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Nonlinear Analysis and Variational Problems pp 235–258Cite as

A Panoramic View on Projected Dynamical Systems

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Part of the Springer Optimization and Its Applications book series (SOIA,volume 35)

Abstract

The theory of generalized projections both in non-pivot Hilbert spaces and strictly convex and smooth Banach spaces is developed and the related theory of projected dynamical systems is highlighted. A particular emphasis is given to the equivalence between solutions of variational inequality and critical points of projected dynamical systems.

Keywords

  • Hilbert Space
  • Banach Space
  • Variational Inequality
  • Equilibrium Problem
  • Duality Pairing

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Reference

  1. V. Acary, B. Brogliato, A. Daniilidis, C. Lemaréchal, On the Equivalence Between Complementarity Systems, Projected Systems and Unilateral Differential Inclusions, Rapport de Recherche 5107 I.N.R.I.A, Janvier 2004.

    Google Scholar 

  2. Ya. I. Alber, Metric and generalized projection operators in Banach spaces: proprieties and applications, A. Kartsatos (Ed.), Theory and Applications of Nonlinear Operators of Monotone and Accretive Type, Marcel Dekker, New York, 1996, 15–50.

    Google Scholar 

  3. Ya. I. Alber, Decomposition theorem in Banach spaces, Field Inst. Comm. 25, 2000, 77–99.

    MathSciNet  Google Scholar 

  4. Ya. I. Alber, J-C. Yao, On the projection dynamical systems in Banach spaces, Taiwanese Journal of Mathematics, (to appear).

    Google Scholar 

  5. J-P. Aubin, Analyse Fonctionelle appliquée, Editions PUF, 1987.

    Google Scholar 

  6. A. Barbagallo, Regularity Results For Evolutionary Variational and Quasi-Variational Inequalities and Applicatios to Dynamic Equilibrium Problems, PhD Thesis, University of Naples, Italy, 2007.

    Google Scholar 

  7. A. Barbagallo, Regularity results for time-dependent variational and quasi-variational inequalities and computational procedures, Math. Models Methods Appl. Sci. 17, (2007),277–304.

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. A. Barbagallo, S. Pia, Weighted variational inequalities in non-pivot Hilbert spaces with applications, Computational Optimization and Applications, to appear.

    Google Scholar 

  9. H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, 1993.

    Google Scholar 

  10. V. Barbu, Th. Precapanu, Convexity and Optimization in Banach Spaces, Romania International Publisher, Bucarest, 1978.

    Google Scholar 

  11. M.G. Cojocaru, Projected Dynamical Systems on Hilbert Spaces, Ph. D. Thesis, Queen’s University, Canada, 2002.

    Google Scholar 

  12. M.G. Cojocaru, L.B. Jonker, Existence of Solutions to Projected Differential Equations in Hilbert Spaces, Proceedings of the American Mathematical Society 132, 2004, 183–193.

    Google Scholar 

  13. M.G. Cojocaru, P. Daniele, A. Nagurney, Projected Dynamical Systems and Evolutionary Variational Inequalities Via Hilbert Spaces with Applications, Journal of Optimization Theory and Applications 127, no. 3, (2005), 549–563.

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. M.G. Cojocaru, P. Daniele, A. Nagurney, Double layered Dynamics: A unified theory of Projected Dynamical Systems and Evolutionary Variational inegualities, European Journal of Operational Research 175, 1, (2006), 494–507.

    CrossRef  MATH  MathSciNet  Google Scholar 

  15. M.G. Cojocaru, P. Daniele, A. Nagurney, Projected Dynamical Systems, Evolutionary Variational Inequalities, Applications, and a Computational Procedure, in Pareto Optimality, Game Theory and Equilibria, Springer, New York, A. Chinchuluun, P.M. Pardalos, A. Migadalas, L. Pitsoulis Eds, 2008, 392–406.

    Google Scholar 

  16. M.G. Cojocaru, L.B. Jonker. Existence of solutions to Projected Differential Equations in Hilbert Spaces, Proceeding of the American Mathematical Society 132, 2004, 183–193.

    Google Scholar 

  17. M.G. Cojocaru, S. Pia. Non-pivot and implicit projected dynamical systems on Hilbert spaces, Preprint.

    Google Scholar 

  18. P. Daniele, Time–Dependent Spatial Price Equilibrium Problem: Existence and Stability Results for the Quantity Formulation Model, Journal of Global Optimization 28, 2004, 283–295.

