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Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Monotonicity and existence of periodic orbits for projected dynamical systems on Hilbert spaces


Author: Monica-Gabriela Cojocaru
Journal: Proc. Amer. Math. Soc. 134 (2006), 793-804
MSC (2000): Primary 34A36, 34C25, 49J40; Secondary 37N40, 34A60
Posted: July 21, 2005
MathSciNet review: 2180897
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Abstract | References | Similar Articles | Additional Information

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

  • [A-C] Aubin, J. P. and Cellina, A., Differential Inclusions, Springer-Verlag, Berlin (1984). MR 0755330 (85j:49010)
  • [B-C] Baiocchi, C. and Capelo, A., Variational and Quasivariational Inequalities. Applications to Free Boundary Problems., J. Wiley and Sons, (1984). MR 0745619 (86e:49018)
  • [Coj1] Cojocaru, M. G., Projected Dynamical Systems on Hilbert Spaces, Ph. D. Thesis, Queen's University, (2002).
  • [Coj2] Cojocaru, M. G., Infinite-dimensional projected dynamics and the 1-dimensional obstacle problem, to appear in J. Funct. Spaces Appl. (2005).
  • [C-J] Cojocaru, M. G. and Jonker, L. B., Projected differential equations in Hilbert spaces, Proc. Amer. Math. Soc., Volume 132, Number 1, 183-193(2004). MR 2021261 (2004k:34011)
  • [C-D-N] 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).
  • [Co] Cornet, B., Existence of Slow Solutions for a Class of Differential Inclusions, J. Math. Anal. Appl. 96 (1983), 130-147. MR 0717499 (86a:34029)
  • [D-Z-N] Dong, J., Zhang, D. and Nagurney, A., A projected dynamical system model of general financial equilibrium with Stability Analysis, Math. Comput. Modelling, Vol. 24, No. 2(1996), 35-444. MR 1403528
  • [D-I] Dupuis, P. and Ishii, H., On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics Stochastics Rep., Vol. 35(1990), 31-62. MR 1110990 (93e:60110)
  • [D-N] Dupuis, P. and Nagurney, A., Dynamical systems and variational inequalities, Ann. Oper. Res. 44, (1993), 9-42. MR 1246835 (94k:49009)
  • [He] Henry, C., An existence theorem for a class of diffferential equations with multivalued right hand sides, J. Math. Anal. Appl. 41 (1973), 179-186. MR 0335906 (49:684)
  • [Hk] Heikkilaa, S., Monotonone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, 181, M. Dekker (1994).
  • [Is-C1] Isac, G. and Cojocaru, M. G., Functions without exceptional family of elements and the solvability of variational inequalities on unbounded sets, Topol. Methods Nonlinear Anal., Vol. 20 (2002), 375-391. MR 1962226 (2003k:49017)
  • [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, Babes-Bolyai University of Cluj-Napoca, Vol. III(2002), 41-62.MR 1929747 (2003k:49018)
  • [Is-C3] 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, J. Func. Spaces Appl., Vol. 2, Number 1 (2004), 71-95. MR 2028181 (2004m:34136)
  • [Ka] Karamardian, S. and Schaible, S., Seven kinds of monotone maps, J. Optim. Theory Appl., Vol. 66, No. 1(1990), 37-46. MR 1061909 (91e:26016)
  • [K-S] Kinderlehrer, D. and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press (1980).MR 0567696 (81g:49013)
  • [K-Z] Krasnoselskii, M. A. and Zabreiko, P. P., Geometrical Methods of Nonlinear Analysis, Springer-Verlag, A Series of Comprehensive Studies in Mathematics, Vol. 263(1984). MR 0736839 (85b:47057)
  • [Mi] Minty, G. J., On variational inequalities for monotone operators, Adv. in Math. 30(1978), 1-7. MR 0511738 (80c:47045)
  • [N-Z1] Nagurney, A. and Zhang, D., On the Stability of Projected Dynamical Systems, J. Optim. Theory Appl. 85 (1995), 97-124.MR 1330844 (96f:34066)
  • [N-Z3] Nagurney, A. and Zhang, D., Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer Academic Publishers (1996).
  • [N-S] Nagurney, A. and Siokos, S., Financial Networks: Statics and Dynamics, Springer-Verlag (1997).
  • [N-D-Z] Nagurney, A., Dupuis, P. and Zhang, D., A Dynamical Systems Approach for Network Oligopolies and Variational Inequalities, Ann. Regional Science 28 (1994), 263-283.
  • [N-T-Z1] Nagurney, A., Takayama, T. and Zhang, D., Projected dynamical systems modelling and computation of spatial networks equilibria, Networks, Vol. 26(1995), 69-85.
  • [Sh] Shapiro, A. S., Existence and differentiability of metric projections in Hilbert spaces, SIAM J. Optim. 4(1994), 130-141. MR 1260410 (94m:90111)
  • [St] Stampacchia, G. Variational Inequalities, Theory and Applications of Monotone Operators, Proc. NATO Advanced Study Institute, Venice 1968, Ed. Oderisi, Gubbio, Italy (1969), 101-192.MR 0425699 (54:13652)
  • [Za] Zarantonello, E., Projections on convex sets in Hilbert space and spectral theory, Contributions to Nonlinear Functional Analysis, Publ. No. 27, Math. Res. Center, Univ. Wisconsin, Academic Press (1971), 237-424. MR 0388177 (52:9014); MR 0388178 (52:9015)
  • [Z-N1] Zhang, D. and Nagurney, A., Stability analysis of an adjustment process for oligopolistic market equilibrium modeled as a projected dynamical system, Optimization, Vol. 36, No. 2(1996), 263-285. MR 1419267
  • [Z-N2] Zhang, D. and Nagurney, A., Formulation, stability and computation of traffic network equilibria as projected dynamical systems, J. Optim. Theory Appl., Vol. 93, No. 2(1997), 417-444. MR 1448567 (98b:90161)

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Additional 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

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08006-8
PII: S 0002-9939(05)08006-8
Received by editor(s): August 12, 2004
Received by editor(s) in revised form: October 6, 2004 and October 18, 2004
Posted: July 21, 2005
Additional Notes: This research was funded by NSERC Discovery Grant No. 045997.
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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