Skip to main content

Projected Dynamical Systems and Evolutionary Variational Inequalities via Hilbert Spaces with Applications1

Journal of Optimization Theory and Applications volume 127, pages 549–563 (2005)Cite this article

Abstract

In this paper, we make explicit the connection between projected dynamical systems on Hilbert spaces and evolutionary variational inequalities. We give a novel formulation that unifies the underlying constraint sets for such inequalities, which arise in time-dependent traffic network, spatial price equilibrium, and a variety of financial equilibrium problems. We emphasize the importance of the results in applications and provide a traffic network numerical example in which we compute the curve of equilibria.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. P. Dupuis A. Nagurney (1993) ArticleTitleDynamical Systems and Variational Inequalities Annals of Operations Research 44 9–42 Occurrence Handle10.1007/BF02073589 Occurrence Handle1246835 Occurrence Handle0785.93044

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Dupuis H. Ishii (1990) ArticleTitleOn Lipschitz Continuity of the Solution Mapping to the Skorokhod Problem with Applications Stochastics and Stochastics Reports 35 31–62 Occurrence Handle1110990

    MathSciNet  Google Scholar 

  3. C. Henry (1973) ArticleTitleAn Existence Theorem for a Class of Differential Equations with Multivalued Right-Hand Sides Journal of Mathematical Analysis and Applications 41 179–186 Occurrence Handle10.1016/0022-247X(73)90192-3 Occurrence Handle0262.49019 Occurrence Handle335906

    Article  MATH  MathSciNet  Google Scholar 

  4. B. Cornet (1983) ArticleTitleExistence of Slow Solutions for a Class of Differential Inclusions Journal of Mathematical Analysis and Application 96 130–147 Occurrence Handle0558.34011 Occurrence Handle717499

    MATH  MathSciNet  Google Scholar 

  5. J.P. Aubin A. Cellina (1984) Differential Inclusions Springer Verlag Berlin, Germany Occurrence Handle0538.34007

    MATH  Google Scholar 

  6. D. Hipfel (1993) The Nonlinear Differential Complementarity Problem Rensselaer Polytechnic Institute Troy, New York

    Google Scholar 

  7. Isac G., Cojocaru M.G., 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 (to appear)

  8. G. Isac M.G. Cojocaru (2002) ArticleTitleVariational Inequalities, Complementarity Problems, and Pseudomonotonicity: Dynamical Aspects Seminar on Fixed-Point Theory, Babes-Bolyai University, Cluj-Napoca, Romania 3 41–62 Occurrence Handle1929747 Occurrence Handle1025.37020

    MathSciNet  MATH  Google Scholar 

  9. Cojocaru M.G., (2005). Infinite-Dimensional Projected Dynamics and the 1-Dimensional Obstacle Problem to appear in Journal of Function Spaces and applications.

  10. M.G. Cojocaru L.B. Jonker (2004) ArticleTitleExistence of Solutions to Projected Differential Equations in Hilbert Space Proceedings of the American Mathematical Society 132 183–193 Occurrence Handle10.1090/S0002-9939-03-07015-1 Occurrence Handle2021261 Occurrence Handle1055.34118

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Nagurney (1993) Network Economics: A Variational Inequality Approach Kluwer Academic Publishers Dordrecht, Netherlands Occurrence Handle0873.90015

    MATH  Google Scholar 

  12. A.A. Cournot (1897) Researches into the Mathematical Principles of the Theory of Wealth, 1838 MacMillan London, England

    Google Scholar 

  13. S. Enke (1951) ArticleTitleEquilibrium among Spatially Separated Markets: Solution by Electronic Analogue Econometrica 10 40–47

    Google Scholar 

  14. P.A. Samuelson (1952) ArticleTitleSpatial Price Equilibrium and Linear Programming American Economic Review 42 283–303

    Google Scholar 

  15. T. Takayama G.G. Judge (1971) Spatial and Temporal Price and Allocation Models North-Holland Amsterdam, Netherlands

    Google Scholar 

  16. M.J. Beckmann C.B. McGuire C.B. Winsten (1956) Studies in the Economics of Transportation Yale University Press New Haven, Connecticut

    Google Scholar 

  17. J.G. Wardrop (1952) ArticleTitleSome Theoretical Aspects of Road Traffic Research Proceedings of the Institution of Civil Engineers 1 325–378

    Google Scholar 

  18. M.J. Smith (1979) ArticleTitleThe Existence, Uniqueness, and Stability of Traffic Equilibrium Transportation Research 38 295–304

    Google Scholar 

  19. S. Dafermos (1980) ArticleTitleTraffic Equilibrium and Variational Inequalities Transportation Science 14 42–54 Occurrence Handle639734

