Integral equations with contrasting kernels
In this paper we study integral equations of the form $x(t)=a(t)-\int^t_0 C(t,s)x(s)ds$ with sharply contrasting kernels typified by $C^*(t,s)=\ln (e+(t-s))$ and $D^*(t,s)=[1+(t-s)]^{-1}$. The kernel assigns a weight to $x(s)$ and these kernels have exactly opposite effects of weighting. Each type...
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| Format: | Article |
| Language: | English |
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University of Szeged
2008-01-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=292 |
| _version_ | 1797830753247035392 |
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| author | Theodore Burton |
| author_facet | Theodore Burton |
| author_sort | Theodore Burton |
| collection | DOAJ |
| description | In this paper we study integral equations of the form $x(t)=a(t)-\int^t_0 C(t,s)x(s)ds$ with sharply contrasting kernels typified by $C^*(t,s)=\ln (e+(t-s))$ and $D^*(t,s)=[1+(t-s)]^{-1}$. The kernel assigns a weight to $x(s)$ and these kernels have exactly opposite effects of weighting. Each type is well represented in the literature. Our first project is to show that for $a\in L^2[0,\infty)$, then solutions are largely indistinguishable regardless of which kernel is used. This is a surprise and it leads us to study the essential differences. In fact, those differences become large as the magnitude of $a(t)$ increases.
The form of the kernel alone projects necessary conditions concerning the magnitude of $a(t)$ which could result in bounded solutions. Thus, the next project is to determine how close we can come to proving that the necessary conditions are also sufficient.
The third project is to show that solutions will be bounded for given conditions on $C$ regardless of whether $a$ is chosen large or small; this is important in real-world problems since we would like to have $a(t)$ as the sum of a bounded, but badly behaved function, and a large well behaved function. |
| first_indexed | 2024-04-09T13:41:08Z |
| format | Article |
| id | doaj.art-01c9676a22954071a58533d466840278 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:41:08Z |
| publishDate | 2008-01-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-01c9676a22954071a58533d4668402782023-05-09T07:52:58ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752008-01-012008212210.14232/ejqtde.2008.1.2292Integral equations with contrasting kernelsTheodore Burton0Northwest Research Institute, Port Angeles, WA, U.S.A.In this paper we study integral equations of the form $x(t)=a(t)-\int^t_0 C(t,s)x(s)ds$ with sharply contrasting kernels typified by $C^*(t,s)=\ln (e+(t-s))$ and $D^*(t,s)=[1+(t-s)]^{-1}$. The kernel assigns a weight to $x(s)$ and these kernels have exactly opposite effects of weighting. Each type is well represented in the literature. Our first project is to show that for $a\in L^2[0,\infty)$, then solutions are largely indistinguishable regardless of which kernel is used. This is a surprise and it leads us to study the essential differences. In fact, those differences become large as the magnitude of $a(t)$ increases. The form of the kernel alone projects necessary conditions concerning the magnitude of $a(t)$ which could result in bounded solutions. Thus, the next project is to determine how close we can come to proving that the necessary conditions are also sufficient. The third project is to show that solutions will be bounded for given conditions on $C$ regardless of whether $a$ is chosen large or small; this is important in real-world problems since we would like to have $a(t)$ as the sum of a bounded, but badly behaved function, and a large well behaved function.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=292 |
| spellingShingle | Theodore Burton Integral equations with contrasting kernels Electronic Journal of Qualitative Theory of Differential Equations |
| title | Integral equations with contrasting kernels |
| title_full | Integral equations with contrasting kernels |
| title_fullStr | Integral equations with contrasting kernels |
| title_full_unstemmed | Integral equations with contrasting kernels |
| title_short | Integral equations with contrasting kernels |
| title_sort | integral equations with contrasting kernels |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=292 |
| work_keys_str_mv | AT theodoreburton integralequationswithcontrastingkernels |