  In this paper we study the zero-sum and nonzero-sum differential games with
not assuming Isaacs condition. Along with the partition $\pi$ of the time
interval $[0,T]$, we choose the suitable random non-anticipative strategy with
delay to study our differential games with asymmetric information. Using
Fenchel transformation, we prove that the limits of the upper value function
$W^\pi$ and lower value function $V^\pi$ coincide when the mesh of partition
$\pi$ tends to 0. Moreover, we give a characterization for the Nash equilibrium
payoff (NEP, for short) of our nonzero-sum differential games without Isaacs
condition, then we prove the existence of the NEP of our games. Finally, by
considering all the strategies along with all partitions, we give a new
characterization for the value of our zero-sum differential game with
asymmetric information under some equivalent Isaacs condition.
