An extremal problem with constraints which, under sufficiently general conditions, is in some sense equivalent to the problem of finding a feasible strategy minimizing the guaranteed value of a terminal cost functional in an antagonistic differential game is investigated. The following results for the extremal problem under consideration are obtained: necessary optimality conditions; sufficient conditions for a linear-concave case; a class of problems for which necessary conditions coincide with sufficient conditions is characterized. Based on these results existence theorems in different classes of strategies are proved.