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

Abstract

The multiple-set split feasibility problem(MSSFP) was introduced by Censor([9]) is stated as finding a point $x \in \cap_N^{i=1}$. such that $Ax \in \cap_M^{j=1}Q_j$. where N and M are positive integers, ${C_1,… ,C_N }$ and ${Q_1,… ,Q_M }$ 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).