Abstract
Suppose that in a real Hilbert space H, the variational inequality problem with Lipschitzian and pseudomonotone mapping A and the common fixed-point problem of a finite family of nonexpansive mappings and a quasi-nonexpansive mapping with a demiclosedness property are represented by the notations VIP and CFPP, respectively. In this article, we suggest two Mann-type inertial subgradient extragradient iterations for finding a common solution of the VIP and CFPP. Our iterative schemes require only calculating one projection onto the feasible set for every iteration, and the strong convergence theorems are established without the assumption of sequentially weak continuity for A. Finally, in order to support the applicability and implementability of our algorithms, we make use of our main results to solve the VIP and CFPP in two illustrating examples.
Highlights
In a real Hilbert space (H, k · k), equipped with the inner product h·, ·i, we assume that C is a nonempty closed convex subset and PC is the metric projection of H onto C
We denote by VI(C, A) the solution set of the variational inequality problem (VIP)
Throughout this paper, we assume that C is a nonempty closed convex subset of a real Hilbert space H
Summary
In a real Hilbert space (H, k · k), equipped with the inner product h·, ·i, we assume that C is a nonempty closed convex subset and PC is the metric projection of H onto C. Thong and Hieu [22] designed two inertial subgradient extragradient algorithms with linesearch process for solving a VIP with monotone and Lipschitz continuous mapping A and a FPP of quasi-nonexpansive mapping T with a demiclosedness property in H. Under appropriate conditions, they established the weak convergence results for the suggested algorithms. Our algorithms require only computing one projection onto the feasible set C per iteration, and the strong convergence theorems are established without the assumption of sequentially weak continuity for A on C.
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