8.
We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium
N*=1/(
a+∑
i=0mbi) of the following differential equation with piecewise constant arguments:
where
r(
t) is a nonnegative continuous function on [0,+∞),
r(
t)0, ∑
i=0mbi>0,
bi0,
i=0,1,2,…,
m, and
a+∑
i=0mbi>0. These new conditions depend on
a,
b0 and ∑
i=1mbi, and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case
m=0 and
r(
t)≡
r>0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms:
where
r(
t) is a nonnegative continuous function on [0,+∞),
r(
t)0, 1−
ax−
g(
x,
x,…,
x)=0 has a unique solution
x*>0 and
g(
x0,
x1,…,
xm)
C1[(0,+∞)×(0,+∞)××(0,+∞)].
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