Abstract :
In this paper, we improve contractivity conditions of solutions for the positive equilibrium
N∗ = 1
a+ m
i =0 bi
of the following differential equation with piecewise constant arguments:
⎧⎪
⎨⎪
⎩
dN(t)
dt = N(t)r(t) 1−aN(t) −
m
i=0
biN(n − i) , n t 0 and N(−j) = N−j 0, j = 1, 2, . . . ,m,
where r(t) is a nonnegative continuous function on [0,+∞), r(t) ≡ 0, m
i =0 bi > 0, bi 0,
i = 0, 1, 2, . . . ,m, and a + b0 > m
i =1 bi . In particular, for the case a = 0 and m 1, we
really improve the known three type conditions of the contractivity for solutions of this
model (see for example, [Y. Muroya, A sufficient condition on global stability in a logistic
equation with piecewise constant arguments, Hokkaido Math. J. 32 (2003) 75–83]). For the
other case a = 0 and m 1, under the condition mj
=1 b j − 2b0 < a ( mj
=1 b j)/(1 +
b0/ mj
=0 b j ), the obtained result partially improves the known results on the contractivity
of solutions for the positive equilibrium of this model given by the author [Y. Muroya,
Persistence, contractivity and global stability in logistic equations with piecewise constant
delays, J. Math. Anal. Appl. 270 (2002) 602–635] and others
