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Unbounded upper and lower solution method for third-order boundary-value problems on the half-line. (English) Zbl 1190.34026
The authors consider the nonlinear boundary value problem
\[ \begin{gathered} u'''(t)+ a(t) f(t,u(t), u'(t), u''(t))= 0,\quad t\in (0,+\infty),\\ u(0)= u'(0)= 0,\quad u''(+\infty)= 0,\end{gathered}\tag{1} \] where \(a: (0,+\infty)\to (0,+\infty)\), \(f: [0,+\infty)\times \mathbb{R}^3\to \mathbb{R}\) are continuous.
By using the upper and lower solutions method, the authors present sufficient conditions for the existence of solutions to (1).
Note: The authors point out that no other works on boundary value problems on the half-line for third-order differential equation by the other researchers have been considered by them as a prototyp. Such problem has been considered, for example, in the work of A. I. Kolosov and S. V. Kolosova [On two-sided approximations in the solution of the Falkner-Skan problem. Mat. Fiz. 23, 63–67 (1978; Zbl 0447.34014)].

34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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