Abstract
In this paper, we propose several new iterative algorithms to solve the split feasibility problem in the Hilbert spaces. By virtue of new analytical techniques, we prove that the iterative sequence generated by these iterative procedures converges to the solution of the split feasibility problem which is the best close to a given point. In particular, the minimum-norm solution can be found via our iteration method.
Highlights
The split feasibility problem (SFP) was first introduced by Censor and Elfving [ ] in the finite-dimensional space, which could be formulated as follows: Finding x ∈ C, such that Ax ∈ Q, ( . )where C and Q are nonempty closed convex subset of Hilbert space H and H, respectively
The split feasibility problem ( . ) has received much attention because it can be used to model the problem in signal and image processing, and it is strongly related to some general problems, such as the convex feasibility problem [ ], the multiple-set split feasibility problem [ ], the split equality problem [ ], the split common fixed point problem [ ], etc
Yang [ ] proposed a relaxed CQ algorithm for solving the SFP ( . ) in which the orthogonal projections PC and PQ are replaced by PCn and PQn, respectively, that is, the orthogonal projections onto two half spaces Cn and Qn
Summary
The split feasibility problem (SFP) was first introduced by Censor and Elfving [ ] in the finite-dimensional space, which could be formulated as follows: Finding x ∈ C, such that Ax ∈ Q, ( . )where C and Q are nonempty closed convex subset of Hilbert space H and H , respectively. In order to obtain strong convergence, Xu [ ] proposed the following algorithm which was inspired by the Halpern iteration method. He proved that the sequence {xn} converges strongly to the projection of u onto the solution set of the SFP Based on the Tikhonov regularization method, Xu [ ] proved the following iterative sequence converges strongly to the minimum-norm solution of the SFP
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