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Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem. (English) Zbl 1236.47066
Summary: 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 the 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.
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
65J20Improperly posed problems; regularization (numerical methods in abstract spaces)
65J22Inverse problems (numerical methods in abstract spaces)
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