Some modified extragradient methods for solving split feasibility and fixed point problems (Q1938353)

Summary: We consider and study the modified extragradient methods for finding a common element of the solution set \(\Gamma\) of a split feasibility problem (SFP) and the fixed point set \(\text{Fix}(S)\) of a strictly pseudocontractive mapping \(S\) in the setting of infinite-dimensional Hilbert spaces. We propose an extragradient algorithm for finding an element of \(\text{Fix}(S) \cap \Gamma\) where \(S\) is strictly pseudocontractive. It is proven that the sequences generated by the proposed algorithm converge weakly to an element of \(\text{Fix}(S) \cap \Gamma\). We also propose another extragradient-like algorithm for finding an element of \(\text{Fix}(S) \cap \Gamma\) where \(S : C \rightarrow C\) is nonexpansive. It is shown that the sequences generated by the proposed algorithm converge strongly to an element of \(\text{Fix}(S) \cap \Gamma\).