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

Published electronically:
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.

**[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****[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****[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]**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**, 10.1090/S0002-9939-03-07015-1**[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]**Bernard Cornet,*Existence of slow solutions for a class of differential inclusions*, J. Math. Anal. Appl.**96**(1983), no. 1, 130–147. MR**717499**, 10.1016/0022-247X(83)90032-X**[D-Z-N]**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**, 10.1016/0895-7177(96)00088-X**[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****[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**, 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****[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]**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****[Is-C2]**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****[Is-C3]**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**, 10.1155/2004/543714**[Ka]**S. Karamardian and S. Schaible,*Seven kinds of monotone maps*, J. Optim. Theory Appl.**66**(1990), no. 1, 37–46. MR**1061909**, 10.1007/BF00940531**[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****[K-Z]**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****[Mi]**George J. Minty,*On variational inequalities for monotone operators. I*, Adv. in Math.**30**(1978), no. 1, 1–7. MR**511738**, 10.1016/0001-8708(78)90128-7**[N-Z1]**D. Zhang and A. Nagurney,*On the stability of projected dynamical systems*, J. Optim. Theory Appl.**85**(1995), no. 1, 97–124. MR**1330844**, 10.1007/BF02192301**[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]**Alexander Shapiro,*Existence and differentiability of metric projections in Hilbert spaces*, SIAM J. Optim.**4**(1994), no. 1, 130–141. MR**1260410**, 10.1137/0804006**[St]**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****[Za]**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**

Eduardo H. Zarantonello,*Projections on convex sets in Hilbert space and spectral theory. II. Spectral theory*, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 343–424. MR**0388178****[Z-N1]**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**, 10.1080/02331939608844183**[Z-N2]**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**, 10.1023/A:1022610325133

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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

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

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.