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

Abstract

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 Γ 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 ) ∩ Γ . 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 ) ∩ Γ under mild assumptions.