On the non-exponential decay to equilibrium of solutions of nonlinear scalar Volterra integro-differential equations
We study the rate of decay of solutions of the scalar nonlinear Volterra equation \[ x'(t)=-f(x(t))+ \int_{0}^{t} k(t-s)g(x(s))\,ds,\quad x(0)=x_0 \] which satisfy $x(t)\to 0$ as $t\to\infty$. We suppose that $xg(x)>0$ for all $x\not=0$, and that $f$ and $g$ are continuous, continuously diff...
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| Format: | Article |
| Language: | English |
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University of Szeged
2004-08-01
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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=169 |
| _version_ | 1797830799336144896 |
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| author | John Appleby D. W. Reynolds |
| author_facet | John Appleby D. W. Reynolds |
| author_sort | John Appleby |
| collection | DOAJ |
| description | We study the rate of decay of solutions of the scalar nonlinear Volterra equation
\[
x'(t)=-f(x(t))+ \int_{0}^{t} k(t-s)g(x(s))\,ds,\quad x(0)=x_0
\]
which satisfy $x(t)\to 0$ as $t\to\infty$. We suppose that $xg(x)>0$ for all $x\not=0$, and that
$f$ and $g$ are continuous, continuously differentiable in some interval $(-\delta_1,\delta_1)$ and $f(0)=0$, $g(0)=0$. Also, $k$ is a continuous, positive, and integrable function, which is assumed to be subexponential in the sense that $k(t-s)/k(t)\to 1$ as $t\to\infty$ uniformly for $s$ in compact intervals. The principal result of the paper asserts that $x(t)$ cannot converge to $0$ as $t\to\infty$ faster than $k(t)$. |
| first_indexed | 2024-04-09T13:41:54Z |
| format | Article |
| id | doaj.art-aaec3c2fc3544105bbb187a1e72576ef |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:41:54Z |
| publishDate | 2004-08-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-aaec3c2fc3544105bbb187a1e72576ef2023-05-09T07:52:57ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752004-08-012003311410.14232/ejqtde.2003.6.3169On the non-exponential decay to equilibrium of solutions of nonlinear scalar Volterra integro-differential equationsJohn Appleby0D. W. Reynolds1School of Mathematical Sciences, Dublin City University, Dublin 9, IrelandCMDE, School of Mathematical Sciences, Dublin City University, Dublin 9, IrelandWe study the rate of decay of solutions of the scalar nonlinear Volterra equation \[ x'(t)=-f(x(t))+ \int_{0}^{t} k(t-s)g(x(s))\,ds,\quad x(0)=x_0 \] which satisfy $x(t)\to 0$ as $t\to\infty$. We suppose that $xg(x)>0$ for all $x\not=0$, and that $f$ and $g$ are continuous, continuously differentiable in some interval $(-\delta_1,\delta_1)$ and $f(0)=0$, $g(0)=0$. Also, $k$ is a continuous, positive, and integrable function, which is assumed to be subexponential in the sense that $k(t-s)/k(t)\to 1$ as $t\to\infty$ uniformly for $s$ in compact intervals. The principal result of the paper asserts that $x(t)$ cannot converge to $0$ as $t\to\infty$ faster than $k(t)$.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=169 |
| spellingShingle | John Appleby D. W. Reynolds On the non-exponential decay to equilibrium of solutions of nonlinear scalar Volterra integro-differential equations Electronic Journal of Qualitative Theory of Differential Equations |
| title | On the non-exponential decay to equilibrium of solutions of nonlinear scalar Volterra integro-differential equations |
| title_full | On the non-exponential decay to equilibrium of solutions of nonlinear scalar Volterra integro-differential equations |
| title_fullStr | On the non-exponential decay to equilibrium of solutions of nonlinear scalar Volterra integro-differential equations |
| title_full_unstemmed | On the non-exponential decay to equilibrium of solutions of nonlinear scalar Volterra integro-differential equations |
| title_short | On the non-exponential decay to equilibrium of solutions of nonlinear scalar Volterra integro-differential equations |
| title_sort | on the non exponential decay to equilibrium of solutions of nonlinear scalar volterra integro differential equations |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=169 |
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