On some approach to construction of $\varepsilon$-equilibrium in noncooperative $n$-person games associated with partial differential equations

The paper is devoted to justification of some comparatively simple algorithm for construction of the Nash $\varepsilon$-equilibrium in noncooperative $n$-person games associated with evolutionary semilinear partial differential equations. Here, we consider piecewise program strategies, the time step is assumed to be fixed and controls are assumed to be piecewise constant vectors with values from a given compact set in a finite dimensional space. The main idea of the algorithm consists in the approximation of an original game by a finite multistep perfect information game with subsequent applying of the Kuhn algorithm. The basis of the algorithm under study consists in two assertions concerning, firstly, the total preservation of global solvability of controlled distributed parameter systems, and, secondly, the continuous dependence of solutions to them on piecewise constant controls, both having been proved by the author formerly. The paper is conducted on the example of the first boundary value problem associated with a parabolic second order equation of a rather general form.