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On the segment $I=[a,b]$ consider the problem $$u^{\text{'}}\left(t\right)=f\left(u\right)\left(t\right),u\left(a\right)=c,$$ where $fC(I,ℝ)\to L(I,ℝ)$ is a continuous, in general nonlinear operator satisfying Carathéodory condition, and $c\in ℝ$. 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.

Nonimprovable, in a sense sufficient conditions guaranteeing the unique solvability of the problem $$u^{\text{'}}\left(t\right)=\ell \left(u\right)\left(t\right)+q\left(t\right),u\left(a\right)=c,$$ where $\ell C(I,ℝ)\to L(I,ℝ)$ is a linear bounded operator, $q\in L(I,ℝ)$, and $c\in ℝ$, are established.