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**Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation.**
*(English)*
Zbl 0763.34009

The paper deals with the second order three-point nonlinear boundary value problem (1) \(u''=f(x,u(x),u'(x))-e(x)\), \(0<x<1\), \(u(0)=0\), \(u(\eta)=u(1)\), where \(f\) is a Carathéodory function and \(e\) is a Lebesgue integrable function. Provided \(f\) has at most linear growth in its phase variables, the author establishes conditions for the existence of solutions to (1) and for the uniqueness of problem (1). The proofs are based on the topological degree theory and the Leray-Schauder continuation theorem. A priori estimates are obtained by Wirtinger-type inequalities.

Reviewer: I.Rachůnková (Olomouc)

### MSC:

34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |

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

### Keywords:

second order three-point nonlinear boundary value problem; existence; uniqueness; topological degree; Leray-Schauder continuation theorem; Wirtinger-type inequalities
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\textit{C. P. Gupta}, J. Math. Anal. Appl. 168, No. 2, 540--551 (1992; Zbl 0763.34009)

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### References:

[1] | Kiguradze, I. T.; Lomtatidze, A. G., In certain boundary value problems for second-order linear ordinary differential equations with singularities, J. Math. Anal. Appl., 101, 325-347 (1984) · Zbl 0559.34012 |

[2] | Mawhin, J., Topological degree methods in nonlinear boundary value problems, (“NSF-CBMS Regional Conference Series in Math.,” No. 40 (1979), Amer. Math. Soc.,: Amer. Math. Soc., Providence, RI) · Zbl 0414.34025 |

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