On the existence of positive solutions of integrodifferential equations. (English) Zbl 0672.45007
This paper is concerned with the existence of positive solutions of the integrodifferential equation
\[
(- 1)^{n+1}x^{(n)}(t)+\int^{t}_{c}K(t-s)x(s)ds=0\quad (c=0\;or\;c=- \infty),
\]
where n is a positive integer and K is a nonnegative continuous function on the interval \([0,\infty)\). More precisely, sufficient conditions are derived for this equation to have positive solutions which tend to zero at \(\infty\). For the special case where \(n=2,\) necessary conditions, and necessary and sufficient conditions are obtained for the existence of positive solutions which are bounded at \(\infty\).
Reviewer: Ch.G.Philos
MSC:
| 45J05 | Integro-ordinary differential equations |
| 45M05 | Asymptotics of solutions to integral equations |
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