Necessary conditions for the existence of global solutions of systems of higher-order nonlinear ordinary differential equations and inequalities (Q1768968)

The paper concerns the existence times \(T\in(1,+\infty]\) of the solutions \(x:[1,T)\to \mathbb{R}\), \(y:[1,T)\to \mathbb{R}\) to a system which consists of the differential inequalities \[ x^{[k]}(t)\geq a(t)| y(t)|^r| x(t)|^s,\quad y^{[l]}(t)\geq b(t)| y(t)|^p| x(t)|^q, \] and of the initial inequalities \[ x^{[k-1]}(1)>0, \quad y^{[l-1]}(1)>0, \] (the trivial solutions, which occur only in case both \(k=1\), \(l=1\), are excluded). Sufficient conditions are provided for the relation \(T<+\infty\) to be always satisfied (cf. J. Hay [Differ. Equ. 38, No.3, 362--368; translation from Differ. Uravn. 38, No.3, 344-350 (2002; Zbl 1033.34014)]).