Journal of Global Optimization

, Volume 41, Issue 3, pp 465–478 | Cite as

Mixed projection methods for systems of variational inequalities



Let H be a real Hilbert space. Let \(F:D(F) \subseteq H \to H, K : D(K) \subseteq H \to H\) be bounded and continuous mappings where D(F) and D(K) are closed convex subsets of H. We introduce and consider the following system of variational inequalities: find [u *,v *]∈D(F) × D(K) such that \(\left\{\begin{array}{lll}&\langle Fu^* - v^*, x - u^*\rangle \geq 0,\quad x \in D(F),\\ &\langle Kv^* + u^*, y - v^*\rangle \geq 0,\quad y \in D(K)\end{array}\right.\) This system of variational inequalities is closely related to a pseudomonotone variational inequality. The well-known projection method is extended to develop a mixed projection method for solving this system of variational inequalities. No invertibility assumption is imposed on F and K. The operators K and F also need not be defined on compact subsets of H.


Hilbert space Variational inequality Pseudomonotonicity Mixed projection method 

Mathematics Subject Classification (2000)

49J30 47H10 47H17 


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© Springer Science+Business Media, LLC. 2007

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiChina
  2. 2.Department of Applied MathematicsNational Sun Yat-sen UniversityKaohsiungTaiwan

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