Qualitative analysis of positive solutions of first-order functional differential equations of neutral type (Q1968649)

The paper is concerned with the asymptotic behavior of solutions to the first-order neutral differential equation (1) \([x(t)-x(t-\tau)]'+F(t,x(g(t)))=0\) for \(t\geq t_0\geq 0\), where \(\tau\) is a positive constant; \(F(t,x)\in C([t_0,\infty)\times\mathbb R,\mathbb R)\), \(\text{sgn}F(t,x)=\text{sgn}x\), \(F(t,x)\) is nondecreasing in \(x\); \(g(t)\in C([t_0,\infty),\mathbb R)\), \(g(t)\leq t\) for \(t\geq t_0\), \(\lim_{t\to\infty}g(t)=\infty\). A classification of all positive solutions to equation (1) is given and some results known for the equation \([x(t)-x(t-\tau)]'+p(t)x(t-\sigma)=0\) are extended to equation (1). Moreover, some comparison results are derived.