Positive solutions to a singular third-order three-point boundary value problem with an indefinitely signed Green’s function.

*(English)*Zbl 1153.34016The authors consider the third order three-point boundary value problem,

$${u}^{\text{'}\text{'}\text{'}}=a\left(t\right)f(t,u\left(t\right)),\phantom{\rule{1.em}{0ex}}0<t<1,$$

$$u\left(0\right)=u\left(1\right)={u}^{\text{'}\text{'}}\left(\eta \right)=0,$$

where $\eta \in (17/24,1)$. They show that even though the Green function is not of constant sign, under certain conditions the problem does admit positive solutions. Their approach is based on phase plane analysis coupled with the Krasnosel’skii fixed point theorem for cone preserving operators.

Reviewer: Eric R. Kaufmann (Little Rock)

##### MSC:

34B18 | Positive solutions of nonlinear boundary value problems for ODE |

34B10 | Nonlocal and multipoint boundary value problems for ODE |

34B15 | Nonlinear boundary value problems for ODE |