Asymptotic behavior of solutions of an \(n\)th order differential equation (Q795213)

The equation is (1) \(u^{(n)}+f(t,u)=0\). Theorem. Suppose k is an integer, 0\(\leq k\leq n-1\), and \(\phi>0\) is continuous and nonincreasing on \((0,\infty)\). If \(k\neq 0\), suppose there is a \(\gamma<1\) such that \(t^{\gamma}\phi(t)\) is nondecreasing. Let p be a given polynomial of degree \(<n\), and suppose there are constants M, \(T>0\) such that f is continuous on the set \(\Omega =\{(t,u)| t\geq T\), \(| u-p(t)| \leq M\phi(t)t^ k\}\) and \(|(f(t,u_ 1)-f(t,u_ 2)| \leq g(t)| u_ 1-u_ 2|\) if \((t,u_ i)\in \Omega (i=1,2)\), where \(g\in C[T,\infty)\), and (2) \(\int^{\infty}_{t}s^{n- 1}g(s)\phi(s)ds=O(\phi(t))\) (\(t\to \infty)\). Suppose also that (3) \(\int^{\infty}_{t}s^{n-k-1}f(s,p(s))ds=O(\phi(t))\) (where this integral may converge conditionally). Then, if M is sufficiently large, (1) has a solution \(u_ 0\) such that (4) \(u_ 0^{(r)}(t)=p^{(r)}(t)+O(\phi(t)t^{k-r})\) (0\(\leq r\leq n-1)\); moreover, if (2) and (3) hold with ''o'' instead of ''O'', then so does (4), with no restriction on \(M>0\).