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 Mat. Teor. Igr Pril., 2015, Volume 7, Issue 3, Pages 48–78 (Mi mgta163)

On existence of the Nash equilibrium in a differential game associated with elliptic equations: the monotone case

Andrey V. Chernovab

a Nizhnii Novgorod State Technical University
b Nizhnii Novgorod State University

Abstract: This work continues investigations concerning the problem of conditions sufficient for existence of the Nash equilibrium in noncooperative $n$-person games associated with semilinear elliptic partial differential equations of the second order. Unlike the previous author's paper having been published on this subject, now the controls may be present explicitly in integrands of the cost functionals. Moreover, the requirements to right-hand sides of equations are distinguished. Lastly, now it is used another on principle method of proving which is based (rather than on the convexity of reachable sets) on establishing convex character of dependence of the state on the control for each equation at the expense of convexity of the right-hand side with respect to the pair state–control and also of the requirements providing with the monotonicity of the nonlinear resolving operator. As an auxiliary results of a specific interest we prove theorems concerning comparison of solutions to semilinear elliptic equations and continuous character of dependence of the state on the control.

Keywords: noncooperative $n$-person game, Nash equilibrium, semilinear elliptic PDEs of the second order.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 172702.Â.49.21.0003

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Citation: Andrey V. Chernov, “On existence of the Nash equilibrium in a differential game associated with elliptic equations: the monotone case”, Mat. Teor. Igr Pril., 7:3 (2015), 48–78

Citation in format AMSBIB
\Bibitem{Che15} \by Andrey~V.~Chernov \paper On existence of the Nash equilibrium in a differential game associated with elliptic equations: the monotone case \jour Mat. Teor. Igr Pril. \yr 2015 \vol 7 \issue 3 \pages 48--78 \mathnet{http://mi.mathnet.ru/mgta163} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. V. Chernov, “On total preservation of solvability for a controlled Hammerstein type equation with non-isotone and non-majorized operator”, Russian Math. (Iz. VUZ), 61:6 (2017), 72–81
2. Andrei V. Chernov, “O nekotorykh podkhodakh k poisku ravnovesiya po Neshu v vognutykh igrakh”, MTIP, 9:2 (2017), 62–104
3. A. V. Chernov, “Differentiation of the functional in a parametric optimization problem for a coefficient of a semilinear elliptic equation”, Differ. Equ., 53:4 (2017), 551–562
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