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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-τ)]+a 0 K(s)x(t-s)ds=0 vanish at least once on (-,) where a(0,), c[0,1), τ[0,). If K is ultimately non-increasing and K¬0 on some subinterval of [0,) then it is shown that nontrivial solutions x(t) will possess such zeros (called zero crossings) provided the associated characteristic equation, viz., λ(1-ce -λτ )+a 0 K(s)e -λs ds=0 possesses no real roots.

The authors prove that all nontrivial solutions x(t) have zero crossings if a 0 K(s)sds>(1-c)/e or (a+c) 0 K(s)ds>1/e. They consider also the logistic integrodifferential equation dN(t)/dt=rN(t)[1-C -1 0 K(s)N(t-s)ds] where r,C(0,), N(0)>0, N(s)0 for s(-,0], K:[0,)[0,) and 0 K(s)ds=1.

If K is ultimately non-increasing and K¬0 on some subinterval of [0,) then all positive solutions N(t) possess ”level crossings”, i.e., N(t * )=C for some t * (-,), if r 0 K(s)e λs ds>λ for λ(0,).

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