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.