On the balanced pantograph equation of mixed type

G. Derfel; B. van Brunt

doi:10.3842/umzh.v75i12.7654

G. Derfel Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva, Israel
B. van Brunt SMCS, Massey University, Palmerston North, New Zealand

DOI: https://doi.org/10.3842/umzh.v75i12.7654

Keywords: functional- differential equations with rescaling, pantograpgh equation, stochastic difference equation, archetypal equation

Abstract

UDC 517.9

We consider the balanced pantograph equation (BPE) $y^{\prime}(x)+y(x)=\sum_{k=1}^{m}p_{k}y(a_{k}x),$ where $a_{k}, p_{k} >0$ and $\sum_{k=1}^{m}p_{k} =1.$ It is known that if $K=\sum_{k=1}^{m}p_{k}\ln a_{k} \leq 0$ then, under mild technical conditions, the BPE does not have bounded solutions that are not constant, whereas for $K>0$ these solutions exist. In the present paper, we deal with a BPE of mixed type, i.e., $a_{1}<1<a_{m},$ and prove that, in this case, the BPE has a nonconstant solution $y$ and that $y(x)\sim cx^{\sigma}$ as $x\to \infty,$ where $c>0$ and $\sigma$ is the unique positive root of the characteristic equation $P(s)=1-\sum_{k=1}^{m}p_{k}a_{k}^{-s}=0.$ We also show that $y$ is unique (up to a multiplicative constant) among the solutions of the BPE that decay to zero as $x\to \infty.$

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