On General Mixed Variational Inequalities
 Muhammad Aslam Noor,
 Khalida Inayat Noor,
 Huma Yaqoob
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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 fixedpoint 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.
 Brezis, H. (1973) Operateurs Maximaux Monotone et Semigroupes de Contractions dans les Espace d’Hilbert. NorthHolland, Amsterdam
 Cristescu, G., Lupsa, L. (2002) NonConnected Convexities and Applications. Kluwer Academic, Dordrecht
 Daniele, P., Giannessi, F., Maugeri, A. (2003) Equilibrium Problems and Variational Models. Kluwer Academic, London
 Dong, J., Zhang, D., Nagurney, A. (1996) A projected dynamical systems model of general financial equilibrium with stability analysis. Math. Comput. Appl. 24: pp. 3544
 Dupuis, P., Nagurney, A. (1993) Dynamical systems and variational inequalities with applications. Ann. Oper. Res. 44: pp. 1942 CrossRef
 Friesz, T.L., Bernstein, D.H., Mehta, N.J., Tobin, R.L., Ganjliazadeh, S. (1994) Daytoday dynamic network disequilibrium and idealized traveler information systems. Oper. Res. 42: pp. 11201136 CrossRef
 Friesz, T.L., Bernstein, D.H., Stough, R. (1996) Dynamic systems, variational inequalities and control theoretic models for predicting timevarying urban network flows. Trans. Sci. 30: pp. 1431 CrossRef
 Giannessi, F., Maugeri, A. (1995) Variational Inequalities and Network Equilibrium Problems. Plenum Press, New York
 Giannessi, F., Maugeri, A., Pardalos, P.M. (2001) Equilibrium Problems, Nonsmooth Optimization and Variational Inequalities Problems. Kluwer Academic, Dordrecht
 Glowinski, R., Lions, J.L., Tremolieres, R. (1981) Numerical Analysis of Variational Inequalities. NorthHolland, Amsterdam
 Hu, X., Wang, J. (2006) Solving pseudomonotone variational inequalities and pseudoconvex optimization problems using the projection neural network. IEEE Trans. Neural Netw. 17: pp. 14871499 CrossRef
 Mosco, U. (1976) Implicit variational methods and quasi variational inequalities. Nonlinear Operators and the Calculus of Variations. Springer, Berlin, pp. 83126 CrossRef
 Nagurney, A., Zhang, D. (1995) Projected Dynamical Systems and Variational Inequalities with Applications. Kluwer Academic, Dordrecht
 Aslam Noor, M.: On variational inequalities. PhD Thesis, Brunel University, London, UK (1975)
 Aslam Noor, M. (1988) General variational inequalities. Appl. Math. Lett. 1: pp. 119121 CrossRef
 Aslam Noor, M. (1993) WienerHopf equations and variational inequalities. J. Optim. Theory Appl. 79: pp. 197206 CrossRef
 Aslam Noor, M. (1997) Some recent advances in variational inequalities, Part I. Basic concepts. New Zealand J. Math. 26: pp. 5380
 Aslam Noor, M. (1997) Some recent advances in variational inequalities, Part II. Other concepts. New Zealand J. Math. 26: pp. 229255
 Aslam Noor, M. (1999) Some algorithms for general monotone mixed variational inequalities. Math. Comput. Model. 29: pp. 19
 Aslam Noor, M. (2000) New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251: pp. 217229 CrossRef
 Aslam Noor, M. (2002) Resolvent dynamical systems for mixed variational inequalities. Korean J. Comput. Appl. Math. 9: pp. 1526
 Aslam Noor, M. (2002) A WienerHopf dynamical system for variational inequalities. New Zealand J. Math. 31: pp. 173182
 Aslam Noor, M. (2003) New extragradienttype methods for general variational inequalities. J. Math. Anal. Appl. 277: pp. 379395 CrossRef
 Aslam Noor, M. (2004) Some developments in general variational inequalities. Appl. Math. Comput. 152: pp. 199277 CrossRef
