A uniqueness and existence theorem for a singular third-order boundary value problem on \([0,\infty)\).

*(English)*Zbl 1021.34020Summary: It is proved that the singular third-order boundary value problem
\[
y'''=f(y),\quad y(0) = 0,\quad y(+\infty) = 1,\quad y'(+\infty) = y''(+\infty) = 0,
\]
has a unique solution. Here, \(f(y) = (1-y)^{\lambda}g(y)\), \(\lambda > 0\), \(g(y)\) is positive and continuous on \((0,1]\). The problem arises in the study of draining and coating flows.

##### MSC:

34B40 | Boundary value problems on infinite intervals for ordinary differential equations |

34B16 | Singular nonlinear boundary value problems for ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

34B60 | Applications of boundary value problems involving ordinary differential equations |

34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |

76S05 | Flows in porous media; filtration; seepage |

##### Keywords:

singular third-order boundary value problem; singular nonlinear second-order initial value problem; positive solution; uniqueness; existence
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\textit{D. Jiang} and \textit{R. P. Agarwal}, Appl. Math. Lett. 15, No. 4, 445--451 (2002; Zbl 1021.34020)

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##### References:

[1] | Wang, J.Y.; Zhang, Z.X., A boundary value problem from draining and coating flows involving a third-order differential equation, Zamp, 49, 506-513, (1998) · Zbl 0907.34016 |

[2] | Tuck, E.O.; Schwartz, L.W., A numerical and asymptotic study of some third-order differential equations relevant to draining and some coating flows, SIAM rev., 32, 453-469, (1990) · Zbl 0705.76062 |

[3] | Bernis, F.; Peletier, L.A., Two problems from draining flows involving third-order differential equations, SIAM J. math. anal., 27, 515-527, (1996) · Zbl 0845.34033 |

[4] | Troy, W.C., Solutions of third-order differential equations relevant to draining and some coating flows, SIAM J. math. anal., 24, 155-171, (1993) · Zbl 0807.34030 |

[5] | Agarwal, R.P.; O’Regan, D., Positive solutions to singular initial value problems with sign changing non-linearities, Mathl. comput. modelling, 28, 3, 31-39, (1998) · Zbl 1098.34500 |

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