    MathSciNet  Google Scholar 

  20. M. Patriksson (1994) The Traffic Assignment Problem: Models and Methods VSP BV Utrecht, Netherlands

    Google Scholar 

  21. A. Nagurney D. Zhang (1995) ArticleTitleOn the Stability of Projected Dynamical Systems Journal of Optimization Theory and Applications 85 97–124 Occurrence Handle1330844 Occurrence Handle0837.93063

    MathSciNet  MATH  Google Scholar 

  22. A. Nagurney D. Zhang (1996) Projected Dynamical Systems and Variational Inequalities with Applications Kluwer Academic Publishers Boston, Massachusetts

    Google Scholar 

  23. A. Nagurney D. Zhang (1997) ArticleTitleFormulation, Stability, and Computation of Traffic Network Equilibria as Projected Dynamical Systems Journal of Optimization Theory and Applications 93 417–444 Occurrence Handle1448567 Occurrence Handle0899.90085

    MathSciNet  MATH  Google Scholar 

  24. A. Nagurney D. Zhang (1997) ArticleTitleProjected Dynamical Systems in the Formulation, Stability Analysis, and Computation of Fixed Demand Traffic Network Equilibria Transportation Science 31 147–158 Occurrence Handle0886.90072

    MATH  Google Scholar 

  25. A. Nagurney J. Dong D. Zhang (1996) ArticleTitleA Projected Dynamical Systems Model of General Financial Equilibrium with Stability Analysis Mathematical and Computer Modelling 24 35–44 Occurrence Handle1403528 Occurrence Handle0858.90020

    MathSciNet  MATH  Google Scholar 

  26. A. Nagurney S. Siokos (1997) Financial Networks: Statics and Dynamics Springer Verlag Heidelberg, Germany

    Google Scholar 

  27. A. Nagurney K. Ke (2001) ArticleTitleFinancial Networks with Intermediation Quantitative Finance 1 309–317 Occurrence Handle1904818

    MathSciNet  Google Scholar 

  28. A. Nagurney J. Cruz (2004) ArticleTitleDynamics of International Financial Networks with Risk Management Quantitative Finance 156 194–212 Occurrence Handle2071837 Occurrence Handle1044.90009

    MathSciNet  MATH  Google Scholar 

  29. A. Nagurney J. Cruz D. Matsypura (2003) ArticleTitleDynamics of Global Supply Chain Supernetworks Mathematical and Computer Modelling 37 963–983 Occurrence Handle10.1016/S0895-7177(03)00112-2 Occurrence Handle1989519 Occurrence Handle1080.90012

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Nagurney J. Dong D. Zhang (2002) ArticleTitleA Supply Chain Network Equilibrium Model Transportation Research 38E 281–303

    Google Scholar 

  31. M.G. Cojocaru (2004) Dynamics of Human Migration Preprint, University of Guelph Guelph, Ontario, Canada

    Google Scholar 

  32. G. Isac V.A. Bulasky V.V. Kalashnikov (2002) Complementarity, Equilibrium, Efficiency, and Economics Kluwer Academic Publishers Dordrecht, Netherlands Occurrence Handle1081.90001

    MATH  Google Scholar 

  33. D. Kinderlehrer G. Stampacchia (1980) An Introduction to Variational Inequalities and Their Applications Academic Press New York, NY Occurrence Handle0457.35001

    MATH  Google Scholar 

  34. J.L. Lions G. Stampacchia (1967) ArticleTitleVariational Inequalities Communications in Pure and Applied Mathematics 22 493–519 Occurrence Handle216344

    MathSciNet  Google Scholar 

  35. H. Brezis (1967) Inequations d’Evolution Abstraites Comptes Rendus de l’Academie des Sciences Paris

    Google Scholar 

  36. J. Steinbach (1998) ArticleTitleOn a Variational Inequality Containing a Memory Term with an Application in Electro-Chemical Machining Journal of Convex Analysis 5 63–80 Occurrence Handle0908.49011 Occurrence Handle1649441

    MATH  MathSciNet  Google Scholar 

  37. P. Daniele A. Maugeri W. Oettli (1998) ArticleTitleVariational Inequalities and Time-Dependent Traffic Equilibria Comptes Rendus de l’Academie des Sciences, Paris, Serie I 326 1059–1062 Occurrence Handle1647182 Occurrence Handle0910.90131

    MathSciNet  MATH  Google Scholar 

  38. P. Daniele A. Maugeri W. Oettli (1999) ArticleTitleTime-Dependent Variational Inequalities Journal of Optimization Theory and Applications 104 543–555 Occurrence Handle1727254

