An extragradient method for solving split feasibility and fixed point problems

The purpose of this paper is to introduce and analyze an extragradient method with regularization for finding a common element of the solution set ΓΓ of the split feasibility problem and the set Fix(S) of fixed points of a nonexpansive mapping SS in the setting of infinite-dimensional Hilbert spaces. Combining the regularization method and the extragradient method due to Nadezhkina and Takahashi [N. Nadezhkina, W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128 (2006) 191–201], we propose an iterative algorithm for finding an element of Fix(S)∩Γ. We prove that the sequences generated by the proposed algorithm converge weakly to an element of Fix(S)∩Γ under mild conditions.