Iterative algorithms approach to variational inequalities and fixed point problems (Q410249)

Summary: We consider a general variational inequality and fixed point problem, which is to find a point \(x^\ast\) with the property that (GVF): \(x^\ast \in \text{GVI} (C, A)\) and \(g(x^\ast) \in \text{Fix}(S)\) where \(\text{GVI}(C, A)\) is the solution set of some variational inequality \(\text{Fix}(S)\) is the fixed points set of nonexpansive mapping \(S\), and \(g\) is a nonlinear operator. Assume the solution set \(\Omega\) of (GVF) is nonempty. For solving (GVF), we suggest the following method \(g(x_{n+1}) = \beta g(x_n) + (1 - \beta) SP_C[\alpha_n F(x_n) + (1 - \alpha_n)(g(x_n) - \lambda Ax_n)]\), \(n \geq 0\). It is shown that the sequence \(\{x_n\}\) converges strongly to \(x^\ast \in \Omega\) which is the unique solution of the variational inequality \(\langle F(x^\ast) - g(x^\ast), g(x) - g(x^\ast)\rangle \leq 0\), for all \(x \in \Omega\).