On the asymptotic behavior of solutions of a higher order linear differential equation. (English) Zbl 0888.34008
The paper deals with the existence of unbounded solutions of the equation
\[
(1)\quad y^{(2n)}+ p(t)y= 0\qquad\text{and} \qquad(2)\quad y^{(2n+1)}+ p(t)y= 0,
\]
respectively, where \(p(t)\) is a real-valued continuous function on \([0,\infty)\) and such that \((-1)^np(t)> 0\), \(|p(t)|< M\) for (1) and \((-1)^{n+ 1}p(t)> 0\) and \(|p(t)|< M\) for (2).
Following U. Elias let \(\sigma(c_0,c_1,\dots, c_n)\) denote the number of sign changes in the sequence \(c_0,c_1,\dots, c_n\) of non-zero numbers. For \(y(t)\in C^n(0, \infty)\) such that \(y(t)\neq 0\) on \((0,\infty)\) let \[ S(y, x^+)= \lim_{t\to x^+} \sigma(y(t), -y'(t),\dots, (-1)^ny^{(n)}(t)) \] and \[ S(y, x^-)= \lim_{t\to x^-} \sigma(y(t), y'(t),\dots, y^{(n)}(t)). \] The solution of (1) or of (2) is from \(S_k\) provided \[ \lim_{x\to\infty} S(y, x^+)= k. \] The main result for equation (1) is the following: There are linearly independent solutions \(y_i\), \(i=n,n+1,\dots, 2n-1\), of (1) so that every nontrivial linear combination of them is unbounded. For \(i= n,n+1,\dots, 2n-1\), \(y_i\) can be chosen to have a zero at \(x= a\) of order exactly \(i\). For \(p(t)> 0\), \(i= n,n+2,\dots, 2n-2\), \(y_i\) and \(y_{i+1}\) and are in \(S_{i+1}\) while for \(p(t)< 0\), \(y_{2n-1}\) is in \(S_{2n}\), \(y_i\) and \(y_{i+1}\) are in \(S_{i+1}\) for \(i= n,n+2,\dots, 2n-3\). Another result is the following: Let be \((-1)^np(t)> 0\), \(|p(t)|< M\) and \(y(t)\) a solution of (1) such that \[ F(y(t))= \sum^{n- 1}_{i=0} (-1)^i y^{(n-i- 1)}(t) y^{(n+ i)}(t)> 0 \] for \(t>a\). Then \(y(t)\) is unbounded. Similar theorems are stated for the solutions of the equation (2). The paper generalizes results of many authors and of the author himself.
Following U. Elias let \(\sigma(c_0,c_1,\dots, c_n)\) denote the number of sign changes in the sequence \(c_0,c_1,\dots, c_n\) of non-zero numbers. For \(y(t)\in C^n(0, \infty)\) such that \(y(t)\neq 0\) on \((0,\infty)\) let \[ S(y, x^+)= \lim_{t\to x^+} \sigma(y(t), -y'(t),\dots, (-1)^ny^{(n)}(t)) \] and \[ S(y, x^-)= \lim_{t\to x^-} \sigma(y(t), y'(t),\dots, y^{(n)}(t)). \] The solution of (1) or of (2) is from \(S_k\) provided \[ \lim_{x\to\infty} S(y, x^+)= k. \] The main result for equation (1) is the following: There are linearly independent solutions \(y_i\), \(i=n,n+1,\dots, 2n-1\), of (1) so that every nontrivial linear combination of them is unbounded. For \(i= n,n+1,\dots, 2n-1\), \(y_i\) can be chosen to have a zero at \(x= a\) of order exactly \(i\). For \(p(t)> 0\), \(i= n,n+2,\dots, 2n-2\), \(y_i\) and \(y_{i+1}\) and are in \(S_{i+1}\) while for \(p(t)< 0\), \(y_{2n-1}\) is in \(S_{2n}\), \(y_i\) and \(y_{i+1}\) are in \(S_{i+1}\) for \(i= n,n+2,\dots, 2n-3\). Another result is the following: Let be \((-1)^np(t)> 0\), \(|p(t)|< M\) and \(y(t)\) a solution of (1) such that \[ F(y(t))= \sum^{n- 1}_{i=0} (-1)^i y^{(n-i- 1)}(t) y^{(n+ i)}(t)> 0 \] for \(t>a\). Then \(y(t)\) is unbounded. Similar theorems are stated for the solutions of the equation (2). The paper generalizes results of many authors and of the author himself.
Reviewer: M.Švec (Bratislava)
MSC:
34A30 | Linear ordinary differential equations and systems |
34D05 | Asymptotic properties of solutions to ordinary differential equations |