ABSTRACT
By designating col ( u~t), t4t), ... , ~)) as the vector u(t), we see that a solution of (2)-(3) or {4} can also be regarded as a vector sequence {u(t)}~-~F· Furthermore, such a sequence satisfies the delay vector recurrence relation
where
A = ( ~2 0 0
or the delay vector recurrence relation
{5)
72 S. S. Cheng and R. Medina
0 1 0 0
Suppose there is some nonnegative integer T such that a solution u = { u~t)} of (2)-{3) satisfies u~e) > 0 for 1 ~ i ~ n and t ~ T, then u is said to be eventually positive. An eventually negative solution is similarly defined. The double sequence u is said to be oscillatory if it is neither eventually positive nor eventually negative. If we call a vector v positive (denoted by v > 0) when all its components are positive, then clearly a solution u of (2}-(3) is eventually positive if, and only if the vector sequence { u<e>} is eventually positive. Therefore, (2)-(3) has an eventually positive solution if, and only if, the relation {5) has an eventually positive solution.
