Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Existence of solutions to projected differential equations in Hilbert spaces


Authors: Monica-Gabriela Cojocaru and Leo B. Jonker
Journal: Proc. Amer. Math. Soc. 132 (2004), 183-193
MSC (2000): Primary 34A12, 34A36; Secondary 34A60, 49J40
Published electronically: May 22, 2003
MathSciNet review: 2021261
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove existence and uniqueness of integral curves to the (discontinuous) vector field that results when a Lipschitz continuous vector field on a Hilbert space of any dimension is projected on a non-empty, closed and convex subset.


References [Enhancements On Off] (What's this?)

  • [A-C] Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR 755330 (85j:49010)
  • [B-C] Claudio Baiocchi and António Capelo, Variational and quasivariational inequalities, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984. Applications to free boundary problems; Translated from the Italian by Lakshmi Jayakar. MR 745619 (86e:49018)
  • [Cj] COJOCARU, M. G., Projected Dynamical Systems on Hilbert Spaces, Ph.D. Thesis, Queen's University, 2002.
  • [D-S] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162 (90g:47001a)
    Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral theory. Selfadjoint operators in Hilbert space; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1963 original; A Wiley-Interscience Publication. MR 1009163 (90g:47001b)
    Nelson Dunford and Jacob T. Schwartz, Linear operators. Part III, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral operators; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1971 original; A Wiley-Interscience Publication. MR 1009164 (90g:47001c)
  • [D-I] Paul Dupuis and Hitoshi Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics Stochastics Rep. 35 (1991), no. 1, 31–62. MR 1110990 (93e:60110)
  • [D-N] Paul Dupuis and Anna Nagurney, Dynamical systems and variational inequalities, Ann. Oper. Res. 44 (1993), no. 1-4, 9–42. Advances in equilibrium modeling, analysis and computation. MR 1246835 (94k:49009), http://dx.doi.org/10.1007/BF02073589
  • [He] Claude Henry, An existence theorem for a class of differential equations with multivalued right-hand side, J. Math. Anal. Appl. 41 (1973), 179–186. MR 0335906 (49 #684)
  • [Hk] Seppo Heikkilä and V. Lakshmikantham, Monotone iterative techniques for discontinuous nonlinear differential equations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 181, Marcel Dekker, Inc., New York, 1994. MR 1280028 (95d:34002)
  • [Hi] HIPFEL, D., The Nonlinear Differential Complementarity Problem, Ph. D. Thesis, Rensselaer Polytechnic Institute (1993).
  • [Ra] Donald H. Hyers, George Isac, and Themistocles M. Rassias, Topics in nonlinear analysis & applications, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. MR 1453115 (98g:47001)
  • [Is] George Isac, Complementarity problems, Lecture Notes in Mathematics, vol. 1528, Springer-Verlag, Berlin, 1992. MR 1222647 (94h:49002)
  • [Is-C1] ISAC, G. and COJOCARU, M. G., The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems, preprint, 2002.
  • [Is-C2] ISAC, G. and COJOCARU, M. G., 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, September 2002, Romania), Babes-Bolyai University of Cluj-Napoca, Vol. III (2002), 41-62.
  • [K-S] David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696 (81g:49013)
  • [Na1] Anna Nagurney, Network economics: a variational inequality approach, Advances in Computational Economics, vol. 1, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 1205777 (93m:90002)
  • [Na2] NAGURNEY, A. and ZHANG, D., Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer Academic Publishers (1996).
  • [Na3] Anna Nagurney and Ding Zhang, On the stability of an adjustment process for spatial price equilibrium modeled as a projected dynamical system, J. Econom. Dynam. Control 20 (1996), no. 1-3, 43–62. MR 1364984 (96j:90021), http://dx.doi.org/10.1016/0165-1889(94)00843-2
  • [Na4] NAGURNEY, A., DUPUIS P. and ZHANG, D., A dynamical systems approach for network oligopolies and variational inequalities, Annals of Regional Science 28, (1994), 263-283.
  • [Na5] NAGURNEY, A. and SIOKOS, S., Financial Networks: Statics and Dynamics, Springer-Verlag, New York, 1997.
  • [Na6] NAGURNEY, A., TAKAYAMA, T. and ZHANG, D., Projected dynamical systems, modeling and computation of spatial network equilibria, Networks 26, (1995), 69-85.
  • [P] PAPPALARDO, M. and PASSACANTANDO, M., Stability for equilibrium problems: from variational inequalities to dynamical systems, J. Opt. Theory Appl. 113, (2002), 567-582.
  • [S] Alexander Shapiro, Existence and differentiability of metric projections in Hilbert spaces, SIAM J. Optim. 4 (1994), no. 1, 130–141. MR 1260410 (94m:90111), http://dx.doi.org/10.1137/0804006
  • [Sw] Charles Swartz, An introduction to functional analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 157, Marcel Dekker, Inc., New York, 1992. MR 1156078 (93c:46002)
  • [Z] Eduardo H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. I. Projections on convex sets, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 237–341. MR 0388177 (52 #9014)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34A12, 34A36, 34A60, 49J40

Retrieve articles in all journals with MSC (2000): 34A12, 34A36, 34A60, 49J40


Additional Information

Monica-Gabriela Cojocaru
Affiliation: Department of Mathematics and Statistics, Jeffery Hall, Room 207, Queen’s University, Kingston, Ontario, Canada K7M 2W8
Address at time of publication: Department of Mathematics and Statistics, Room 536 MacNaughton Building, University of Guelph, Guelph, Ontario, Canada N1G 2W1
Email: monica@mast.queensu.ca

Leo B. Jonker
Affiliation: Department of Mathematics and Statistics, Jeffery Hall, Room 508, Queen’s University, Kingston, Ontario, Canada K7M 2W8
Email: leo@mast.queensu.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07015-1
PII: S 0002-9939(03)07015-1
Received by editor(s): June 27, 2002
Received by editor(s) in revised form: September 9, 2002
Published electronically: May 22, 2003
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2003 American Mathematical Society