Behaviour of solutions of a third order differential equation (Q1315070)

The authors study properties of proper and regular non-oscillatory solutions of the equation (1) \(x'''+ f(t) g(x,x',x'') =0\), where (i) \(f\) is continuous on \((a,\infty)\), \(a>-\infty\), and \(f(t)>0\) for all \(t\in (a,\infty)\), (ii) \(g\) is continuous on \(R^ 3\) and \(g(y_ 0, y_ 1, y_ 2)>0\) for all \((y_ 0, y_ 1, y_ 2)\in R^ 3\) with \(y_ 0\neq 0\). A function \(x\in C^ 3\langle T_ x,\infty)\) \((T_ x\geq a)\) being a solution of (1) is said to be a) a proper solution if \(\sup\{| x(t)|\): \(t\geq T\}>0\) for every \(T>T_ x\); b) a regular solution if it is proper and if for some \(T\geq T_ x\) it does not have a triple null point in \(\langle T,\infty)\).