##
A note on the Cauchy problem for first order linear differential equations
with a deviating argument

##
*Robert Hakl and Alexander Lomtatidze*

**Address.** Robert Hakl, Mathematical Institute, Czech Academy of Sciences,
branch Brno, Zizkova 22, 616 62 Brno, CZECH REPUBLIC
Alexander Lomtatidze, Department of Mathematical Analysis, Faculty of
Sciences, Masaryk University, Janackovo nam. 2a, 662 95 Brno, CZECH REPUBLIC

**E-mail:** hakl@ipm.cz

bacho@math.muni.cz

**Abstract.** Conditions for the existence and uniqueness of a solution
of the Cauchy problem $$ u'(t)=p(t)u(\tau(t))+q(t)\,,\qquad u(a)=c\,, $$
established in [2], are formulated more precisely and refined for the special
case, where the function $\tau$ maps the interval $]a,b[$ into some subinterval
$[\tau_0,\tau_1]\subseteq[a,b]$, which can be degenerated to a point.

**AMSclassification.** 34K10.

**Keywords.** First order equation, differential equation with deviating
arguments, initial value problems.