Unbounded positive solutions to higher-order nonlinear functional-differential equations (Q5950591)

The authors deal with the higher-order nonlinear functional-differential equation NEWLINE\[NEWLINEx^{(n)}(t)+ \sigma f(t, x(g(t)))= 0,\quad t\geq t_0,\tag{1}NEWLINE\]NEWLINE where \(n\geq 2\), \(\sigma= 1\) or \(\sigma= -1\), \(g: [t_0,\infty)\to \mathbb{R}\) is a continuous and nondecreasing function such that \(\lim_{t\to\infty} g(t)= \infty\), and \(f: [t_0,\infty)\times \mathbb{R}\to \mathbb{R}\) is a continuous function such that NEWLINE\[NEWLINEf(t, x)\geq 0,\qquad (t,x)\in [t_0,\infty)\times (0,\infty).NEWLINE\]NEWLINE Let \(k\) be an integer such that \(0< k< n\) and \((-1)^{n-k-1}\sigma= 1\). Denote by \(P_k\) the set of positive solutions \(x\) to (1) for which NEWLINE\[NEWLINEx^{(i)}(t)> 0,\quad t\geq T_x,\quad 0\leq i\leq k-1,\quad (-1)^{i-k}x^{(i)}(t)> 0,\quad t\geq T_x,\quad k\leq i\leq n,NEWLINE\]NEWLINE for some sufficiently large \(T_x\).NEWLINENEWLINENEWLINEThe main results here provide necessary and sufficient conditions for the existence of a solution \(x\in P_k\) with the additional properties NEWLINE\[NEWLINE\lim_{t\to\infty} x^{(k)}(t)= 0\quad\text{and}\quad \lim_{t\to\infty} x^{(k-1)}(t)= \infty.NEWLINE\]