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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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