    CrossRef  MATH  MathSciNet  Google Scholar 

  19. P. Daniele, Evolutionary Variational Inequalities and Economic Models for Demand Supply Markets, M3AS: Mathematical Models and Methods in Applied Sciences 4 (13), 2003,471–489.

    CrossRef  MathSciNet  Google Scholar 

  20. P. Daniele, Variational Inequalities for Evolutionary Financial Equilibrium, in Innovations in Financial and Economic Networks, Edward Elgar Publishing, Cheltenham, England, A. Nagurney, Editor, 2003, 84–108.

    Google Scholar 

  21. P. Daniele, S. Giuffré, G. Idone, A. Maugeri, Infinite Dimentional Duality and Applications, Mathematische Annalen 339 (1), 2007, 221–239.

    CrossRef  MATH  MathSciNet  Google Scholar 

  22. P. Daniele, A. Maugeri, On Dynamical Equilibrium Problems and Variational Inequalities, in Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Kluwer Academic Publishers, Dordrecht, The Netherlands, F. Giannessi, A. Maugeri, and P. Pardalos, Editors, 2001, 59–69.

    Google Scholar 

  23. P. Daniele, A. Maugeri, W. Oettli, Time-Dependent Traffic Equilibria, Journal of Optimization Theory and Applications 103, no. 3, 1999, 543–555.

    CrossRef  MATH  MathSciNet  Google Scholar 

  24. P. Daniele, A. Maugeri, W. Oettli, Variational Inequalities and Time–Dependent Traffic Equilibria, C. R. Acad. Sci. Paris t. 326, serie I, 1998, 1059–1062.

    Google Scholar 

  25. J. Diestel, Geometry of Banach Spaces - Selected topics, Springer-Verlag, Berlin, 1975.

    MATH  Google Scholar 

  26. P. Dupuis, H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics and Stochastics Reports 35, 1990, 31–62.

    MathSciNet  Google Scholar 

  27. P. Dupuis, A. Nagurney, Dynamical Systems and Variational Inequalities, Annals of Operations Research 44, 1993, 9–42.

    CrossRef  MATH  MathSciNet  Google Scholar 

  28. S. Giuffré, G. Idone, S. Pia, Some Classes of Projected Dynamical Systems in Banach Spaces and Variational Inequalities, Journal of Global Optimization 40, (1–3), 2008, 119–128.

    CrossRef  MATH  Google Scholar 

  29. S. Giuffré, G. Idone, S. Pia, Projected Dynamical Systems and Variational inequalities equivalence results, Journal of Nonlinear and Convex Analysis 7, no. 3, 2006, 453–463.

    MATH  MathSciNet  Google Scholar 

  30. S. Giuffré, S. Pia, Wireless Communication Densities and User-Oriented Traffic Equilibrium Problem, Proceedings of IEEE 2008 Wireless Communications, Networking and Mobile Computing (WICOM 2008), Dalian, China, October 12–14, 2008.

    Google Scholar 

  31. S. Giuffré, S. Pia, Weighted Traffic Equilibrium problem in non-pivot Hilbert spaces, Nonlinear Analysis, 2009, doi: 10.1016/j.na.2009.03.044.

    Google Scholar 

  32. G. Isac, M.G. Cojocaru, Variational Inequalities, Complementarity Problems and Pseudo-Monotonicity. Dynamical Aspects, in “Seminar on Fixed Point Theory Cluj-Napoca,” Proceedings of the International Conference on Nonlinear Operators, Differential Equations and Applications, Babes-Bolyai University of Cluj-Napoca, Vol. III, 2002, 41–62.

    Google Scholar 

  33. G. Isac, M.G. Cojocaru, The Projection Operator in a Hilbert Space and its Directional Derivative. Consequences for the Theory of Projected Dynamical Systems, Journal of Function Spaces and Applications 2, 2004, 71–95.

    MATH  MathSciNet  Google Scholar 

  34. S. Heikkila, Monotonone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics 181, Marcel Dekker, New York, 1994.

    Google Scholar 

  35. A. Nagurney, Network Economics: A Variational Inequality Approach, Second and Revised Edition, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.