 Aslam Noor, M. (2003) Mixed quasi variational inequalities. Appl. Math. Comput. 146: pp. 553578 CrossRef
 Aslam Noor, M. (2004) Fundamentals of mixed quasi variational inequalities. Int. J. Pure Appl. Math. 15: pp. 137258
 Aslam Noor, M. (2006) Fundamentals of equilibrium problems. Math. Inequal. Appl. 9: pp. 529566
 Aslam Noor, M. (2006) Merit functions for general variational inequalities. J. Math. Anal. Appl. 316: pp. 736752 CrossRef
 Aslam Noor, M. (2006) Projectionproximal methods for general variational inequalities. J. Math. Anal. Appl. 318: pp. 5362 CrossRef
 Aslam Noor, M. (2008) Differentiable nonconvex functions and general variational inequalities. Appl. Math. Comput. 199: pp. 623630 CrossRef
 Aslam Noor, M.: Variational inequalities and applications. Lecture Notes, Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan (2007)
 Aslam Noor, M.: Mixed variational inequalities and nonexpansive mappings. In: Th.M. Rassias (ed.) Inequalities and Applications (2008)
 Aslam Noor, M., Bnouhachem, A. (2005) Selfadaptive methods for mixed quasi variational inequalities. J. Math. Anal. Appl. 312: pp. 514526 CrossRef
 Aslam Noor, M., Inayat Noor, K. (2004) Selfadaptive projection algorithms for general variational inequalities. Appl. Math. Comput. 151: pp. 659670 CrossRef
 Aslam Noor, M., Inayat Noor, K., Rassias, Th.M. (1993) Some aspects of variational inequalities. J. Comput. Appl. Math. 47: pp. 285312 CrossRef
 Aslam Noor, M., Inayat Noor, K., Rassias, T.M. (1998) Setvalued resolvent equations and mixed variational inequalities. J. Math. Anal. Appl. 220: pp. 741759 CrossRef
 Aslam Noor, M., Huang, Z. (2007) Threestep methods for nonexpansive mappings and variational inequalities. Appl. Math. Comput. 187: pp. 680685 CrossRef
 Patriksson, M. (1998) Nonlinear Programming and Variational Inequalities: A Unified Approach. Kluwer Academic, Dordrecht
 Pitonyak, A., Shi, P., Shillor, M. (1990) On an iterative method for variational inequalities. Numer. Math. 58: pp. 231244 CrossRef
 Shi, P. (1991) Equivalence of variational inequalities with WienerHopf equations. Proc. Am. Math. Soc. 111: pp. 339346 CrossRef
 Stampacchia, G. (1964) Formes bilineaires coercivities sur les ensembles coercivities sur les ensembles convexes. C.R. Acad. Sci. Paris 258: pp. 44134416
 Xia, Y.S. (2004) Further results on global convergence and stability of globally projected dynamical systems. J. Optim. Theory Appl. 122: pp. 627649 CrossRef
 Xia, Y.S. (2005) On convergence conditions of an extended projection neural network. Neural Comput. 17: pp. 515525 CrossRef
 Xia, Y.S., Wang, J. (2000) A recurrent neural network for solving linear projection equations. Neural Network 13: pp. 337350 CrossRef
 Xia, Y.S., Wang, J. (2000) On the stability of globally projected dynamical systems. J. Optim. Theory Appl. 106: pp. 129150 CrossRef
 Zhang, D., Nagurney, A. (1995) On the stability of the projected dynamical systems. J. Optim. Theory Appl. 85: pp. 97124 CrossRef
 Title
 On General Mixed Variational Inequalities
 Journal

Acta Applicandae Mathematicae
Volume 110, Issue 1 , pp 227246
 Cover Date
 20100401
 DOI
 10.1007/s1044000894024
 Print ISSN
 01678019
 Online ISSN
 15729036
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Variational inequalities
 Nonconvex functions
 Fixedpoint problem
 Resolvent operator
 Resolvent equations
 Projection operator
 Convergence
 Dynamical systems
 49J40
 90C33
 Industry Sectors
 Authors

 Muhammad Aslam Noor ^{(1)}
 Khalida Inayat Noor ^{(1)}
 Huma Yaqoob ^{(1)}
 Author Affiliations

 1. Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan