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Necessary and sufficient conditions for ”zero crossing” in integrodifferential equations. (English) Zbl 0735.45007
The authors establish conditions under which all solutions $x(t)$ of the integrodifferential equation $(d/dt)[x(t)-cx(t-\tau)]+a\int\sb 0\sp \infty K(s)x(t-s)ds=0$ vanish at least once on $(-\infty,\infty)$ where $a\in(0,\infty)$, $c\in[0,1)$, $\tau\in[0,\infty)$. If $K$ is ultimately non-increasing and $K\not\equiv0$ on some subinterval of $[0,\infty)$ then it is shown that nontrivial solutions $x(t)$ will possess such zeros (called zero crossings) provided the associated characteristic equation, viz., $\lambda(1-ce\sp{-\lambda\tau})+a\int\sb 0\sp \infty K(s)e\sp{- \lambda s}ds=0$ possesses no real roots.\par\noindent The authors prove that all nontrivial solutions $x(t)$ have zero crossings if $a\int\sb 0\sp \infty K(s)s ds>(1-c)/e$ or $(a+c)\int\sb 0\sp \infty K(s)ds>1/e$. They consider also the logistic integrodifferential equation $dN(t)/dt=rN(t)[1-C\sp{-1}\int\sb 0\sp \infty K(s)N(t-s)ds]$ where $r,C\in(0,\infty)$, $N(0)>0$, $N(s)\ge0$ for $s\in(-\infty,0]$, $K:[0,\infty)\to[0,\infty)$ and $\int\sb 0\sp \infty K(s)ds=1$. If $K$ is ultimately non-increasing and $K\not\equiv0$ on some subinterval of $[0,\infty)$ then all positive solutions $N(t)$ possess ”level crossings”, i.e., $N(t\sp*)=C$ for some $t\sp*\in(- \infty,\infty)$, if $r\int\sb 0\sp \infty K(s)e\sp{\lambda s}ds>\lambda$ for $\lambda\in(0,\infty)$.

MSC:
45J05Integro-ordinary differential equations
45M15Periodic solutions of integral equations
92D25Population dynamics (general)
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References:
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