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**A Dirichlet type multi-point boundary value problem for second order ordinary differential equations.**
*(English)*
Zbl 0847.34018

The author considers the second order ordinary differential equation (1) \(x'' (t)= f(t, x(t), x'(t))+ e(t)\), \(0< t< 1\), where \(f: [0, 1] \times \mathbb{R}^2\to \mathbb{R}\) satisfies the Carathéodory conditions and \(e\in L^1 [0, 1]\). He proves the existence of a solution to (1) which satisfies the multipoint boundary value condition (2) \(x(0)= 0\), \(x(1)= \sum^{m- 1}_{i= 1} a_i x(\xi_i)\), with \(a_i\in \mathbb{R}\), \(\xi_i\in (0, 1)\), \(i= 1, 2, \dots, m-2\), \(0< \xi_1< \xi_2< \dots< \xi_{m-2}< 1\). He directs his attention at the nonresonant case \(\sum^{m-2}_{i=1} a_i \xi_i\neq 1\). The existence results are proved by means of Mawhin’s version of the Leray-Schauder continuation theorem and complete the earlier ones by the author, S. K. Ntouyas and P. Ch. Tsamatos in [Nonlinear Anal., Theory Methods Appl. 23, No. 11, 1427-1436 (1994; Zbl 0815.34012)].

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 |

### Citations:

Zbl 0815.34012
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\textit{C. P. Gupta}, Nonlinear Anal., Theory Methods Appl. 26, No. 5, 925--931 (1996; Zbl 0847.34018)

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### References:

[1] | Gupta, C. P.; Ntouyas, S. K.; Tsamatos, P. C.H., On an m-point boundary value problem for second order ordinary differential equations, Nonlinear Analysis, 23, 11, 1427-1436 (1994) · Zbl 0815.34012 |

[2] | Il’in, V. A.; Moiseev, E. I., Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, Diff. Eqns., 23, 7, 803-810 (1987) · Zbl 0668.34025 |

[3] | Il’in, V. A.; Moiseev, E. I., Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator, Diff. Eqns., 23, 8, 979-987 (1987) · Zbl 0668.34024 |

[4] | Bitsadze, A. V., On the theory of nonlocal boundary value problems, Soviet Math. Dokl., 30, 1, 8-10 (1984) · Zbl 0586.30036 |

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[6] | Bicadze, A. V.; Samarskii, A. A., On some simple generalizations of linear elliptic boundary problems, Soviet Math. Dokl., 10, 2, 298-400 (1969) · Zbl 0187.35501 |

[7] | Gupta, C. P., A second order \(m\)-point boundary value problem at resonance, Nonlinear Analysis, 24, 10, 1483-1489 (1995) · Zbl 0824.34023 |

[8] | Gupta C.P., Solvability of a multi-point boundary value problems at resonance, Results Math, (to appear).; Gupta C.P., Solvability of a multi-point boundary value problems at resonance, Results Math, (to appear). |

[9] | Gupta C.P., Existence theorems for a second order \(m\)-point boundary value problem at resonance, Int. J. Math. math. Sci. (to appear).; Gupta C.P., Existence theorems for a second order \(m\)-point boundary value problem at resonance, Int. J. Math. math. Sci. (to appear). · Zbl 0839.34027 |

[10] | Gupta, C. P.; Ntouyas, S. K.; Tsamatos, P. CH., Existence results for \(m\)-point boundary value problems, Diff. Eqns Dynam. Syst., 2, 4, 289-298 (1994) · Zbl 0877.34019 |

[11] | Gupta, C. P.; Ntouyas, S. K.; Tsamatos, P. CH., Solvability of an \(m\)-point boundary value problem for second order ordinary differential equations, J. math. Analysis Applic., 189, 575-584 (1995) · Zbl 0819.34012 |

[12] | Mawhin, J., Toplogical degree methods in nonlinear boundary value problems, (NSF-CBMS Regional Conference Series in Math. No. 40 (1979), Am. Math. Soc.: Am. Math. Soc. New York) · Zbl 0414.34025 |

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