On the segment $I=[a,b]$ consider the problem \[ u^{\prime }(t)=f(u)(t) , \quad u(a)=c, \] where $f\:C(I,\mathbb{R})\rightarrow L(I,\mathbb{R})$ is a continuous, in general nonlinear operator satisfying Carathéodory condition, and $c\in \mathbb{R}$. The effective sufficient conditions guaranteeing the solvability and unique solvability of the considered problem are established. Examples verifying the optimality of obtained results are given, as well.