Solvability for second-order three-point boundary value problems on a half-line. (English) Zbl 1123.34307

Summary: This work is concerned with the existence of a solution to the second-order three-point boundary value problem on the half-line \[ \begin{cases} x''(t)+f(t,x(t), x'(t))=0,\;0<t<+\infty,\\ x(0)=\alpha x(\eta),\;\lim_{t\to+\infty}x'(t)=0, \end{cases} \] where \(\alpha\in \mathbb{R}\), \(\alpha\neq 1\) and \(\eta\in(0,+\infty)\) are given. After the discussion of the Green function for the corresponding homogeneous system on the half-line, we establish some criteria for the existence of solutions to the system discussed with suitable conditions imposed on \(f\). The results are obtained by the Leray-Schauder continuation theorem.


34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
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