Convergence of a hybrid iterative scheme for fixed points of nonexpansive maps, solutions of equilibrium, and variational inequalities problems (Q355691): Difference between revisions

Summary: Let \(K\) be a closed, convex, and nonempty subset of a real \(q\)-uniformly smooth Banach space \(E\) which is also uniformly convex. For some \(\kappa > 0\), let \(T_i : K \to E\), \(i \in \mathbb N\), be a family of nonexpansive maps and \(A : K \to E\) be a \(\kappa\)-inverse strongly accretive map. Let \(G : K \times K \to \mathbb R\) be a bifunction satisfying some conditions. Let \(P_K\) be a nonexpansive projection of \(E\) onto \(K\). For some fixed real numbers \(\delta \in (0, 1)\), \(\lambda \in (0, (q\kappa/d_q)^{1/(q-1)})\), and arbitrary but fixed vectors \(x_1, u \in E\), let \(\{x_n\}\) and \(\{y_n\}\) be sequences generated by \(G(y_n, \eta) + (1/r)\langle \eta - y_n, j_q(y_n - x_n)\rangle \geq 0\) for all \(\eta \in K\), \(x_{n+1} = \alpha_nu + (1 - \delta)(1 - \alpha_n)x_n + \delta \sum_{i \geq 1} \sigma_{in}T_iP_K(y_n - \lambda Ay_n)\), \(n \geq 1\), where \(r \in (0, 1)\) is fixed, and \(\{\alpha_n\}, \{\sigma_{i, n}\} \subset (0, 1)\) are sequences satisfying appropriate conditions. If \(F := [\cap^\infty_{i=1}F(T_i)] \cap \text{VI}(K, A) \cap \text{EP}(G) \neq \emptyset\), under some mild conditions, we prove that the sequences \(\{x_n\}\) and \(\{y_n\}\) converge strongly to some element in \(F\).