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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| Datum: | 2024 |
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| Sprache: | Englisch |
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Institute of Mathematics, NAS of Ukraine
2024
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/7654 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512692849082368 |
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| author | Derfel, G. Brunt, B. van Derfel, G. Brunt, B. van |
| author_facet | Derfel, G. Brunt, B. van Derfel, G. Brunt, B. van |
| author_sort | Derfel, G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-06-19T00:34:56Z |
| description | 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 $x\to \infty.$ |
| doi_str_mv | 10.3842/umzh.v75i12.7654 |
| first_indexed | 2026-03-24T03:32:50Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-7654 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:32:50Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-76542024-06-19T00:34:56Z On the balanced pantograph equation of mixed type On the balanced pantograph equation of mixed type Derfel, G. Brunt, B. van Derfel, G. Brunt, B. van functional- differential equations with rescaling pantograpgh equation stochastic difference equation archetypal equation Functional differential equations 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 $x\to \infty.$ УДК 517.9 Про рівняння рівноважного пантографа мішаного типу Розглянуто рівняння збалансованого пантографа (РЗП) $y^{\prime}(x)+y(x)=\sum_{k=1}^{m}p_{k}y(a_{k}x),$ де $a_{k}, p_{k} >0$ і $\sum_{k=1}^{m}p_{k} =1.$ Відомо, що якщо $K=\sum_{k=1}^{m}p_{k}\ln a_{k} \leq 0,$ то за м’яких технічних умов РЗП не має обмежених розв’язків, які не є сталими; водночас у випадку $K>0$ такі розв'язки існують. У цій статті ми маємо справу з РЗП мішаного типу, тобто $a_{1}<1<a_{m},$ і доводимо, що в цьому випадку РЗП має несталий розв’язок $y$ і, крім того, $y(x)\sim cx^{\sigma}$ при $x\to \infty,$ де $c>0,$ а $\sigma$ --- єдиний додатний корінь характеристичного рівняння $P(s)=1-\sum_{k=1}^{m}p_{k}a_{k}^{- s}=0.$ Також показано, що $y$ є єдиним (з точністю до мультиплікативної константи) серед розв’язків РЗП, що спадають до нуля при $x\to \infty.$ Institute of Mathematics, NAS of Ukraine 2024-01-02 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7654 10.3842/umzh.v75i12.7654 Ukrains’kyi Matematychnyi Zhurnal; Vol. 75 No. 12 (2023); 1627 - 1634 Український математичний журнал; Том 75 № 12 (2023); 1627 - 1634 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7654/9931 Copyright (c) 2024 Gregory Derfel |
| spellingShingle | Derfel, G. Brunt, B. van Derfel, G. Brunt, B. van On the balanced pantograph equation of mixed type |
| title | On the balanced pantograph equation of mixed type |
| title_alt | On the balanced pantograph equation of mixed type |
| title_full | On the balanced pantograph equation of mixed type |
| title_fullStr | On the balanced pantograph equation of mixed type |
| title_full_unstemmed | On the balanced pantograph equation of mixed type |
| title_short | On the balanced pantograph equation of mixed type |
| title_sort | on the balanced pantograph equation of mixed type |
| topic_facet | functional- differential equations with rescaling pantograpgh equation stochastic difference equation archetypal equation Functional differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7654 |
| work_keys_str_mv | AT derfelg onthebalancedpantographequationofmixedtype AT bruntbvan onthebalancedpantographequationofmixedtype AT derfelg onthebalancedpantographequationofmixedtype AT bruntbvan onthebalancedpantographequationofmixedtype |