The theory of generalized projections both in non-pivot Hilbert spaces and strictly convex and smooth Banach spaces is developed and the related theory of projected dynamical systems is highlighted. A particular emphasis is given to the equivalence between solutions of variational inequality and critical points of projected dynamical systems.
- Hilbert Space
- Banach Space
- Variational Inequality
- Equilibrium Problem
- Duality Pairing
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Unable to display preview. Download preview PDF.
V. Acary, B. Brogliato, A. Daniilidis, C. Lemaréchal, On the Equivalence Between Complementarity Systems, Projected Systems and Unilateral Differential Inclusions, Rapport de Recherche 5107 I.N.R.I.A, Janvier 2004.
Ya. I. Alber, Metric and generalized projection operators in Banach spaces: proprieties and applications, A. Kartsatos (Ed.), Theory and Applications of Nonlinear Operators of Monotone and Accretive Type, Marcel Dekker, New York, 1996, 15–50.
Ya. I. Alber, Decomposition theorem in Banach spaces, Field Inst. Comm. 25, 2000, 77–99.
Ya. I. Alber, J-C. Yao, On the projection dynamical systems in Banach spaces, Taiwanese Journal of Mathematics, (to appear).
J-P. Aubin, Analyse Fonctionelle appliquée, Editions PUF, 1987.
A. Barbagallo, Regularity Results For Evolutionary Variational and Quasi-Variational Inequalities and Applicatios to Dynamic Equilibrium Problems, PhD Thesis, University of Naples, Italy, 2007.
A. Barbagallo, Regularity results for time-dependent variational and quasi-variational inequalities and computational procedures, Math. Models Methods Appl. Sci. 17, (2007),277–304.
A. Barbagallo, S. Pia, Weighted variational inequalities in non-pivot Hilbert spaces with applications, Computational Optimization and Applications, to appear.
H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, 1993.
V. Barbu, Th. Precapanu, Convexity and Optimization in Banach Spaces, Romania International Publisher, Bucarest, 1978.
M.G. Cojocaru, Projected Dynamical Systems on Hilbert Spaces, Ph. D. Thesis, Queen’s University, Canada, 2002.
M.G. Cojocaru, L.B. Jonker, Existence of Solutions to Projected Differential Equations in Hilbert Spaces, Proceedings of the American Mathematical Society 132, 2004, 183–193.
M.G. Cojocaru, P. Daniele, A. Nagurney, Projected Dynamical Systems and Evolutionary Variational Inequalities Via Hilbert Spaces with Applications, Journal of Optimization Theory and Applications 127, no. 3, (2005), 549–563.
M.G. Cojocaru, P. Daniele, A. Nagurney, Double layered Dynamics: A unified theory of Projected Dynamical Systems and Evolutionary Variational inegualities, European Journal of Operational Research 175, 1, (2006), 494–507.
M.G. Cojocaru, P. Daniele, A. Nagurney, Projected Dynamical Systems, Evolutionary Variational Inequalities, Applications, and a Computational Procedure, in Pareto Optimality, Game Theory and Equilibria, Springer, New York, A. Chinchuluun, P.M. Pardalos, A. Migadalas, L. Pitsoulis Eds, 2008, 392–406.
M.G. Cojocaru, L.B. Jonker. Existence of solutions to Projected Differential Equations in Hilbert Spaces, Proceeding of the American Mathematical Society 132, 2004, 183–193.
M.G. Cojocaru, S. Pia. Non-pivot and implicit projected dynamical systems on Hilbert spaces, Preprint.
P. Daniele, Time–Dependent Spatial Price Equilibrium Problem: Existence and Stability Results for the Quantity Formulation Model, Journal of Global Optimization 28, 2004, 283–295.
P. Daniele, Evolutionary Variational Inequalities and Economic Models for Demand Supply Markets, M3AS: Mathematical Models and Methods in Applied Sciences 4 (13), 2003,471–489.
P. Daniele, Variational Inequalities for Evolutionary Financial Equilibrium, in Innovations in Financial and Economic Networks, Edward Elgar Publishing, Cheltenham, England, A. Nagurney, Editor, 2003, 84–108.
P. Daniele, S. Giuffré, G. Idone, A. Maugeri, Infinite Dimentional Duality and Applications, Mathematische Annalen 339 (1), 2007, 221–239.
P. Daniele, A. Maugeri, On Dynamical Equilibrium Problems and Variational Inequalities, in Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Kluwer Academic Publishers, Dordrecht, The Netherlands, F. Giannessi, A. Maugeri, and P. Pardalos, Editors, 2001, 59–69.
P. Daniele, A. Maugeri, W. Oettli, Time-Dependent Traffic Equilibria, Journal of Optimization Theory and Applications 103, no. 3, 1999, 543–555.
