On General Mixed Variational Inequalities
In this paper, 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 generalizations of the previous known results.
KeywordsVariational inequalities Nonconvex functions Fixed-point problem Resolvent operator Resolvent equations Projection operator Convergence Dynamical systems
Mathematics Subject Classification (2000)49J40 90C33
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- 1.Brezis, H.: Operateurs Maximaux Monotone et Semigroupes de Contractions dans les Espace d’Hilbert. North-Holland, Amsterdam (1973) Google Scholar
- 9.Giannessi, F., Maugeri, A., Pardalos, P.M.: Equilibrium Problems, Nonsmooth Optimization and Variational Inequalities Problems. Kluwer Academic, Dordrecht (2001) Google Scholar
- 14.Aslam Noor, M.: On variational inequalities. PhD Thesis, Brunel University, London, UK (1975) Google Scholar
- 19.Aslam Noor, M.: Some algorithms for general monotone mixed variational inequalities. Math. Comput. Model. 29, 1–9 (1999) Google Scholar
- 31.Aslam Noor, M.: Variational inequalities and applications. Lecture Notes, Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan (2007) Google Scholar
- 32.Aslam Noor, M.: Mixed variational inequalities and nonexpansive mappings. In: Th.M. Rassias (ed.) Inequalities and Applications (2008) Google Scholar
- 38.Patriksson, M.: Nonlinear Programming and Variational Inequalities: A Unified Approach. Kluwer Academic, Dordrecht (1998) Google Scholar