On Volterra functional operator games on a given set

The paper is devoted to obtaining the sufficient conditions of $\varepsilon$-equilibrium in the sense of piecewise program strategies in antagonistic games associated with nonlinear controlled functional operator equations and cost functional of a general enough form. The concept of piecewise program strategies is defined on the base of a concept of Volterra set chain for operators involved in the equations controlled by the opponent players. The reduction of controlled distributed parameter systems to an equation of the type under study is illustrated by examples.