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

Andrey V. Chernov

Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 7 (2015) no. 3, pp. 48-78.

Résumé

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.

Détail

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

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