J. Šremr and P. Šremr

On a Two Point Boundary Problem for First Order Linear Differential Equations with a Deviating Argument

abstract:
The aim of the paper is to study the question on the existence and uniqueness of a solution of the problem
$$
u'(t)=p(t)u(\tau(t))+q(t),\qquad u(a)+\lambda u(b)=c,
$$
where $p,q:[a,b]\to R$ are Lebesgue integrable functions, $\tau:[a,b]\to[a,b]$ is a measurable function, and $\lambda,c\in R$. More precisely, some solvability conditions established in \cite{hls,hls2,hls7} are refined for the special case, where $\tau$ maps the segment $[a,b]$ into some subsegment $[\tau_0,\tau_1]\subseteq[a,b]$.

Mathematics Subject Classification: 34K10.

Key words and phrases: First order linear differential equations with deviating argument, two point boundary value problem, unique solvability.