On equations of the program iteration method (Q2284198)

The author consider a standard differential game from the point of view of the so called program iteration method going back to \textit{A. G. Chentsov} [Sov. Math., Dokl. 16, 1404--1408 (1975; Zbl 0351.90094); translation from Dokl. Akad. Nauk SSSR 224, 1272--1275 (1975)], which defines the minmax- and the maxmin-values $w_+$ (resp. $w_+$) of the game as the limit of this iteration method applied to two distinct operators $\Phi_+$ and $\Phi_-$. In the present paper it is shown that both, $w_-$ and $w_+$ are respectively the smallest and the largest function which are fixed points for both $\Phi_+$ and $\Phi_-$. If Isaacs' assumption holds the set of these fixed points is a singleton and the value functions coincide.