A variable Krasnosel'skii–Mann algorithm and the multiple-set split feasibility problem
Hong‐Kun Xu · Inverse Problems · 2006
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.