The authors establish conditions under which all solutions of the integrodifferential equation vanish at least once on where , , . If is ultimately non-increasing and on some subinterval of then it is shown that nontrivial solutions will possess such zeros (called zero crossings) provided the associated characteristic equation, viz., possesses no real roots.
The authors prove that all nontrivial solutions have zero crossings if or . They consider also the logistic integrodifferential equation where , , for , and .
If is ultimately non-increasing and on some subinterval of then all positive solutions possess ”level crossings”, i.e., for some , if for .