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Two-step projection methods for a system of variational inequality problems in Banach spaces. (English) Zbl 1260.47085
Summary: Let \(C\) be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space \(E\) and let \(\Pi _{C }\) be a sunny nonexpansive retraction from \(E\) onto \(C\). Let the mappings \({T, S: C \to E}\) be \(\gamma _{1}\)-strongly accretive, \(\mu _{1}\)-Lipschitz continuous and \(\gamma _{2}\)-strongly accretive, \(\mu _{2}\)-Lipschitz continuous, respectively. For arbitrarily chosen initial point \({x^0 \in C}\), define the sequences \(\{x ^{k }\}\) and \(\{y ^{k }\}\) by \[ \begin{aligned} y^k & = \Pi_C[x^k-\eta S(x^k)], \\ x^{k+1} & = (1-\alpha^k)x^k+\alpha^k\Pi_C[y^k-\rho T(y^k)],\quad k\geq 0, \end{aligned} \] where \(\{\alpha ^{k }\}\) is a sequence in [0,1] and \(\rho ,\eta \) are two positive constants. Under some mild conditions, we prove that the sequences \(\{x ^{k }\}\) and \(\{y ^{k }\}\) converge to \(x^\ast\) and \(y^\ast\), respectively, where \((x^\ast, y^\ast)\) is a solution of the following system of variational inequality problems in Banach spaces: \[ \begin{cases} \langle \rho T(y^\ast)+x^\ast-y^\ast,j(x-x^\ast)\rangle\geq 0, \quad \forall x \in C, \\ \langle \eta S(x^\ast)+y^\ast-x^\ast,j(x-y^\ast)\rangle\geq 0,\quad\forall x \in C.\end{cases} \] Our results extend the main results in [R. U. Verma, Appl. Math. Lett. 18, No. 11, 1286–1292 (2005; Zbl 1099.47054)] from Hilbert spaces to Banach spaces. We also obtain some corollaries which include some results in the literature as special cases.

MSC:
47J25 Iterative procedures involving nonlinear operators
47H05 Monotone operators and generalizations
47H10 Fixed-point theorems
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