Periodic solutions of Volterra integral equations
Consider the system of equationsx(t)=f(t)+∫−∞tk(t,s)x(s)ds, (1)andx(t)=f(t)+∫−∞tk(t,s)g(s,x(s))ds. (2)Existence of continuous periodic solutions of (1) is shown using the resolvent function of the kernel k. Some important properties of the resolvent function including its uniqueness...
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| Format: | Article |
| Language: | English |
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Wiley
1988-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S016117128800095X |
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| author | M. N. Islam |
| author_facet | M. N. Islam |
| author_sort | M. N. Islam |
| collection | DOAJ |
| description | Consider the system of equationsx(t)=f(t)+∫−∞tk(t,s)x(s)ds, (1)andx(t)=f(t)+∫−∞tk(t,s)g(s,x(s))ds. (2)Existence of continuous periodic solutions of (1) is shown using the resolvent function of the kernel k. Some important properties of the resolvent function including its uniqueness are obtained in the process. In obtaining periodic solutions of (1) it is necessary that the resolvent of k is integrable in some sense. For a scalar convolution kernel k some explicit conditions are derived to determine whether or not the resolvent of k is integrable. Finally, the existence and uniqueness of continuous periodic solutions of (1) and (2) are btained using the contraction mapping principle as the basic tool. |
| format | Article |
| id | doaj-art-04b1e680a9da4d438d3980c8f206cbec |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1988-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-04b1e680a9da4d438d3980c8f206cbec2025-02-03T05:52:21ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251988-01-0111478179210.1155/S016117128800095XPeriodic solutions of Volterra integral equationsM. N. Islam0Department of Mathematics, University of Dayton, Dayton 45469, OH, USAConsider the system of equationsx(t)=f(t)+∫−∞tk(t,s)x(s)ds, (1)andx(t)=f(t)+∫−∞tk(t,s)g(s,x(s))ds. (2)Existence of continuous periodic solutions of (1) is shown using the resolvent function of the kernel k. Some important properties of the resolvent function including its uniqueness are obtained in the process. In obtaining periodic solutions of (1) it is necessary that the resolvent of k is integrable in some sense. For a scalar convolution kernel k some explicit conditions are derived to determine whether or not the resolvent of k is integrable. Finally, the existence and uniqueness of continuous periodic solutions of (1) and (2) are btained using the contraction mapping principle as the basic tool.http://dx.doi.org/10.1155/S016117128800095XVolterra integral equationperiodic solutionresolventintegrability of resolventlimit equation. |
| spellingShingle | M. N. Islam Periodic solutions of Volterra integral equations International Journal of Mathematics and Mathematical Sciences Volterra integral equation periodic solution resolvent integrability of resolvent limit equation. |
| title | Periodic solutions of Volterra integral equations |
| title_full | Periodic solutions of Volterra integral equations |
| title_fullStr | Periodic solutions of Volterra integral equations |
| title_full_unstemmed | Periodic solutions of Volterra integral equations |
| title_short | Periodic solutions of Volterra integral equations |
| title_sort | periodic solutions of volterra integral equations |
| topic | Volterra integral equation periodic solution resolvent integrability of resolvent limit equation. |
| url | http://dx.doi.org/10.1155/S016117128800095X |
| work_keys_str_mv | AT mnislam periodicsolutionsofvolterraintegralequations |