An Implicit Iteration Method for Variational Inequalities over the Set of Common Fixed Points for a Finite Family of Nonexpansive Mappings in Hilbert Spaces

Abstract

We introduce a new implicit iteration method for finding a solution for a variational inequality involving Lipschitz continuous and strongly monotone mapping over the set of common fixed points for a finite family of nonexpansive mappings on Hilbert spaces.

Systems of variational inequalities with hierarchical variational inequality constraints in Banach spaces

“…Finally, under very mild conditions, we prove the strong convergence of the proposed methods by using V-mappings instead of W-ones. Our results improve and extend the corresponding results announced by some others, e.g., Ceng et al [7] and Buong and Phuong [5].…”

“…In [5], motivated by methods (2.5) and (2.7), Buong and Phuong introduced a mapping V k , defined by…”

“…In [32], Zeng and Yao introduced an implicit method that converges weakly to a solution of a variational inequality. Ceng et al [9] extended the result from nonexpansive mappings to Lipschitz pseudocontractive mappings and strictly pseudocontractive mappings on H. Very recently, Buong and Phuong [5] introduced two new implicit iterative algorithms, which converge strongly in Banach spaces without weakly continuous duality mapping. These methods are two different combinations of the steepest-descent method with the V-mapping, a composition, and a convex combination.…”

Strong convergence of implicit and explicit iterations for a class of variational inequalities in Banach spaces

“…Recently, in order to obtain the strong convergence and decrease the strictness of the condition on λ k , Buong and Anh [4] proposed the following implicit iteration method:…”

“…Yamada's method is subsequently extended and modified to solve more complex problems, containing finite or infinite nonexpansive mappings (see, e.g., [3,6,40] and references therein). In [40], based on the Yamada result, Zeng and Yao introduced an implicit method that converges weakly to a solution of a variational inequality, involving a Lipschitz continuous and strongly monotone mapping in a Hilbert space H, where the feasible set is that of common fixed points of a finite family of nonexpansive mappings on H. In [7], Ceng et al extended this result from nonexpansive mappings to Lipschitz pseudocontractive mappings and strictly pseudocontractive mappings on H. Recently, in [4], Buong and Anh proposed a strongly convergent implicit method, which is a modification of Yamada's result. In the case where the feasible set is that of common fixed points of an infinite family of nonexpansive mappings on H, based on the W-mapping (see [29]) and Moudafi's viscosity approximation method (see [23]), in [16,17], Kikkawa and Takahashi studied an implicit iteration scheme that converges strongly to a solution of the stated problem.…”