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

Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E and let Π C be a sunny nonexpansive retraction from E onto C. Let the mappings $${T, S: C \to E}$$ be γ 1 -strongly accretive, μ 1 -Lipschitz continuous and γ 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 {α k } is a sequence in [0,1] and ρ, η 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. Copyright Springer Science+Business Media, LLC. 2013