Annals of Functional Analysis

Dynamical systems for quasi variational inequalities

Awais Gul Khan, Muhammad Aslam Noor, and Khalida Inayat Noor

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Abstract

In this paper, using the projection operator, we introduce two new dynamical systems for extended general quasi variational inequalities. These dynamical systems are called extended general implicit projected dynamical system and extended general implicit Wiener-Hopf dynamical system. We prove that these new dynamical systems converge globally exponentially to a unique solution of the extended general quasi variational inequalities under some suitable conditions. Some special cases are also discussed. The ideas and technique of this paper may stimulate further research.

Article information

Source
Ann. Funct. Anal., Volume 6, Number 1 (2015), 193-209.

Dates
First available in Project Euclid: 19 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.afa/1419001459

Digital Object Identifier
doi:10.15352/afa/06-1-14

Mathematical Reviews number (MathSciNet)
MR3297796

Zentralblatt MATH identifier
1316.47058

Subjects
Primary: 46T99
Secondary: 47H05: Monotone operators and generalizations 49J40: Variational methods including variational inequalities [See also 47J20]

Keywords
Dynamical systems Wiener-Hopf equations global convergence quasi-variational inequalities

Citation

Noor, Muhammad Aslam; Noor, Khalida Inayat; Khan, Awais Gul. Dynamical systems for quasi variational inequalities. Ann. Funct. Anal. 6 (2015), no. 1, 193--209. doi:10.15352/afa/06-1-14. https://projecteuclid.org/euclid.afa/1419001459


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References

  • Q.H. Ansari, J. Balooee and J.-C. Yao, Extended general nonlinear quasi-variational inequalities and projection dynamical systems, Taiwanese J. Math. 17 (2013), no. 4, 1321-1352.
  • A.S. Antipin, M. Jacimovic and N. Mijailovic, A second-order continuous method for solving quasi-variational inequalities, Comput. Math. Math. Phy. 51 (2013), no. 11, 1856–1863.
  • A. Baiocchi and A. Capelo, Variational and Quasi-Variational Inequalities, J. Willey and Sons, New York, 1984.
  • A. Bensoussan and J.L. Lions, Applications Des Inequations Variationnelles En Control Eten Stochastique, Dunod, Paris, 1978.
  • M.-G. Cojocaru, P. Daniele and A. Nagurney, Projected dynamical systems and evolutionary variational inequalities via Hilbert spaces with applications, J. Optim. Theory Appl. 127 (2005), no. 3, 549–563.
  • J. Dong, D. Zhang and A. Nagurney, A projected dynamical systems model of general financial equilibrium with stability analysis, Math. Comput. Modell. 24 (1996), no. 2, 35–44.
  • P. Dupuis and A. Nagurney, Dynamical systems and variational inequalities, Ann. Oper. Res. 44 (1993), no. 1, 7–42.
  • F. Facchinei, C. Kanzow and S. Sagratella, Solving quasi-variational inequalities via their KKT conditions, Math. Program. Ser. A. (2013), doi:10.1007/s10107-013-0637-0.
  • T.L. Friesz, D. Bernstein, N.J. Mehta, R.L. Tobin and S. Ganjalizadeh, Day-to-day dynamic network disequilibria and idealized traveler information systems, Oper. Res. 42 (1994), no. 6, 1120–1136.
  • T.L. Friesz, D. Bernstein and R. Stough, Dynamic systems, variational inequalities and control theoretic models for predicting time-varying urban network flows, Trans. Sci. 30 (1996), no. 1, 14–31.
  • A.S. Kravchuk and P.J. Neittaanmki, Variational and Quasi-variational Inequalities in Mechanics, Springer,Dordrecht, Holland, 2007.
  • F. Lenzen, F. Becker, J. Lellmann, S. Petra and C. Schn örr, A class of quasi-variational inequalities for adaptive image denoising and decomposition, Comput. Optim. Appl. 54 (2013), no. 2, 371–398.
  • Q. Liu and J. Cao, A recurrent neural network based on projection operator for extended general variational inequalities. Systems , IEEE. Trans. Syst. Man Cybern. Part B Cybern. 40 (2010), no. 3, 928–938.
  • Q. Liu and Y. Yang, Global exponential system of projection neural networks for system of generalized variational inequalities and related nonlinear minimax problems, Neurocomputing. 73 (2010), no. 10, 2069–2076.
  • A. Nagurney and A.D. Zhang, Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer Academic, Boston, 1996.
  • Z.D. Mitrovic, Some existence results on a class of inclusions. Ann. Funct. Anal. 2 (2011), no. 2, 51–58.
  • M.A. Noor, Quasi variational inequalities, Appl. Math. Lett. 1 (1988), no. 4, 367–370.
  • M.A. Noor, A Wiener–Hopf dynamical system for variational inequalities, New Zealand J. Math. 31 (2002), 173–182.
  • M.A. Noor, Implicit dynamical systems and quasi variational inequalities, Appl. Math. Comput. 134 (2003), no. 1, 69–81.
  • M.A. Noor, Existence results for quasi variational inequalities, Banach J. Math. Anal. 1 (2007), no. 2, 186–194.
  • M.A. Noor, Auxiliary principle technique for extended general variational inequalities, Banach J. Math. Anal. 2 (2008), no. 1, 33–39.
  • M.A. Noor and K.I. Noor, Sensitivity analysis of some quasi variational inequalities, J. Adv. Math. Studies. 6 (2013), no. 1, 43–52.
  • M.A. Noor and K.I. Noor, Some new classes of quasi split feasibility problems, Appl. Math. Inform. Sci. 7 (2013), no. 4, 1547–1552.
  • M.A. Noor, K.I. Noor and A.G. Khan, Some iterative schemes for solving extended general quasi variational inequalities, Appl. Math. Inform. Sci. 7 (2013), no. 3, 917–925.
  • Y. Shehu, Iterative methods for fixed points and equilibrium problems, Ann. Funct. Anal. 1 (2010), no. 2, 121–132.
  • P. Shi, Equivalence of variational inequalities with Wiener–Hopf equations, Proc. Amer. Math. Soc. 111 (1991), no. 2, 339–346.
  • G. Stampacchia, Formes bilineaires coercivites sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413–4416.
  • Z.-M. Wang, M.K. Kang and Y.J. Cho, Convergence theorems based on the shrinking projection method for hemi-relatively nonexpansive mappings, variational inequalities and equilibrium problems, Banach J. Math. Anal. 6 (2012), no. 1, 11–34.
  • Y. Xia and J. Wang, On the stability of globally projected dynamical systems, J. Optim. Theory Appl. 106 (2000), no. 1, 129–150.