The paper deals with variational properties of fixed points for contraction-type operators. Undersuitable conditions, the unique xed point of a vector-valued operator is a Nash-type equilibrium of thecorresponding energy functionals. This is achieved by an iterative scheme based on Ekeland’s variationalprinciple. An application to periodic solutions for second order dierential systems is given to illustratethe theory.
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
Nash-type equilibrium; fxed point; critical point; Ekeland principle; periodic solution; secondorder system
R. Precup, Nash-type equilibria and periodic solutions to nonvariational systems, Adv. Nonlinear Anal., 3 (2014), no. 4, 197-207, https://doi.org/10.1515/anona-2014-0006
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Advances Nonlinear Analysis
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 E. Bishop and R. R. Phelps, The support functionals of a convex set, Proc. Sympos. Pure Math. 7(1963), 27–35.
 I. Ekeland, On the variational principle,J. Math. Anal. Appl.47(1974), 324–353.
 M. Frigon, On some generalizations of Ekeland’s principle and inward contractions in gauge spaces,J. Fixed Point Theory Appl. 10 (2011), 279–298.
 A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
 J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, Berlin, 1989.
 S. Park, Generalizations of the Nash equilibrium theorem in the KKM theory, Fixed Point Theory Appl. 2010 (2010), ArticleID 234706.
 J.-P. Penot, The drop theorem, the petal theorem and Ekeland’s variational principle, Nonlinear Anal. 10(1986), 813–822.
 A. I. Perov, On Cauchy problem for a system of ordinary differential equations, Pviblizhen. Met. Reshen. Differ. Uravn. 2(1964), 115–134.
 R. Precup,Methods in Nonlinear Integral Equations, Springer-Verlag, Amsterdam, 2002.
 R. Precup, The role of matrices that are convergent to zero in the study of semilinear operator systems, Math. Comput. Modelling 49 (2009), 703–708.
 R. Precup, On a bounded critical point theorem of Schechter, Stud. Univ. Babeş–Bolyai Math. 58 (2013), 87–95.
 M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser-Verlag, Basel, 1999.
 S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331(2007), 506–515