A note on the Cauchy problem for first order linear differential equations with a deviating argument. (English) Zbl 1087.34043

Hakl, Robert; Lomtatidze, Alexander

Arch. Math., Brno 38, No. 1, 61-71 (2002).

This paper, from a series of results successively describing the basic properties of solutions of functional differential equations, is devoted to the Cauchy problem \[ u'(t)=p(t)u(\tau (t))+q(t),\qquad u(a)=c, \] where \(p\) and \(q\) are Lebesgue integrable on \([a,b]\) real functions, \(c\in \mathbb R\), and \(\tau :[a,b]\to [\tau _0,\tau _1]\) is a measurable function. Here \([\tau _0,\tau _1]\subseteq [a,b]\) can be degenerated to a point.
The obtained effective criteria are in some sense nonimprovable, which is shown by several examples.

Reviewer: Bedřich Pŭža (Brno)

MSC:

34K10

Boundary value problems for functional-differential equations

Keywords:

first order equation; differential equation with deviating arguments; initial value problems