Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: A projective splitting algorithm for solving generalized mixed variational inequalities | Xia and Zou Journal of Inequalities and Applications 2011 2011 27 http content 2011 1 27 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access A projective splitting algorithm for solving generalized mixed variational inequalities Fu-quan Xia 1 and Yun-zhi Zou2 Correspondence fuquanxia@sina. com department of Mathematics Sichuan Normal University Chengdu Sichuan 610066 P. R. China Full list of author information is available at the end of the article SpringerOpen0 Abstract In this paper a projective splitting method for solving a class of generalized mixed variational inequalities is considered in Hilbert spaces. We investigate a general iterative algorithm which consists of a splitting proximal point step followed by a suitable orthogonal projection onto a hyperplane. Moreover in our splitting algorithm we only use the individual resolvent mapping I pkcf -1 and never work directly with the operator T df where ụk is a positive real number T is a set-valued mapping and df is the sub-differential of function f. We also prove the convergence of the algorithm for the case that T is a pseudomonotone set-valued mapping and f is a non-smooth convex function. 2000 Mathematics Subject Classification 90C25 49D45 49D37. Keywords projective splitting method generalized mixed variational inequality pseudomonotonicity 1 Introduction Let X be a nonempty closed convex subset of a real Hilbert space H T X 2H be a set-valued mapping and f H - be a lower semi-continuous proper convex function. We consider a generalized mixed variational inequality problem GMVIP find x e X such that there exists w e T x satisfying W y - x f y - f x 0 Vy e X. The GMVIP has enormous applications in many areas such as mechanics optimization equilibrium etc. For details we refer to 1-3 and the references therein. It has therefore been widely studies by many authors recently. For example by Rockafel-lar 4 Tseng 5 Xia
