Continuity, compactness, fixed points, and integral equations
An integral equation, $x(t)=a(t)-\int^t_{-\infty} D(t,s)g(x(s))ds$ with $a(t)$ bounded, is studied by means of a Liapunov functional. There results an a priori bound on solutions. This gives rise to an interplay between continuity and compactness and leads us to a fixed point theorem of Schaefer ty...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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University of Szeged
2002-01-01
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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=90 |
| Summary: | An integral equation, $x(t)=a(t)-\int^t_{-\infty} D(t,s)g(x(s))ds$ with $a(t)$ bounded, is studied by means of a Liapunov functional. There results an a priori bound on solutions. This gives rise to an interplay between continuity and compactness and leads us to a fixed point theorem of Schaefer type. It is a very flexible fixed point theorem which enables us to show that the solution inherits properties of $a(t)$, including periodic or almost periodic solutions in a Banach space. |
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| ISSN: | 1417-3875 1417-3875 |