Strong Convergence Theorem of the CQ Algorithm for the Multiple-Set Split Feasibility Problem

The multiple-set split feasibility problem(MSSFP) was introduced by Censor([9]) is stated as finding a point x ∈ ∩_i=1^NC_i. such that Ax ∈ ∩_j=1^MQ_j. where N and M are positive integers, {C₁,⋯,C_N} and {Q₁,⋯,Q_M} are closed convex subset of Hilbert H₁ and H₂, respectively, and A is a linear bounded operator from H₁ to H₂. MSSFP can be applied to the problem of intensity-modulated radiation therapy in medical care. In this paper, we discuss iterative methods for solving MSSFP in Hilbert spaces. We present modifications of the CQ algorithm in such a way that strong convergence is guaranteed and the limit is a minimum norm solution of MSSFP. Our iterative methods modifies and improves some methods in literature HK Xu(Inverse Problems, vol.20, no.1, pp. 103-120,2004) and FH Wang, HK Xu(Journal of Inequalities and Applications, 2010).