Relaxed Implicit Extragradient-like 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 implicit extragradient-like methods for finding a common element of the solution set Γ of the SFP and the set Fix(S) of fixed points of a nonexpansive mapping S. Combining Mann’s implicit iterative method and Korpelevich’s extragradient method, we propose two implicit iterative algorithms for finding an element of Fix(S) ∩ Γ . On one hand, for S = I, the identity mapping, we derive the strong convergence of one implicit 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 implicit iterative algorithm to an element of Fix(S)∩Γ under mild assumptions.