Asymptotic speed of spreading of a nonlocal delayed equation without quasimonotonicity

This paper is concerned with the asymptotic speed of spreading of a nonlocal delayed equation, which does not satisfy the local quasimonotonicity. By constructing auxiliary undelayed equations, the asymptotic speed of spreading is established. In particular, for such a nonmonotonic equation, the asymptotic speed of spreading is the same as that in the corresponding undelayed equation.

A Final Result on the Oscillation of Solutions of the Linear Discrete Delayed EquationΔx(n)=−p(n)x(n−k)with a Positive Coefficient

A linear(k+1)th-order discrete delayed equationΔx(n)=−p(n)x(n−k)wherep(n)a positive sequence is considered forn→∞. This equation is known to have a positive solution if the sequencep(n)satisfies an inequality. Our aim is to show that, in the case of the opposite inequality forp(n), all solutions of the equation considered are oscillating forn→∞.