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Solvability for second-order three-point boundary value problems at resonance on a half-line. (English) Zbl 1136.34034

Summary: This paper deals with the solvability and uniqueness of the second-order three-point boundary value problems at resonance on a half-line

${x}^{\text{'}\text{'}}\left(t\right)=f\left(t,x\left(t\right),{x}^{\text{'}}\left(t\right)\right),\phantom{\rule{1.em}{0ex}}0
$x\left(0\right)=x\left(\eta \right),\phantom{\rule{1.em}{0ex}}\underset{t\to +\infty }{lim}{x}^{\text{'}}\left(t\right)=0,$

and

${x}^{\text{'}\text{'}}\left(t\right)=f\left(t,x\left(t\right),{x}^{\text{'}}\left(t\right)\right)+e\left(t\right),\phantom{\rule{1.em}{0ex}}0
$x\left(0\right)=x\left(\eta \right),\phantom{\rule{1.em}{0ex}}\underset{t\to +\infty }{lim}{x}^{\text{'}}\left(t\right)=0,$

where $f:\left[0,+\infty \right]×{ℝ}^{2}\to ℝ,$ $e:\left[0,+\infty \right]\to ℝ$ are continuous and $\eta \in \left(0,+\infty \right)$. By using the coincidence degree theory, we establish some existence and uniqueness criteria.

##### MSC:
 34B40 Boundary value problems for ODE on infinite intervals 34B10 Nonlocal and multipoint boundary value problems for ODE
##### Keywords:
coincidence degree theory; infinite intervals
##### References:
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