Property \(A\) of the \((n+1)^{th}\) order differential equation \(\left [\frac 1{r_1(t)}\left (x^{(n)}(t)+p(t)x(t)\right )\right ]' = f(t,x(t),\cdots ,x^{(n)}(t))\). (English) Zbl 1072.34034

Asymptotic properties of nonoscillatory solutions of \[ \left [{1\over r_{1}(t)} (x^{(n)} + p(t) x)\right ]' = f(t, x, \cdots , x^{(n)}) \tag{1} \] are investigated in case \(r\) and \(p\) are bounded from bellow and from above by positive constants and \(f\) fulfils the sign condition \(f (t, x_1, \dots , x_{n+1}) x_1 \leq 0\). Hence, the operator \(x^{(n)} + p x\) is oscillatory and this case is very rare studied. Sufficient conditions are given for every nonoscillatory proper solution \(x\) of (1) to be the solution of \(x^{(n)} + p x = \alpha (t)\, sgn\, x(t)\) with a positive function \(\alpha \) (depending on \(x\)), \(\lim _{t\to \infty } \alpha (t) = 0\).


34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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