On the balanced pantograph equation of mixed type
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 doe...
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| Date: | 2024 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2024
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7654 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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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| DOI: | 10.3842/umzh.v75i12.7654 |