P. Daniele, A. Maugeri, W. Oettli, Variational Inequalities and Time–Dependent Traffic Equilibria, C. R. Acad. Sci. Paris t. 326, serie I, 1998, 1059–1062.
J. Diestel, Geometry of Banach Spaces - Selected topics, Springer-Verlag, Berlin, 1975.
P. Dupuis, H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics and Stochastics Reports 35, 1990, 31–62.
P. Dupuis, A. Nagurney, Dynamical Systems and Variational Inequalities, Annals of Operations Research 44, 1993, 9–42.
S. Giuffré, G. Idone, S. Pia, Some Classes of Projected Dynamical Systems in Banach Spaces and Variational Inequalities, Journal of Global Optimization 40, (1–3), 2008, 119–128.
S. Giuffré, G. Idone, S. Pia, Projected Dynamical Systems and Variational inequalities equivalence results, Journal of Nonlinear and Convex Analysis 7, no. 3, 2006, 453–463.
S. Giuffré, S. Pia, Wireless Communication Densities and User-Oriented Traffic Equilibrium Problem, Proceedings of IEEE 2008 Wireless Communications, Networking and Mobile Computing (WICOM 2008), Dalian, China, October 12–14, 2008.
S. Giuffré, S. Pia, Weighted Traffic Equilibrium problem in non-pivot Hilbert spaces, Nonlinear Analysis, 2009, doi: 10.1016/j.na.2009.03.044.
G. Isac, M.G. Cojocaru, 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.
G. Isac, M.G. Cojocaru, The Projection Operator in a Hilbert Space and its Directional Derivative. Consequences for the Theory of Projected Dynamical Systems, Journal of Function Spaces and Applications 2, 2004, 71–95.
S. Heikkila, Monotonone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics 181, Marcel Dekker, New York, 1994.
A. Nagurney, Network Economics: A Variational Inequality Approach, Second and Revised Edition, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.
M.A. Noor, Generalized multivalued quasi-variational inequalities. II, Comput. Math. Appl. 35, no. 5, 1998, 63–78.
M.A. Noor, Generalized multivalued quasi-variational inequalities, Comput. Math. Appl. 31, no. 12, 1996, 1–13.
M.A. Noor, Implicit dynamical systems and quasi variational inequalities, Applied Math. and Comput. 134, 2003, 69–81.
M. Patriksson, R.T. Rockafellar, Variational Geometry and Equilibrium, Equilibrium problems and Variational Models, Kluwer Academic Publishers, Dordrecht, 2003, 347–367.
F. Raciti, Equilibria Trajectories as Stationary Solutions of Infinite Dimensional Projected Dynamical Systems, Applied Mathematics Letters 17, 2004, 153–158.
C. Ratti, R.M. Pulselli, S. Williams, D. Frenchman, Mobile Landscapes: Using location data from cell phones for urban analysis, Environment and Planning B. Planning and Design, 2006, 727–748.
C. Ratti, A. Sevtsuk, S. Huang, R. Pailer, Mobile Landscapes: Graz in Real Time, in Proceedings of the 3rd Symposium on LBS & TeleCartography, 28–30 November, 2007, Vienna, Austria.
R.T. Rockafellar, R. J-B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998.
W. Song, Z. Cao, The generalized Decomposition Theorem in Banach spaces and Its applications, J. Approx. Theory 129, 2004,167–181.
G. Tian, J. Zhou, Quasi-variational inequality with non-compact sets, J. Math. Anal. Appl. 160, 1991, 583–595.
F. Tinti, Numerical solution for pseudomonotone variational inequality problems by extragradient methods. Variational analysis and applications, Nonconvex Optim. Appl. 79, 2005, 1101–1128.
E.H. Zarantonello, Projections on Convex sets in Hilbert space and spectral theory, Contributions to Nonlinear Functional Analysis, Mathematics Research Center, Madison, April 12–14 1971 (E.H. Zarantonello, ed.) Academic Press, New York, 1971, 237–424.
E.H. Zarantonello, Projectors on convex sets in reflexive Banach spaces, Technical Summary report 1768, Winsconsin Univ. Madison Mathematics Research Center, 1977.
Editors and Affiliations
Dedicated to the memory of Professor George Isac
Rights and permissions
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Daniele, P., Giuffré, S., Maugeri, A., Pia, S. (2010). A Panoramic View on Projected Dynamical Systems. In: Pardalos, P., Rassias, T., Khan, A. (eds) Nonlinear Analysis and Variational Problems. Springer Optimization and Its Applications, vol 35. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0158-3_17
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-0157-6
Online ISBN: 978-1-4419-0158-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)