Ultimately positive (negative) solutions to a differential inclusion of order \(n\) (Q1327728)

A differential inclusion of the form \(L_ n x(t)\in F(t,x(\varphi(t))\), \(n>1\), is considered, where \(L_ n x(t)\) is the \(n\)th quasiderivative of \(x(t)\) with respect to continuous functions \(a_ i: [t_ 0,\infty)\to (0,\infty)\), \(\int^ \infty_{t_ 0} a^{-1}_ i(t)dt= \infty\). \(L_ n\) are defined by \(L_ 0 x(t)= a_ 0(t)x(t)\), \(L_ i x(t)= a_ i(t)(L_{i-1} x(t))'\). The values of \(F\) are compact intervals and it is upper semicontinuous. Moreover, \(F(t,0)= \{0\}\), \(F(t,x) x<0\) (or \(F(t,x)x>0\)) for \(x\neq 0\). Under some special assumptions the existence of nonoscillatory solutions is proven such that they are asymptotic to the solutions of \(L_ n y(t)= 0\). The conditions are also given implying the existence of nonoscillatory solutions which are not asymptotic to any solutions of \(L_ n y(t)= 0\) -- this is combined with the solutions being negative or positive.