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Zbl 1236.47066
Ceng, L.-C.; Ansari, Q.H.; Yao, J.-C.
Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 4, 2116-2125 (2012). ISSN 0362-546X

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 $\varGamma$ of SFP and the set $\text{Fix}(S)$ 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 $\text{Fix}(S)\cap\varGamma$. 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 $\text{Fix}(S)\cap\varGamma$ under mild assumptions.
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H09 Mappings defined by "shrinking" properties
65J20 Improperly posed problems (numerical methods in abstract spaces)
65J22 Inverse problems

Keywords: split feasibility problems; fixed point problems; relaxed extragradient methods; nonexpansive mappings; minimum-norm solutions; demiclosedness principle

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