The Perron integral and existence and uniqueness theorems for a first order nonlinear differential equation

The Perron integral is used to establish an existence and uniqueness theorem concerning the initial value problem $y’(t) = f(t,y((t))$, and $y({t_0}) = \alpha$, for $t$ on the interval $I = \{ t|0 \leqq t \leqq 1\}$. The existence and uniqueness of the solution is obtained by use of a generalized Lipschitz condition, and a Picard sequence which is equiabsolutely continuous on $I$. Also, we prove a theorem on the uniqueness of solution by a generalization of Gronwall’s inequality.