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Global existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations. (English) Zbl 1017.34005
The differential equation under consideration is \(u''+ f(t,u,u')= 0\). A typical result is the following. If there exist continuous positive nondecreasing functions \(h\), \(p_1,p_2: \mathbb{R}_+\to \mathbb{R}_+\) such that \[ |f(t,u,v)|\leq h(t)\Biggl[p_1\Biggl({|u|\over t}\Biggr)+ p_2(|v|)\Biggr], \] where \(h(s)\) satisfies \(\int^\infty_{h_0} h(s) ds= H< \infty\). Assume also that \(G(+\infty)= +\infty\), where, for \(x\geq t_0\), \[ G(x)= \int^x_{t_0} {dx\over p_1(s)+ p_2(s)}. \] Then, for every \(u_0,u_1\in \mathbb{R}\), the initial value problem \[ u''+ f(t,u,u')= 0,\quad t\geq t_0,\quad u(t_0)= u_0,\quad u'(t_0)= u_1, \] has at least one solution \(u(t)\) defined on \([t_0,+\infty)\) with the asymptotic representation \(u(t)= at+ o(t)\) as \(t\to+\infty\), where \(a\) is a real constant. Further results with variations of these conditions are investigated in some detail.
Reviewer: P.Smith (Keele)

MSC:
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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