Differential equations with bounded positive Green's functions and generalized Aizerman's hypothesis (Q1890027)

The author shows that if \(b_k(t) \geq 0\) for \(t \geq 0, k = 1, \dots , n\), and if the roots of the polynomial \(P(\lambda) = \lambda^n + c_1 \lambda^{n-1} + \cdots + c_n, c_k > 0\) for \(k = 1, \dots, n,\) lie in the open left half-plane, and if \[ G_0(t) = \frac{1}{2\pi} \int_{-\infty}^\infty \! e^{i \omega t} P^{-1}(i \omega) \, d \omega \geq 0 \] and \[ K(t,s) = \frac{1}{2\pi}\int_{-\infty}^\infty \! e^{i \omega t} P^{-1}(i \omega) Q(i \omega, s) \, d\omega \geq 0 \] for \(s, t \geq 0\), where \(Q(z, s) = \sum_{k=1}^n b_k(s) z^{n-k}\), then Green's function \(W( \cdot, \cdot)\) for the equation \[ P(D)x(t) = \sum_{k=0}^{n-1} \frac{d^k}{dt^{k}}\big ( b_{n-k}(t) x(t) \big )=0, D = \frac{d}{dt}, t \geq 0, \] with \[ \frac{\partial^k}{\partial t^k} W(t, \tau) = 0, k = 0, \dots, n-2; \frac{\partial^{n-1}}{\partial t^{n-1}} W(t, \tau) = 1, \] is positive. In addition, the author establishes upper and lower bounds for Green's function on the half-line. Furthermore, the author shows that if \(F(y_1, y_2, \dots , y_n, t) \geq 0\) for \(t \geq 0, y_k \geq 0\), then \[ P(D) x(t) = \sum_{k=0} \frac{d^k}{dt^{k}} \big ( b_{n-k}(t) x(t) \big ) + F(x(t), x'(t), \dots, x^{(n-1)}(t), t)=0 \] has nonnegative solutions. Finally, the author shows that if the Green's function is positive, then the nonlinear equation having separated nonautonomous linear parts satisfy the generalized Aizerman hypothesis on absolute stability. This paper will be of interest to anyone studying sign conditions of Green's functions and positive solutions.