Simultaneous multiple capture of evaders in a simple group pursuit problem

The present paper deals with the problem of simple pursuit of group of $m$ evaders $(m\geqslant1)$ with equal opportunities. We say that a multiple capture in the problem of pursuit of one evader $(m=1)$ holds if the specified number of pursuers catch him, possibly at different times. The problem of the simultaneous capture of one evader requires that capture moments coincide. We introduce the concept of multiple simultaneous capture of the whole group of evaders $(m\geqslant2).$ We say that the simultaneous multiple capture of the whole group of evaders holds if the simultaneous multiple capture of every evader holds in the same time. We obtain necessary and sufficient conditions for simultaneous multiple capture of the whole group of evaders in terms of initial positions of the participants.