Convergence of explicit iterative algorithms for solving a class of variational inequalities

摘要

Tian proposed a general iterative algorithm for finding a solution for variational inequalities over the set of fixed points of a nonexpansive mapping on Hilbert spaces and obtained the strong convergence theorem [M. Tian, Nonlinear Analysis, 73 (2010) 689-694]. Zhou et al. proposed a simpler explicit iterative algorithm for finding a solution for variational inequalities over the set of common fixed points of a finite family of nonexpansive mappings on Hilbert spaces and proved the strong convergence [H. Y. Zhou, P. Y. Wang, J. Optim. Theory Appl.09 November 2013]. In this paper, we firstly give a new proof of Tian's convergence theorem, which is much more simpler than Tian's original proof. Then we improve the main convergence result of Zhou et al., more precisely, using a recent new lemma, we prove the strong convergence of this algorithm under more weaker conditions (indeed, one of the original conditions is removed). Based on the two results, a more general algorithm is then proposed for solving a more general class of variational inequalities over the set of common fixed points of a finite family of nonexpansive mappings on Hilbert spaces and its strong convergence is proved. Finally, some extensions to our main results have been obtained. Our results in this paper extend and improve ones of Tian and Zhou et al.