Iterative methods for the split common fixed point problem in Hilbert spaces

AbstractThe split common fixed point problem is an inverse problem that consists in finding an element in a fixed point set such that its image under a bounded linear operator belongs to another fixed point set. Recently Censor and Segal proposed an efficient algorithm for solving such a problem. However, to employ their algorithm, one needs to know prior information on the norm of the bounded linear operator. In this paper we propose a new algorithm that does not need any prior information of the operator norm, and we establish the weak convergence of the proposed algorithm under some mild assumptions. MSC:47J25, 47J20, 49N45, 65J15.

[1]  Abdellatif Moudafi,et al.  A note on the split common fixed-point problem for quasi-nonexpansive operators , 2011 .

[2]  Y. Censor,et al.  A unified approach for inversion problems in intensity-modulated radiation therapy , 2006, Physics in medicine and biology.

[3]  Yair Censor,et al.  The Split Common Null Point Problem , 2012 .

[4]  Hong-Kun Xu,et al.  Cyclic algorithms for split feasibility problems in Hilbert spaces , 2011 .

[5]  Yair Censor,et al.  The Split Common Fixed Point Problem for Directed Operators. , 2010, Journal of convex analysis.

[6]  C. Byrne,et al.  A unified treatment of some iterative algorithms in signal processing and image reconstruction , 2003 .

[7]  Hong-Kun Xu,et al.  Solving the split feasibility problem without prior knowledge of matrix norms , 2012 .

[8]  Heinz H. Bauschke,et al.  On Projection Algorithms for Solving Convex Feasibility Problems , 1996, SIAM Rev..

[9]  Yair Censor,et al.  A multiprojection algorithm using Bregman projections in a product space , 1994, Numerical Algorithms.

[10]  Abdellatif Moudafi,et al.  Split Monotone Variational Inclusions , 2011, J. Optim. Theory Appl..

[11]  Abdellatif Moudafi,et al.  The split common fixed-point problem for demicontractive mappings , 2010 .

[12]  Yair Censor,et al.  Algorithms for the Split Variational Inequality Problem , 2010, Numerical Algorithms.

[13]  C. Byrne,et al.  Iterative oblique projection onto convex sets and the split feasibility problem , 2002 .

[14]  W. A. Kirk,et al.  Topics in Metric Fixed Point Theory , 1990 .