$\varepsilon$-positional strategies in the theory of differential pursuit games and the invariance of a constant multivalued mapping in the heat conductivity problem

In this paper, we consider two problems. In the first problem, we prove that if the assumption from the paper [1] and one additional condition on the parameters of the game hold, then the pursuit can be finished in any neighborhood of the terminal set. To complete the game, an $\varepsilon$-positional pursuit strategy is constructed.
In the second problem, we study the invariance of a given multivalued mapping with respect to the system with distributed parameters. The system is described by the heat conductivity equation containing additive control terms on the right-hand side.