Two-step projection methods for a system of variational inequality problems in Banach spaces

• Authors: Yonghong Yao;Yeong-Cheng Liou;Shin Min Kang
• Venue: Journal of Global Optimization
• Year: 2013

Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E and let 驴 C be a sunny nonexpansive retraction from E onto C. Let the mappings $${T, S: C \to E}$$ be 驴 1-strongly accretive, μ 1-Lipschitz continuous and 驴 2-strongly accretive, μ 2-Lipschitz continuous, respectively. For arbitrarily chosen initial point $${x^0 \in C}$$ , compute the sequences {x k } and {y k } such that $${\begin{array}{ll} \quad y^k = \Pi_C[x^k-\eta S(x^k)],\\ x^{k+1} = (1-\alpha^k)x^k+\alpha^k\Pi_C[y^k-\rho T(y^k)],\quad k\geq 0, \end{array}}$$ where {驴 k } is a sequence in [0,1] and 驴, 驴 are two positive constants. Under some mild conditions, we prove that the sequences {x k } and {y k } converge to x* and y*, respectively, where (x*, y*) is a solution of the following system of variational inequality problems in Banach spaces: $${\left\{\begin{array}{l}\langle \rho T(y^*)+x^*-y^*,j(x-x^*)\rangle\geq 0, \quad\forall x \in C,\\\langle \eta S(x^*)+y^*-x^*,j(x-y^*)\rangle\geq 0,\quad\forall x \in C.\end{array}\right.}$$ Our results extend the main results in Verma (Appl Math Lett 18:1286---1292, 2005) from Hilbert spaces to Banach spaces. We also obtain some corollaries which include some results in the literature as special cases.