A Nash type solution for hemivariational inequality systems. (English) Zbl 1229.47103
The authors prove an existence result for a general class of hemivariational inequality systems using the Ky Fan version of the KKM theorem [K. Fan, Math. Ann. 266, 519–537 (1984; Zbl 0515.47029)] or Tarafdar fixed points [E. Tarafdar, J. Math. Anal. Appl. 128, 475–479 (1987; Zbl 0644.47050)]. As application, an infinite-dimensional version is given for the existence result of Nash generalized derivative points recently introduced by A. Kristály [Proc. Am. Math. Soc. 138, No. 5, 1803–1810 (2010; Zbl 1189.91018)]. An application is also proposed to a general hemivariational inequality system.
Reviewer: Dian K. Palagachev (Bari)
MSC:
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 49N70 | Differential games and control |
References:
| [1] | Breckner, B. E.; Varga, Cs., A multiplicity result for gradient-type systems with non-differentiable term, Acta Math. Hungar., 118, 85-104 (2008) · Zbl 1164.47063 |
| [2] | Breckner, B. E.; Horváth, A.; Varga, Cs., A multiplicity result for a special class of gradient-type systems with non-differentiable term, Nonlinear Anal. TMA, 70, 606-620 (2009) · Zbl 1161.47046 |
| [3] | Kristály, A., An existence result for gradient-type systems with a nondifferentiable term on unbounded strips, J. Math. Anal. Appl., 229, 186-204 (2004) · Zbl 1129.35423 |
| [4] | Kristály, A., Existence of two non-trivial solutions for a class of quasilinear elliptic variational systems on stripe-like domains, Proc. Edinb. Math. Soc., 48, 1-13 (2005) |
| [5] | Kristály, A., Location of Nash equilibria: a Riemannian geometrical approach, Proc. Amer. Math. Soc., 138, 1803-1810 (2010) · Zbl 1189.91018 |
| [6] | Lisei, H.; Varga, Cs., Multiple solutions for gradient elliptic systems with nonsmooth boundary conditions, Mediterr. J. Math. (2010) |
| [7] | Motreanu, D.; Rădulescu, V., Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems. Nonconvex Optimization and its Applications (2003), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 1040.49001 |
| [8] | Motreanu, D.; Panagiotopoulos, P. D., Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 1060.49500 |
| [9] | Kristály, A.; Rădulescu, V.; Varga, Cs., (Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems. Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyklopedia of Mathematics, vol. 136 (2010), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1206.49002 |
| [10] | Fan, K., Some properties of convex sets related to fixed point theorem, Math. Ann., 266, 519-537 (1984) · Zbl 0515.47029 |
| [11] | Tarafdar, E., A fixed point theorem equivalent to the Fan-Knaster-Kuratwski-Mazurkiewicz theorem, J. Math. Anal. Appl., 128, 475-497 (1987) · Zbl 0644.47050 |
| [12] | Clarke, F. H., Optimization and Nonsmooth Analysis (1990), SIAM: SIAM Philadelphia · Zbl 0727.90045 |
| [13] | Nash, J., Equilibrium points in \(n\)-person games, Proc. Nat. Acad. Sci. USA, 36, 48-49 (1950) · Zbl 0036.01104 |
| [14] | Nash, J. F., Non-cooperative games, Ann. of Math., 54, 2, 286-295 (1951) · Zbl 0045.08202 |
| [15] | Kassay, G.; Kolumbán, J.; Páles, Zs., On Nash stationary points, Publ. Math. Debrecen, 54, 267-279 (1999) · Zbl 0932.49009 |
| [16] | Kristály, A., Hemivariational inequalities systems and applications, Mathematica, 46, 2, 161-168 (2004) · Zbl 1150.35395 |
| [17] | Panagiotopoulos, P. D.; Fundo, M.; Rădulescu, V., Existence theorems of Hartman-Stampacchia type for hemivariational inequalities and applications, J. Global Optim., 15, 41-54 (1999) · Zbl 0951.49018 |
| [18] | Dályai, Z.; Varga, Cs., An existence result for hemivariational inequalities, Electron. J. Differential Equations, 37, 1-17 (2004) · Zbl 1161.49303 |
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