This paper investigates the solution of the initial value problem for first-order impulsive evolution equations in Banach space. Firstly, by using the monotone iterative method, we obtain the existence and uniqueness of the positive mild solution to the non-impulsive evolution equations on a finite interval without assuming the existence of upper and lower solutions and the equicontinuity of semigroup. Secondly, without the continuity and monotonicity of the impulsive function, we establish the existence and uniqueness of the positive mild solution to the impulsive evolution equations on an infinite interval by extending the finite interval, which improve the existing results. Finally, the gained abstract results are applied to the parabolic partial differential equations, which illustrates the validity of the obtained theorem in the application.