    MathSciNet  Google Scholar 

  39. P. Daniele A. Maugeri (2001) On Dynamical Equilibrium Problems and Variational Inequalities F. Giannessi A. Maugeri P.M. Pardalos (Eds) Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models Kluwer Academic Publishers Dordrecht, Netherlands 59–69

    Google Scholar 

  40. P. Daniele (2003) ArticleTitleEvolutionary Variational Inequalities and Economic Models for Demand Supply Markets M3AS: Mathematical Models and Methods in Applied Sciences 4 471–489 Occurrence Handle02084094 Occurrence Handle1976299

    MATH  MathSciNet  Google Scholar 

  41. Daniele P., Time-Dependent Spatial Price Equilibrium Problem: Existence and Stability Results for the Quantity Formulation Model. Journal of Global Optimization, 2002 (to appear)

  42. P. Daniele (2003) Variational Inequalities for Evolutionary Financial Equilibrium A. Nagurney (Eds) Innovations in Financial and Economics Networks Edward Elgar Publishing Cheltenham, England 84–108

    Google Scholar 

  43. L. Scrimali (2004) ArticleTitleQuasi-Variational Inequalities in Transportation Networks Mathematical Models and Methods, in Applied Sciences 14 1541–1560 Occurrence Handle10.1142/S0218202504003714 Occurrence Handle1069.90026 Occurrence Handle2095302

    Article  MATH  MathSciNet  Google Scholar 

  44. M.C.J. Bliemer P.H.L. Bovy (2003) ArticleTitleQuasivariational Inequality Formulation of the Multiclass Dynamic Traffic Assignment Problem Transportation Research 37 501–519

    Google Scholar 

  45. B. Ran D.E. Boyce (1996) Modeling Dynamic Transportation Networks: An Intelligent System-Oriented Approach EditionNumber2 Springer Verlag Berlin, Germany Occurrence Handle0898.90004

    MATH  Google Scholar 

  46. C. Baiocchi A. Capelo (1984) Variational and Quasivariational Inequalities: Applications to Free Boundary Problems John Wiley and Sons New York, NY Occurrence Handle0551.49007

    MATH  Google Scholar 

  47. J. Gwinner (2003) Time-Dependent Variational Inequalities: Some Recent Trends P. Daniele F. Giannessi A. Maugeri (Eds) Equilibrium Problems and Variational Models Kluwer Academic Publishers Dordrecht, Netherlands 225–264

    Google Scholar 

  48. F. Raciti (2004) ArticleTitleEquilibria Trajectories as Stationary Solutions of Infinite-Dimensional Projected Dynamical Systems Applied Mathematics Letters 17 153–158 Occurrence Handle10.1016/S0893-9659(04)90026-2 Occurrence Handle1058.49008 Occurrence Handle2034761

    Article  MATH  MathSciNet  Google Scholar 

  49. F. Raciti (2003) On the Calculation of Equilibrium in Time-Dependent Traffic Networks P. Daniele F. Giannessi A. Maugeri (Eds) Equilibrium Problems and Variational Models Kluwer Academic Publisher Dordrecht, Netherlands

    Google Scholar 

  50. G. Stampacchia (1969) Variational Inequalities: Theory and Applications of Monotone Operators Edition Oderisi Gubbio, Italy 101–192

    Google Scholar 

  51. S. Dafermos A. Nagurney (1984) ArticleTitleSensitivity Analysis for the General Spatial Economic Equilibrium Problem Operations Research 32 1069–1086 Occurrence Handle770509 Occurrence Handle10.1287/opre.32.5.1069 Occurrence Handle0562.90009

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada

    M. G. Cojocaru (Assistant Professor)

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

    P. Daniele (Associate Professor)

  3. Department of Finance and Operations Management, University of Massachusetts, Amherst, Massachusetts

    A. Nagurney (John F. Smith Memorial Professor)

Authors
  1. M. G. Cojocaru
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Daniele
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Nagurney
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

1This research was funded by the Rockefeller Foundation, Bellagio Center Program. The first author was also supported by NSERC Discovery Grant 045997 and the third author by NSF Grant IIS-0002647. The authors thank Professors George Isac, Antonino Maugeri, and Panos Pardalos for support and inspiration; they are indebted to Professor Antonino Maugeri for assistance throughout their residency at Bellagio, Italy in March 2004.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Cojocaru, M.G., Daniele, P. & Nagurney, A. Projected Dynamical Systems and Evolutionary Variational Inequalities via Hilbert Spaces with Applications1 . J Optim Theory Appl 127, 549–563 (2005). https://doi.org/10.1007/s10957-005-7502-0

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-005-7502-0

Keywords

  • Projected dynamical systems
  • evolutionary variational inequalities
  • traffic network equilibrium
  • spatial price equilibrium
  • financial equilibrium