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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 $\text{Fix}\left(S\right)$ 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 $\text{Fix}\left(S\right)\cap 𝛤$. 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 $\text{Fix}\left(S\right)\cap 𝛤$ under mild assumptions.
##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 65J20 Improperly posed problems; regularization (numerical methods in abstract spaces) 65J22 Inverse problems (numerical methods in abstract spaces)
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