On existence of $\varepsilon$-equilibrium in noncooperative $n$-person games associated with elliptic partial differential equations

The paper is devoted to obtaining the sufficient conditions for existence of the Nash $\varepsilon$-equilibrium in noncooperative $n$-person games associated with semilinear elliptic partial differential equations of the second order. Here, for all players, we consider only program strategies. The basis of the theory constructed consists in some assertion concerning the total preservation of solvability and the uniform solution boundedness of an operator equation of the first kind having been proved by the author formerly by means of a generalization of the method of monotone maps. As an auxiliary result of a specific interest we prove a theorem on convexity of the reachable set of a controlled semilinear elliptic equation.