Abstract
In this paper, we aim to generalize an existing result by obtaining localized solutions within bounded convex sets, while also relaxing specific initial assumptions. To achieve this, we employ an iterative scheme that combines a fixed-point argument based on the Minty-Browder Theorem with a modified version of the Ekeland variational principle for bounded sets. An application to a system of second-order differential equations with Dirichlet boundary conditions is presented.
Authors
Andrei Stan
Faculty of Mathematics and Computer Science, Babeș-Bolyai University
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Keywords
Nash equilibrium; iterative methods; Ekeland variational principle; monotone operator.
Paper coordinates
A. Stan, Localization of Nash-type equilibria for systems with partial variational structure, Journal of Numerical Analysis and Approximation Theory, 52 (2023) no. 2, pp. 253–272, https://doi.org/10.33993/jnaat522-1356
About this paper
Journal
Journal of Numerical Analysis and Approximation Theory
Publisher Name
Publishing House of the Romanian Academy
Print ISSN
2457-6794
Online ISSN
2501-059X
google scholar link
http://doi.org/10.24193/subbmath.2021.2.14.
[2] I. Benedetti, T. Cardinali,R. Precup,Fixed point-critical point hybrid theoremsand application to systems with partial variational structure, J. Fixed Point TheoryAppl.,23(2021) art. no. 63,https://doi.org/10.1007/s11784-021-00852-6.
[3] R. Precup,A critical point theorem in bounded convex sets and localization of Nash-type equilibria of nonvariational systems, J. Math. Anal. Appl.,463(2018), pp. 412–431,https://doi.org/10.1016/j.jmaa.2018.03.035.
[4] R. Precup,Nash-type equilibria and periodic solutions to nonvariational systems, Adv.Nonlinear Anal.,3(2014) no. 4, pp. 197–207,
https://doi.org/10.1515/anona-2014-0006.
[5] M. Beldinski,M. Galewski,Nash type equilibria for systems of non-potential equa-tions, Appl. Math. Comput.,385(2020), pp. 125-456, https://doi.org/10.1016/j.amc.2020.125456.
[6] A. Cournot,The mathematical principles of the theory of wealth, Economic J., 1838.
[7] J. Nash,Non-cooperative games, Ann. Math.,54(1951) no. 2, pp. 286–295, https://doi.org/10.2307/1969529.
[8] A. Berman,R.J. Plemmons,Nonnegative Matrices in the Mathematical Sciences,Academic Press, 1997.
[9] R. Precup,The role of matrices that are convergent to zero in the study of semilinearoperator systems, Math. Comput. Modelling,49(2009) no. 3, pp. 703–708, https://doi.org/10.1016/j.mcm.2008.04.006.
[10] G. Ciarlet,Linear and Nonlinear Functional Analysis with Applications, SIAM,Philadelphia, 2013.
[11] H. Brezis,Functional Analysis, Sobolev Spaces and Partial Differential Equations,Springer, New York, 2011,https://doi.org/10.1007/978-0-387-70914-7.
[12] R. Precup,Linear and Semilinear Partial Differential Equations, De Gruyter, Berlin,2013.
[13] A. Cabada,Green’s Functions in the Theory of Ordinary Differential Equations,Springer, New York, 2014,https://doi.org/10.1007/978-1-4614-9506-2.