One of the most important problems in the theory of differential games is to derive a solvability condition for a given game. In this paper we consider a class of differential games and derive some solvability conditions which throw a light on the fundamental structure of differential games.
The first part treats nonlinear differential games of prescribed duration. By the use of a mapping relating to canonical equations, a condition is derived which removes the possibility of existence of a type of singularity. It is shown that this singularity plays an essential role in the problem of determining the existence of the saddle point. Using this result simple solvability conditions are derived and an explicit form for the saddle point is shown. In the second part, similar but stronger conditions are obtained for a linear differential game whose payoff function is given by a convex function of the terminal state. It is shown that under an appropriate condition, instead of solving the Bellman equation, the value of the game can be calculated by maximizing a function on the unit sphere of the output space whose dimension is usually much less than that of the state space. Two examples are shown in order to illustrate the results. In the conclusion some results obtained previously by the author, concerning the applications of differential games to control problems, are mentioned briefly.

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