Periodic and asymptotically periodic solutions of neutral integral equations
Many results have been obtained for periodic solutions of Volterra integral equations (for instance, [1-3] and references cited therein). Here we consider two systems of neutral integral equations \begin{eqnarray} x(t)=a(t)+\int_0^t D(t,s,x(s))ds+\int_t^\infty E(t,s,x(s))ds, \ t\in R^+ \end{eqnarray...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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University of Szeged
2000-01-01
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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=39 |
| _version_ | 1797830793518645248 |
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| author | Tetsuo Furumochi Theodore Burton |
| author_facet | Tetsuo Furumochi Theodore Burton |
| author_sort | Tetsuo Furumochi |
| collection | DOAJ |
| description | Many results have been obtained for periodic solutions of Volterra integral equations (for instance, [1-3] and references cited therein). Here we consider two systems of neutral integral equations
\begin{eqnarray}
x(t)=a(t)+\int_0^t D(t,s,x(s))ds+\int_t^\infty E(t,s,x(s))ds, \ t\in R^+
\end{eqnarray}
and
\begin{eqnarray}
x(t)=p(t)+\int_{-\infty}^t P(t,s,x(s))ds+\int_t^\infty Q(t,s,x(s))ds, \ t\in R,\
\end{eqnarray}
where $a, \ p, \ D, \ P, \ E$ and $Q$ are at least continuous. Under suitable conditions, if $\phi$ is a given $R^n$-valued bounded and continuous initial function on $[0,t_0)$ or $(-\infty,t_0)$, then both Eq.(1) and Eq.(2) have solutions denoted by $x(t,t_0,\phi)$ with $x(t,t_0,\phi)=\phi(t)$ for $t<t_0$, satisfying Eq.(1) or Eq.(2) on $[t_0,\infty)$. (cf. Burton-Furumochi [4].) A solution $x(t,t_0,\phi)$ may have a discontinuity at $t_0$. |
| first_indexed | 2024-04-09T13:42:52Z |
| format | Article |
| id | doaj.art-94d2dc87bc484a35b7c9fed5b2f643f8 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:42:52Z |
| publishDate | 2000-01-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-94d2dc87bc484a35b7c9fed5b2f643f82023-05-09T07:52:56ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752000-01-0119991011910.14232/ejqtde.1999.5.1039Periodic and asymptotically periodic solutions of neutral integral equationsTetsuo Furumochi0Theodore Burton1Shimane University, Matsue, JapanNorthwest Research Institute, Port Angeles, WA, U.S.A.Many results have been obtained for periodic solutions of Volterra integral equations (for instance, [1-3] and references cited therein). Here we consider two systems of neutral integral equations \begin{eqnarray} x(t)=a(t)+\int_0^t D(t,s,x(s))ds+\int_t^\infty E(t,s,x(s))ds, \ t\in R^+ \end{eqnarray} and \begin{eqnarray} x(t)=p(t)+\int_{-\infty}^t P(t,s,x(s))ds+\int_t^\infty Q(t,s,x(s))ds, \ t\in R,\ \end{eqnarray} where $a, \ p, \ D, \ P, \ E$ and $Q$ are at least continuous. Under suitable conditions, if $\phi$ is a given $R^n$-valued bounded and continuous initial function on $[0,t_0)$ or $(-\infty,t_0)$, then both Eq.(1) and Eq.(2) have solutions denoted by $x(t,t_0,\phi)$ with $x(t,t_0,\phi)=\phi(t)$ for $t<t_0$, satisfying Eq.(1) or Eq.(2) on $[t_0,\infty)$. (cf. Burton-Furumochi [4].) A solution $x(t,t_0,\phi)$ may have a discontinuity at $t_0$.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=39 |
| spellingShingle | Tetsuo Furumochi Theodore Burton Periodic and asymptotically periodic solutions of neutral integral equations Electronic Journal of Qualitative Theory of Differential Equations |
| title | Periodic and asymptotically periodic solutions of neutral integral equations |
| title_full | Periodic and asymptotically periodic solutions of neutral integral equations |
| title_fullStr | Periodic and asymptotically periodic solutions of neutral integral equations |
| title_full_unstemmed | Periodic and asymptotically periodic solutions of neutral integral equations |
| title_short | Periodic and asymptotically periodic solutions of neutral integral equations |
| title_sort | periodic and asymptotically periodic solutions of neutral integral equations |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=39 |
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