On $\varepsilon$-equilibrium in noncooperative functional operator $n$-person games

The paper is devoted to the investigation of noncooperative Volterra functional operator $n$-person games. Functional operator games are understood as games associated with controlled functional operator equations. Such games provide a uniform description for a wide class of differential games associated with evolutionary partial differential equations. We introduce the concept of abstract reachable set for functional operator equations and prove its precompactness under certain assumptions. The precompactness is used to prove the existence of a Nash $\varepsilon$-equilibrium in the sense of piecewise program strategies defined by means of the Volterra property in the games under consideration. As an example of the reduction of a controlled distributed-parameter system to a functional operator equation of the considered type, a mixed-value problem for a semilinear wave equation is considered, and the proposed hypotheses are verified for this equation. An example of a corresponding game statement is given.