Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem

In this paper, we consider the split feasibility problem (SFP) in infinite-dimensional Hilbert spaces, and study the relaxed extragradient methods for finding a common element of the solution set Gamma of SFP and the set Fix(S) of fixed points of a nonexpansive mapping S. Combining Mann's iterative method and Korpelevich's extragradient method, we propose two iterative algorithms for finding an element of Fix(S) boolean AND Gamma. On one hand, for S = I, the identity mapping, we derive the strong convergence of one iterative algorithm to the minimum-norm solution of the SFP under appropriate conditions. On the other hand, we also derive the weak convergence of another iterative algorithm to an element of Fix(S) boolean AND Gamma under mild assumptions. (C) 2011 Elsevier Ltd. All rights reserved.