New optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations. (English) Zbl 1017.34065

Hakl, R.; Lomtatidze, A.; Půža, B.

Math. Bohem. 127, No. 4, 509-524 (2002).

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

Keywords:

linear functional-differential equations; differential equations with deviating arguments; initial value problems