnosel'skii–Mann algorithm generates a sequence {x n} via the formula x n+ 1=(1− α n) x n+ α n T n x 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 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∊∩ N i= 1 C i such that Ax∊∩ 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.