Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E and let Π<Subscript> C </Subscript> be a sunny nonexpansive retraction from E onto C. Let the mappings <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$${T, S: C \to E}$$</EquationSource> </InlineEquation> be γ <Subscript>1</Subscript>-strongly accretive, μ <Subscript>1</Subscript>-Lipschitz continuous and γ <Subscript>2</Subscript>-strongly accretive, μ <Subscript>2</Subscript>-Lipschitz continuous, respectively. For arbitrarily chosen initial point <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${x^0 \in C}$$</EquationSource> </InlineEquation> , compute the sequences {x <Superscript> k </Superscript>} and {y <Superscript> k </Superscript>} such that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${\begin{array}{ll} \quad y^k=\Pi_C[x^k-\eta S(x^k)],\\ x^{k+1}=(1-\alpha^k)x^k+\alpha^k\Pi_C[y^k-\rho T(y^k)],\quad k\geq 0, \end{array}}$$</EquationSource> </InlineEquation> where {α <Superscript> k </Superscript>} is a sequence in [0,1] and ρ, η are two positive constants. Under some mild conditions, we prove that the sequences {x <Superscript> k </Superscript>} and {y <Superscript> k </Superscript>} converge to x* and y*, respectively, where (x*, y*) is a solution of the following system of variational inequality problems in Banach spaces: <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$${\left\{\begin{array}{l}\langle \rho T(y^*)+x^*-y^*,j(x-x^*)\rangle\geq 0, \quad\forall x \in C,\\\langle \eta S(x^*)+y^*-x^*,j(x-y^*)\rangle\geq 0,\quad\forall x \in C.\end{array}\right.}$$</EquationSource> </InlineEquation> Our results extend the main results in Verma (Appl Math Lett 18:1286–1292, <CitationRef CitationID="CR16">2005</CitationRef>) 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
