On an asymptotic property of a nonlinear ordinary differential equation (Q1192430)

In the study of the fifth Painlevé equation we treated an equation of the form \[ x(xu')'={\alpha \over 2} \tanh u \cosh^{-2}u+{\gamma \over 4} x \sinh 2u+{\delta \over 8}x^ 2 \sinh 4u \] \(('=d/dx)\), where \(\alpha\), \(\gamma \in \mathbb{R}\), \(\delta<0\). In [Funkc. Ekvacioj, Ser. Int. 30, 203-224 (1987; Zbl 0654.34049)] we studied an asymptotic behaviour of the solution \(u=u_ 0(x)=u(x_ 0,u_ 0,u_ 0';x)\) \((x_ 0>0,\;u_ 0,\;u_ 0' \in \mathbb{R})\) as \(x \to+\infty\) satisfying an initial condition \(u_ 0(x_ 0)=u_ 0\), \(u_ 0'(x_ 0)=u_ 0'\). In this paper we consider a more general nonlinear equation of the form \(v''+v\Phi(x,v)=0\). Under some assumptions we prove that the solution \(v=V(x)\) satisfying an initial condition as above can be prolonged over the interval \(x_ 0 \leq x<+\infty\), and we give an asymptotic expression of \(V(x)\) as \(x \to+\infty\).