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The nonimprovable sufficient conditions for the unique solvability of the problem u ′(t) = `(u)(t) + q(t), u(a) = c, where ` : C(I; ) → L(I; ) is a linear bounded operator, q ∈ L(I; ), c ∈ , 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.

New optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations