Some Modified Extragradient Methods for Solving Split Feasibility and Fixed Point Problems

Abstract

We consider and study the modified extragradient methods for finding a common element of the solution set $\mathrm{\Gamma }$ 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 $\text{Fix}(S)\cap \mathrm{\Gamma }$ where $S$ is strictly pseudocontractive. It is proven that the sequences generated by the proposed algorithm converge weakly to an element of $\text{Fix}(S)\cap \mathrm{\Gamma }$. We also propose another extragradient-like algorithm for finding an element of $\text{Fix}(S)\cap \mathrm{\Gamma }$ where $S:C\to C$ is nonexpansive. It is shown that the sequences generated by the proposed algorithm converge strongly to an element of $\text{Fix}(S)\cap \mathrm{\Gamma }$.