Nonoscillation of generalized nonautonomous logistic equation with multiple delays (Q675735)

The existence (some sufficient and also necessary and sufficient conditions) of nonoscillatory solutions about \(k\) of the following generalized delay logistic equation is established: \[ x'(t)=r(t)x(t) [a-(x(g_1(t))/k)-(x(g_2(t))/k)-\cdots -(x(g_m(t))/k)]^\alpha,\quad t\geq 0 \] with the following conditions: (i) \(r,g_i\in C([0,\infty)],[0,\infty))\), as \(t\to\infty\) \(g_i(t)<t\), \(g_i(t)\to\infty\), \(g_i(t)\) are nondecreasing for \(t\geq 0\), \(1\leq i\leq m\), \(a>0\in \mathbb{R}\); (ii) \(k>0\) is a constant and \(\alpha\neq 1\) is the rate of two positive odd integers. By substituting \(y(t)=(x(t)/k)-1\) the above equation is transformed into the form \[ y'(t)=-r(t)(1+y(t)) [(a+m)+\sum_{i=1}^m y(g_i(t))]^\alpha \] The main result in the paper is that if \(0<\alpha<1\) then this equation has a nonoscillatory solution if and only if \(\int_0^\infty r(s) ds<\infty\).