Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Existence of solutions to projected differential equations in Hilbert spaces
HTML articles powered by AMS MathViewer

by Monica-Gabriela Cojocaru and Leo B. Jonker PDF
Proc. Amer. Math. Soc. 132 (2004), 183-193 Request permission


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.
  • 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, DOI 10.1007/978-3-642-69512-4
  • 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
  • COJOCARU, M. G., Projected Dynamical Systems on Hilbert Spaces, Ph.D. Thesis, Queen’s University, 2002.
  • John W. Green, Harmonic functions in domains with multiple boundary points, Amer. J. Math. 61 (1939), 609–632. MR 90, DOI 10.2307/2371316
  • 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, DOI 10.1080/17442509108833688
  • 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, DOI 10.1007/BF02073589
  • 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 335906, DOI 10.1016/0022-247X(73)90192-3
  • 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
  • HIPFEL, D., The Nonlinear Differential Complementarity Problem, Ph. D. Thesis, Rensselaer Polytechnic Institute (1993).
  • 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, DOI 10.1142/9789812830432
  • George Isac, Complementarity problems, Lecture Notes in Mathematics, vol. 1528, Springer-Verlag, Berlin, 1992. MR 1222647, DOI 10.1007/BFb0084653
  • 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.
  • 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.
  • 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
  • Anna Nagurney, Network economics: a variational inequality approach, Advances in Computational Economics, vol. 1, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 1205777, DOI 10.1007/978-94-011-2178-1
  • NAGURNEY, A. and ZHANG, D., Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer Academic Publishers (1996).
  • 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, DOI 10.1016/0165-1889(94)00843-2
  • 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.
  • NAGURNEY, A. and SIOKOS, S., Financial Networks: Statics and Dynamics, Springer-Verlag, New York, 1997.
  • NAGURNEY, A., TAKAYAMA, T. and ZHANG, D., Projected dynamical systems, modeling and computation of spatial network equilibria, Networks 26, (1995), 69-85.
  • PAPPALARDO, M. and PASSACANTANDO, M., Stability for equilibrium problems: from variational inequalities to dynamical systems, J. Opt. Theory Appl. 113, (2002), 567-582.
  • Alexander Shapiro, Existence and differentiability of metric projections in Hilbert spaces, SIAM J. Optim. 4 (1994), no. 1, 130–141. MR 1260410, DOI 10.1137/0804006
  • 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
  • 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
Similar Articles
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:
  • Leo B. Jonker
  • Affiliation: Department of Mathematics and Statistics, Jeffery Hall, Room 508, Queen’s University, Kingston, Ontario, Canada K7M 2W8
  • Email:
  • 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
  • © Copyright 2003 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 132 (2004), 183-193
  • MSC (2000): Primary 34A12, 34A36; Secondary 34A60, 49J40
  • DOI:
  • MathSciNet review: 2021261