Existence and asymptotic behavior of positive solutions of neutral differential equations (Q1342487)

The author considers the neutral differential equation \[ {d^ n\over dt^ n} [x(t)- h(t) x(\tau(t))]+ f(t, x(g(t)))= 0, \] where \(h(t)\), \(\tau(t)\), \(g(t)\), \(f(t, u)\) are continuous, \(h(t)\geq 0\), \(0< \tau(t)< t\), \(| f(t, u)|\) is nondecreasing, \(\tau(t)\) is strictly increasing, \(\lim_{t\to\infty} \tau(t)= \lim_{t\to\infty} g(t)= \infty\). The existence and asymptotic behavior of a positive solution \(x\) of this equation such that \(x(t)- h(t) x(\tau(t))/ t^ k\to \text{const}\neq 0\) as \(t\to\infty\), \(k= 1,\dots,n- 1\), are investigated. The author gives a necessary and sufficient condition of positivity of the solution \(x\).