On oscillation and nonoscillation of general functional differential equations (Q1074759)

The n-th order functional differential equation (1) \(L_ nx(t)+f(t,x(t),x(g(t)))=h(t),\) where \(L_ 0x=x\), \(L_ kx=a_ k(L_{k- 1}x)'\), \(k=1,...,n\), \(a_ n=1\), is considered. Conditions which guarantee that any oscillatory solution of (1), satisfying a growth estimate, tends to 0 as \(t\to \infty\) (together with \(L_ kx\), \(k=1,...,n-1)\) are given. As consequences, some nonoscillation criteria for (1) are derived. Moreover, necessary and sufficient conditions so that all oscillatory solutions of \(L_ nx(t)+q(t)f(x(g(t)))=h(t)\) tend to 0 as \(t\to \infty\) are established.