Taiwanese Journal of Mathematics

EXTENDED GENERAL NONLINEAR QUASI-VARIATIONAL INEQUALITIES AND PROJECTION DYNAMICAL SYSTEMS

Qamrul Ansari, Javad Balooee, and Jen-Chih Yao

Full-text: Open access

Abstract

The aim of this paper is to introduce and study a new class of the extended general nonlinear quasi-variational inequalities and a new class of the extended general Wiener-Hopf equations. The equivalence between the extended general nonlinear quasi-variational inequalities and the fixed point problems, and as well as the extended general Wiener-Hopf equations is established. Then by using these equivalences, we discuss the existence and uniqueness of a solution of the extended general nonlinear quasi-variational inequalities. Applying the equivalent alternative formulation and a nearly uniformly Lipschitzian mapping $S$, we define some new $p$-step projection iterative algorithms with mixed errors for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping $S$ which is also a unique solution of the extended general nonlinear quasi-variational inequalities. The convergence analysis of the suggested iterative schemes under some suitable conditions is studied. We also suggest and analyze a class of extended general projection dynamical systems associated with the extended general nonlinear quasi-variational inequalities. We show that the trajectory of the solution of the extended general projection dynamical system converges globally exponential to a unique solution of the extended general nonlinear quasi-variational inequalities. Results obtained in this paper may be viewed as an refinement and improvement of the previously known results.

Article information

Source
Taiwanese J. Math., Volume 17, Number 4 (2013), 1321-1352.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706120

Digital Object Identifier
doi:10.11650/tjm.17.2013.2559

Mathematical Reviews number (MathSciNet)
MR3085514

Zentralblatt MATH identifier
1275.49013

Subjects
Primary: 49J40: Variational methods including variational inequalities [See also 47J20]
Secondary: 47J20: Variational and other types of inequalities involving nonlinear operators (general) [See also 49J40] 47H05: Monotone operators and generalizations

Keywords
variational inequalities fixed point problems nearly uniformly Lipschitzian mappings extended general Wiener-Hopf equations dynamical systems projection operator

Citation

Ansari, Qamrul; Balooee, Javad; Yao, Jen-Chih. EXTENDED GENERAL NONLINEAR QUASI-VARIATIONAL INEQUALITIES AND PROJECTION DYNAMICAL SYSTEMS. Taiwanese J. Math. 17 (2013), no. 4, 1321--1352. doi:10.11650/tjm.17.2013.2559. https://projecteuclid.org/euclid.twjm/1499706120


Export citation

References

  • C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, New York, 1984.
  • A. Bensoussan and J. L. Lions, Application des Inéquations Variationelles en Control et en Stochastiques, Dunod, Paris, 1978.
  • J. Dong, D. Zhang and A. Nagurney, A projected dynamical systems model of general financial equilibrium with stability analysis, Math. Comput. Modell., 24(2) (1996), 35- 44.
  • P. Dupuis and A. Nagurney, Dynamical systems and variational inequalities, Ann. Oper. Res., 44 (1993), 19-42.
  • T. L. Friesz, D. H. Bernstein and R. Stough, Dynamical systems, variationl inequalities and control theoretical models for predicting time-varying unban network flows, Trans. Sci., 30 (1996), 14-31.
  • K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171-174.
  • G. Isac, Complementarity Problems, Lecture Notes in Math., No. 1528, Springer-Verlag, New York, Berlin, 1992.
  • W. A. Kirk and H. K. Xu, Asymptotic pointwise contractions, Nonlinear Anal., 69 (2008), 4706-4712.
  • A. S. Kravchuk and P. J. Neittaanmaki, Variational and Quasi Variational Inequalities in Mechanics, Springer, Dordrecht, Holland, 2007.
  • L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl., 194 (1995), 114-125.
  • R. K. Miller and A. N. Michel, Ordinary Differential Equations, Academic Press, New York, 1982.
  • A. Nagurney and D. Zhang, Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer Academic Publishers, Dordrecht, Holland, 1995.
  • M. A. Noor, Existence results for quasi-variational inequalities, Banach J. Math. Anal., 1 (2007), 186-194.
  • M. A. Noor, On general quasi-variational inequalities, Journal of King Saud University-Science, 24 (2012), 81-88.
  • M. A. Noor, Resolvent dynamical systems for mixed variational inequalities, Korean J. Comput. Appl. Math., 9 (2002), 15-26.
  • M. A. Noor and Z. Huang, Three-step iterative methods for nonexpansive mappings and variational inequalities, Appl. Math. Comput., 187 (2007), 680-685.
  • M. A. Noor, K. I. Noor and H. Yaqoob, On general mixed variational inequalities, Acta Appl. Math., 110(1) (2010), 227-246.
  • X. Qin and M. A. Noor, General Wiener-Hopf equation technique for nonexpansive mappings and general variational inequalities in Hilbert spaces, Appl. Math. Comput., 201 (2008), 716-722.
  • D. R. Sahu, Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces, Comment. Math. Univ. Carolin, 46(4) (2005), 653-666.
  • P. Shi, Equivalence of variational inequalities with Wiener-Hopf equations, Proc. Amer. Math. Soc., 111 (1991), 339-346.
  • A. H. Siddiqi and Q. H. Ansari, An algorithm for a class of quasivariational inequalities, J. Math. Anal. Appl., 145 (1990), 413-418.
  • A. H. Siddiqi and Q. H. Ansari, Strongly nonlinear quasivariational inequalities, J. Math. Anal. Appl., 149 (1990), 444-450.
  • Y. S. Xia, On convergence conditions of an extended projection neural network, Neural Comput., 17 (2005), 515-525.
  • Y. S. Xia and J. Wang, A recurrent neural network for solving linear projection equations, Neural Network, 137 (2000), 337-350.
  • Y. S. Xia and J. Wang, On the stability of globally projected dynamical systems, J. Optim. Theory Appl., 106 (2000), 129-150.
  • D. Zhang and A. Nagurney, On the stability of the projected dynamical systems, J. Optim. Theory Appl., 85 (1985), 97-124.