On the global attractivity for a logistic equation with piecewise constant arguments

In this paper, we consider the following logistic equation with piecewise constant arguments: dN(t)dt=rN(t){1−∑j=0majN([t−j])},t⩾0,m⩾1,N(0)=N0>0,N(−j)=N−j⩾0,j=1,2,…,m, where r>0, a0,a1,…,am⩾0, ∑j=0maj>0, and [x] means the maximal integer not greater than x. The sequence {Nn}n=0∞, where Nn=N(n), n=0,1,2,… , satisfies the difference equation Nn+1=Nnexpr1−∑j=0majNn−j,n=0,1,2,…. Under the condition that the first term a0 dominates the other m coefficients ai, 1⩽i⩽m, we establish new sufficient conditions of the global asymptotic stability for the positive equilibrium N∗=1/(∑j=0maj).