Differential games in a Banach space on a fixed chain

The paper is devoted to obtaining the sufficient conditions for existence of $\varepsilon$-equilibrium in the sense of piecewise program strategies in antagonistic games associated with nonlinear non-autonomous controlled differential equation in a Banach space and cost functional of a general enough form. The concept of piecewise program strategies in such a game is defined on the base of a concept of Volterra set chain for a right-hand operator of the corresponding integral equation controlled by the opponent players and according to a given partition of the time segment. As example we consider the game associated with a nonlinear pseudoparabolic partial differential equation governing the evolution of electric field in a semiconductor.