ACTA issues

Representation of the solutions to linear pantograph equations

József Terjéki

Acta Sci. Math. (Szeged) 60:3-4(1995), 705-713

Abstract. The pantograph equation $$\dot x(t)=A(t)x(t)+B(t)x(\theta(t))$$ is considered, where $A$, $B$ are continuous matrix functions, the lag function $\theta\colon {\msbm R}_+\to{\msbm R}_+$ is continuous and $0\le\theta (t)\le t $ for all $t\ge0$. A representation using the Cauchy matrix of the system $\dot x=A(t)x$ is given for the solutions. This representation gives the well-known Dirichlet series solution for the scalar ``autonomous" cases $$\dot x(t)=ax(t)+bx(pt)\qquad (a, b, p \hbox{ are constants, }0< p< 1).$$

AMS Subject Classification (1991): 34K05

Received January 18, 1995. (Registered under 5660/2009.)