    Google Scholar 

  36. M.A. Noor, Generalized multivalued quasi-variational inequalities. II, Comput. Math. Appl. 35, no. 5, 1998, 63–78.

    CrossRef  MATH  MathSciNet  Google Scholar 

  37. M.A. Noor, Generalized multivalued quasi-variational inequalities, Comput. Math. Appl. 31, no. 12, 1996, 1–13.

    CrossRef  MATH  MathSciNet  Google Scholar 

  38. M.A. Noor, Implicit dynamical systems and quasi variational inequalities, Applied Math. and Comput. 134, 2003, 69–81.

    CrossRef  MATH  Google Scholar 

  39. M. Patriksson, R.T. Rockafellar, Variational Geometry and Equilibrium, Equilibrium problems and Variational Models, Kluwer Academic Publishers, Dordrecht, 2003, 347–367.

    Google Scholar 

  40. F. Raciti, Equilibria Trajectories as Stationary Solutions of Infinite Dimensional Projected Dynamical Systems, Applied Mathematics Letters 17, 2004, 153–158.

    CrossRef  MATH  MathSciNet  Google Scholar 

  41. C. Ratti, R.M. Pulselli, S. Williams, D. Frenchman, Mobile Landscapes: Using location data from cell phones for urban analysis, Environment and Planning B. Planning and Design, 2006, 727–748.

    Google Scholar 

  42. C. Ratti, A. Sevtsuk, S. Huang, R. Pailer, Mobile Landscapes: Graz in Real Time, in Proceedings of the 3rd Symposium on LBS & TeleCartography, 28–30 November, 2007, Vienna, Austria.

    Google Scholar 

  43. R.T. Rockafellar, R. J-B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998.

    CrossRef  MATH  Google Scholar 

  44. W. Song, Z. Cao, The generalized Decomposition Theorem in Banach spaces and Its applications, J. Approx. Theory 129, 2004,167–181.

    CrossRef  MATH  MathSciNet  Google Scholar 

  45. G. Tian, J. Zhou, Quasi-variational inequality with non-compact sets, J. Math. Anal. Appl. 160, 1991, 583–595.

    CrossRef  MATH  MathSciNet  Google Scholar 

  46. F. Tinti, Numerical solution for pseudomonotone variational inequality problems by extragradient methods. Variational analysis and applications, Nonconvex Optim. Appl. 79, 2005, 1101–1128.

    CrossRef  MathSciNet  Google Scholar 

  47. E.H. Zarantonello, Projections on Convex sets in Hilbert space and spectral theory, Contributions to Nonlinear Functional Analysis, Mathematics Research Center, Madison, April 12–14 1971 (E.H. Zarantonello, ed.) Academic Press, New York, 1971, 237–424.

    Google Scholar 

  48. E.H. Zarantonello, Projectors on convex sets in reflexive Banach spaces, Technical Summary report 1768, Winsconsin Univ. Madison Mathematics Research Center, 1977.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Catania, Catania, Italy

    Patrizia Daniele, Antonino Maugeri & Stephane Pia

  2. D.I.M.E.T., Mediterranean University, Reggio Calabria, Italy

    Sofia Giuffré

Authors
  1. Patrizia Daniele
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  2. Sofia Giuffré
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  3. Antonino Maugeri
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  4. Stephane Pia
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Corresponding author

Correspondence to Patrizia Daniele .

Editor information

Editors and Affiliations

  1. Dept. Industrial & Systems, Engineering, University of Florida, Weil Hall 303, Gainesville, 32611-6595, Florida, USA

    Panos M. Pardalos

  2. Zagoras Street 4, Athens, 151 25, Greece

    Themistocles M. Rassias

  3. School of Mathematical Sciences, Rochester Institute of Technology, Lomb Memorial Drive 85, Rochester, 14623-5602, USA

    Akhtar A. Khan

Additional information

Dedicated to the memory of Professor George Isac

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Daniele, P., Giuffré, S., Maugeri, A., Pia, S. (2010). A Panoramic View on Projected Dynamical Systems. In: Pardalos, P., Rassias, T., Khan, A. (eds) Nonlinear Analysis and Variational Problems. Springer Optimization and Its Applications, vol 35. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0158-3_17

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