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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.

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