If \(n\) is even (odd), \(f(t,w_1, w_2, \ldots, w_n) \leq 0 (\geq 0)\) for \(t \in [0, \infty)\), \(0 \leq (-1)^{i - 1} w_i \leq 1\), \(i = 1,2, \ldots, n\), \(f(.,w)\) is not identically zero on any subinterval of \([0, \infty)\) for every fixed \(w \in R^n\) and \(\lim_{w_1 \to 0 +} {f(t, w_1, w_2, \ldots, w_n) \over w_1} = \vartheta\), where \(- \infty < \vartheta \leq 0\) \((0 \leq \vartheta < \infty)\), then the problem \(y^{(n)} + f(t,y,y', \ldots, y^{(n - 1)}) = 0\), \(t \in [0, \infty)\), \((- 1)^iy^{(i)} (t) \geq 0\), \(i = 0,1, \ldots, n - 1\), admits a nontrivial monotone solution \(y\) such that \(0 < y(0) \leq 1\).