Journal of Global Optimization

, Volume 55, Issue 4, pp 801–811 | Cite as

Two-step projection methods for a system of variational inequality problems in Banach spaces



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.


Projection method Accretive mapping Variational inequality Banach spaces 

Mathematics Subject Classification (2000)

47H05 47H10 47J25 


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Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  • Yonghong Yao
    • 1
  • Yeong-Cheng Liou
    • 2
  • Shin Min Kang
    • 3
  1. 1.Department of MathematicsTianjin Polytechnic UniversityTianjinChina
  2. 2.Department of Information ManagementCheng Shiu UniversityKaohsiungTaiwan
  3. 3.Department of Mathematics and the RINSGyeongsang National UniversityJinjuKorea

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