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New optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations. (English) Zbl 1017.34065
Summary: Nonimprovable sufficient conditions for the unique solvability of the problem \[ u'(t)=\ell (u)(t)+q(t),\qquad u(a)=c, \] where \(\ell \: C(I;\mathbb{R})\to L(I;\mathbb{R})\) is a linear bounded operator, \(q\in L(I;\mathbb{R})\), \(c\in \mathbb{R}\), are established which are different from known results. They are interesting especially in the case where the operator \(\ell \) is not of Volterra’s type with respect to the point \(a\).

MSC:
34K05 General theory of functional-differential equations
34K06 Linear functional-differential equations
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