Differential games in a Banach space without discrimination

\language=0 The paper is a continuation of author's research on the question of $\varepsilon$-equilibrium existence in the sense of piecewise program strategies in antagonistic games associated with nonlinear non-autonomous controlled differential equation in the Banach space and cost functional of a general form. Just as in the paper published earlier on this subject, the main result consists in sufficient conditions of $\varepsilon$-equilibrium. The difference is that we investigate the game without discrimination of players and without fixing Volterra chain. Application of results obtained in the paper is illustrated by example of the game associated with a nonlinear pseudoparabolic partial differential equation governing the evolution of electric field in a semiconductor.