A construction of Nash equilibrium based on system of Hamilton–Jacobi equations of special type

The paper is devoted to the Nash program strategies' construction in nonzero-sum differential games with two players applying the solution of a strong coupled system of Hamilton–Jacobi equations. The system of Hamilton–Jacobi equations is of the special type where the first equation of the system doesn't depend on the second one, and the second equation depends on the derivative of the first equation's solution. We show that the solution of the system of Hamilton–Jacobi equations should be considered in the class of multivalued maps. We propose a generalized solution for the system of Hamilton–Jacobi equations and prove the existence theorem for such generalized solution. To conclude we consider an example illustrating the Nash program strategies' construction.