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Zbl 1190.49015
Noor, Muhammad Aslam; Noor, Khalida Inayat; Huma, Yaqoob
On general mixed variational inequalities. (English)
[J] Acta Appl. Math. 110, No. 1, 227-246 (2010). ISSN 0167-8019; ISSN 1572-9036/e

Summary: We introduce and consider a new class of mixed variational inequalities, which is called the general mixed variational inequality. Using the resolvent operator technique, we establish the equivalence between the general mixed variational inequalities and the fixed-point problems as well as resolvent equations. We use this alternative equivalent formulation to suggest and analyze some iterative methods for solving the general mixed variational inequalities. We study the convergence criteria of the suggested iterative methods under suitable conditions. Using the resolvent operator technique, we also consider the resolvent dynamical systems associated with the general mixed variational inequalities. We show that the trajectory of the dynamical system converges globally exponentially to the unique solution of the general mixed variational inequalities. Our methods of proofs are very simple as compared with others' techniques. Results proved in this paper may be viewed as a refinement and important generalization of the previous known results.

MSC 2000:: *49J40 Variational methods including variational inequalities
90C33 Complementarity problems

Keywords: variational inequalities; nonconvex functions; resolvent operator; resolvent equations; projection operator; dynamical systems

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Zentralblatt MATH Berlin [Germany]