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Multiple unbounded positive solutions for three-point BVPs with sign-changing nonlinearities on the positive half-line. (English) Zbl 1195.34042

The authors consider the following second-order nonlinear three-point boundary value problem on the positive half-line

$-{x}^{\text{'}\text{'}}+c{x}^{\text{'}}+\lambda x=f\left(t,x\left(t\right),{x}^{\text{'}}\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in \left(0,\infty \right),$
$x\left(0\right)-\alpha x\left(\eta \right)={a}_{0},\phantom{\rule{1.em}{0ex}}\underset{t\to \infty }{lim}\frac{{x}^{\text{'}}\left(t\right)}{r{e}^{rt}}={b}_{0},$

where ${a}_{0},{b}_{0}$ are nonnegative real numbers, $\alpha \ge 0,$ $\eta >0,$ $c,\lambda$ are real positive constants, $r\in \left(0,c\right),$ and $f:\left(0,\infty \right)×\left[0,\infty \right)×ℝ\to ℝ$ is a Carathéodory function which may change sign. Under general polynomial growth conditions on $f$ the existence of nontrivial single and multiple unbounded positive solutions is proved via fixed point theorems in a cone in a special weighted Banach space.

##### MSC:
 34B40 Boundary value problems for ODE on infinite intervals 34B15 Nonlinear boundary value problems for ODE 34B10 Nonlocal and multipoint boundary value problems for ODE 34B16 Singular nonlinear boundary value problems for ODE 34B18 Positive solutions of nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations
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