WEAK CONVERGENCE THEOREM BY A MODIFIED EXTRAGRADIENT METHOD FOR VARIATIONAL INCLUSIONS,VARIATIONAL INEQUALITIES AND FIXED POINT PROBLEMS

摘要

In this paper, we investigate the problem of finding common solutions of variational inclusions, variational inequalities and fixed point problems in real Hilbert spaces. Motivated by Nadezhkina and Takahashi's extragradient method [N. Nadezhkina, W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128 (2006) 191-201], we propose and analyze a modified extragradient algorithm for finding common solutions. It is proven that three sequences generated by this algorithm converge weakly to the same common solution under very mild conditions by virtue of the Opial condition of Hilbert spaces, the demiclosedness principle for nonexpansive mappings and the coincidence of solutions of variational inequalities with zeros of maximal monotone operators.