The nonimprovable sufficient conditions for the unique solvability of the problem u' (t) = l(u)(t) + q(t), u(a) = c, where l : C(I; R) → L(I; R) is a linear bounded operator, q ∈ L(I; R), c ∈ R, are established which are different from the previous results. More precisely, they are interesting especially in the case where the operator ` is not of Volterra’s type with respect to the point a.
Nonimprovable, in a sense sufficient conditions guaranteeing the unique solvability of the problem \[ u^{\prime }(t)=\ell (u)(t)+q(t), \qquad u(a)=c, \] where $\ell \:C(I,\mathbb R)\rightarrow L(I,\mathbb R)$ is a linear bounded operator, $q\in L(I,\mathbb R)$, and $c\in \mathbb R$, are established.