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

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.