Abstract
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.
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Yao, Y., Liou, YC. & Kang, S.M. Two-step projection methods for a system of variational inequality problems in Banach spaces. J Glob Optim 55, 801–811 (2013). https://doi.org/10.1007/s10898-011-9804-0
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DOI: https://doi.org/10.1007/s10898-011-9804-0