Some classes of projected dynamical systems in Banach spaces and variational inequalities. (English) Zbl 1194.49015

Summary: We introduce some projected dynamical systems based on metric and generalized projection operator in a strictly convex and smooth Banach space. Then we prove that critical points of these systems coincide with the solution of a variational inequality.


49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47N20 Applications of operator theory to differential and integral equations
Full Text: DOI


[1] Alber Ya.I. (1996). Metric and generalized projection operators in Banach spaces: proprieties and applications. In: Kartsatos, A. (eds) Theory and Applications of Nonlinear operators of Monotone and Accretive Type, pp 15–50. Marcel Dekker, New York · Zbl 0883.47083
[2] Alber Ya.I. (2000). Decomposition theorem in banach spaces. Field Inst. Commun. 25: 77–99 · Zbl 0971.46004
[3] Barbu V. and Precapanu Th. (1978). Convexity and Optimization in Banach Spaces. Romania International Publisherd, Bucarest · Zbl 0379.49010
[4] Brezis H. (1993). Analyse Fonctionnelle, Théorie et Applications. Masson, Paris
[5] Cojocaru M.G. (2002). Projected dynamical systems on Hilbert spaces. Ph.D. thesis, Queen’s University Canada
[6] Cojocaru M.G., Daniele P. and Nagurney A. (2005). Projected dynamical systems and evolutionarry (time-dependent) variational inequalities via Hilbert spaces with applications. J. Optim. Theory Appl. 27(3): 1–15
[7] Cojocaru M.G. and Jonker L.B. (2004). Existence of solutions to projected differential equations in Hilbert spaces. Proc. Am. Math. Soc. 132: 183–193 · Zbl 1055.34118
[8] Daniele, P., Giuffre, S., Idone, G., Maugeri, A.: Infinite Dimentional Duality and Applications Mathematische Annalen, Springer, May 2007
[9] Daniele P., Maugeri A. and Oettli W. (1999). Time-dependent traffic equilibria. J. Optim. Theory Appl. 103(3): 543–555 · Zbl 0937.90005
[10] Diestel J. (1975). Geometry of Banach Spaces – Selected topics. Springer, Berlin · Zbl 0307.46009
[11] Giuffré, S., Idone, G., Pia, S.: Projected Dynamical Systems and Variational inequalities equivalence results. J Nonlinear Convex Anal. 7(3) (2006) · Zbl 1118.49021
[12] Gwinner J. (2003). Time dependent variational inequalities-some recent trends. In: Daniele, P., Giannessi, F. and Maugeri, A. (eds) Equilibrium Problems and Variational Models, pp 225–264. Kluwer, USA · Zbl 1069.49005
[13] Isac, G., 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, Barbes-Bolyai University of Cluj-Napoca, 111, September 2002, Romania, pp. 41–62 · Zbl 1025.37020
[14] Isac G. and Cojocaru M.G. (2002). The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems. J. Funct. Spaces Appl. 2: 71–95 · Zbl 1075.47032
[15] Moreau J.J. (1962). Decomposition orthogonale d’un espace de Hilbert selon deux cones mutuellement polaires. C. R. Acad. des Sci. Paris 255: 238–240 · Zbl 0109.08105
[16] Nagurney A. and Zhang D. (1996). Projected Dynamical Systems and Variational Inequalities with Applications. Kluwer, Dordrecht · Zbl 0865.90018
[17] Raciti F. (2004). Equilibria trajectories as stationary solutions of infinite dimensional projected dynamical systems. Appl. Math. Lett. 17: 153–158 · Zbl 1058.49008
[18] Song, W., Cao, Z.: The Generalized Decomposition Theorem in Banach Spaces and Its Applications. Journal of Approximation Theory, Elsevier, Amsterdam (2004) · Zbl 1067.46009
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