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 2024 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.

 

Monotonicity and existence of periodic orbits for projected dynamical systems on Hilbert spaces
HTML articles powered by AMS MathViewer

by Monica-Gabriela Cojocaru
Proc. Amer. Math. Soc. 134 (2006), 793-804
DOI: https://doi.org/10.1090/S0002-9939-05-08006-8
Published electronically: July 21, 2005
PDF | Request permission

Abstract:

We present here results about the existence of periodic orbits for projected dynamical systems (PDS) under Minty-Browder monotonicity conditions. The results are formulated in the general context of a Hilbert space of arbitrary (finite or infinite) dimension. The existence of periodic orbits for such PDS is deduced by means of nonlinear analysis, using a fixed point approach. It is also shown how occurrence of periodic orbits is intimately related to that of critical points (equilibria) of a PDS in certain cases.
References
  • 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).
  • Cojocaru, M. G., Infinite-dimensional projected dynamics and the 1-dimensional obstacle problem, to appear in J. Funct. Spaces Appl. (2005).
  • Monica-Gabriela Cojocaru and Leo B. Jonker, Existence of solutions to projected differential equations in Hilbert spaces, Proc. Amer. Math. Soc. 132 (2004), no. 1, 183–193. MR 2021261, DOI 10.1090/S0002-9939-03-07015-1
  • Cojocaru, M. G., Daniele, P. and Nagurney, A., Projected dynamical systems and evolutionary (time-dependent) variational inequalities on Hilbert spaces with applications, to appear in J. Optim. Theory Appl. (2005).
  • Bernard Cornet, Existence of slow solutions for a class of differential inclusions, J. Math. Anal. Appl. 96 (1983), no. 1, 130–147. MR 717499, DOI 10.1016/0022-247X(83)90032-X
  • J. Dong, D. Zhang, and A. Nagurney, A projected dynamical systems model of general financial equilibrium with stability analysis, Math. Comput. Modelling 24 (1996), no. 2, 35–44. MR 1403528, DOI 10.1016/0895-7177(96)00088-X
  • 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
  • Heikkilaa, S., Monotonone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, 181, M. Dekker (1994).
  • George Isac and M. Gabriela Cojocaru, Functions without exceptional family of elements and the solvability of variational inequalities on unbounded sets, Topol. Methods Nonlinear Anal. 20 (2002), no. 2, 375–391. MR 1962226, DOI 10.12775/TMNA.2002.041
  • George Isac and Monica Gabriela Cojocaru, Variational inequalities, complementarity problems and pseudo-monotonicity. Dynamical aspects, Semin. Fixed Point Theory Cluj-Napoca 3 (2002), 41–62. International Conference on Nonlinear Operators, Differential Equations and Applications (Cluj-Napoca, 2001). MR 1929747
  • George Isac and Monica G. Cojocaru, The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems, J. Funct. Spaces Appl. 2 (2004), no. 1, 71–95. MR 2028181, DOI 10.1155/2004/543714
  • S. Karamardian and S. Schaible, Seven kinds of monotone maps, J. Optim. Theory Appl. 66 (1990), no. 1, 37–46. MR 1061909, DOI 10.1007/BF00940531
  • 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
  • M. A. Krasnosel′skiĭ and P. P. Zabreĭko, Geometrical methods of nonlinear analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 263, Springer-Verlag, Berlin, 1984. Translated from the Russian by Christian C. Fenske. MR 736839, DOI 10.1007/978-3-642-69409-7
  • George J. Minty, On variational inequalities for monotone operators. I, Adv. in Math. 30 (1978), no. 1, 1–7. MR 511738, DOI 10.1016/0001-8708(78)90128-7
  • D. Zhang and A. Nagurney, On the stability of projected dynamical systems, J. Optim. Theory Appl. 85 (1995), no. 1, 97–124. MR 1330844, DOI 10.1007/BF02192301
  • Nagurney, A. and Zhang, D., Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer Academic Publishers (1996).
  • Nagurney, A. and Siokos, S., Financial Networks: Statics and Dynamics, Springer-Verlag (1997).
  • Nagurney, A., Dupuis, P. and Zhang, D., A Dynamical Systems Approach for Network Oligopolies and Variational Inequalities, Ann. Regional Science 28 (1994), 263-283.
  • Nagurney, A., Takayama, T. and Zhang, D., Projected dynamical systems modelling and computation of spatial networks equilibria, Networks, Vol. 26(1995), 69-85.
  • 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
  • Guido Stampacchia, Variational inequalities, Theory and Applications of Monotone Operators (Proc. NATO Advanced Study Inst., Venice, 1968) Edizioni “Oderisi”, Gubbio, 1969, pp. 101–192. MR 0425699
  • 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) Publ. Math. Res. Center Univ. Wisconsin, No. 27, Academic Press, New York, 1971, pp. 237–341. MR 0388177
  • D. Zhang and A. Nagurney, Stability analysis of an adjustment process for oligopolistic market equilibrium modeled as a projected dynamical system, Optimization 36 (1996), no. 3, 263–285. MR 1419267, DOI 10.1080/02331939608844183
  • D. Zhang and A. Nagurney, Formulation, stability, and computation of traffic network equilibria as projected dynamical systems, J. Optim. Theory Appl. 93 (1997), no. 2, 417–444. MR 1448567, DOI 10.1023/A:1022610325133
Similar Articles
Bibliographic Information
  • Monica-Gabriela Cojocaru
  • Affiliation: Department of Mathematics and Statistics, MacNaughton Hall, Room 548, University of Guelph, Guelph, Ontario, Canada N1G 2W1
  • Email: mcojocar@uoguelph.ca
  • Received by editor(s): August 12, 2004
  • Received by editor(s) in revised form: October 6, 2004, and October 18, 2004
  • Published electronically: July 21, 2005
  • Additional Notes: This research was funded by NSERC Discovery Grant No. 045997.
  • Communicated by: Carmen C. Chicone
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 793-804
  • MSC (2000): Primary 34A36, 34C25, 49J40; Secondary 37N40, 34A60
  • DOI: https://doi.org/10.1090/S0002-9939-05-08006-8
  • MathSciNet review: 2180897