### Abstract

We consider a boundary-value problem of the form Lu = ?f(u), where L is a 2m-th order disconjugate ordinary differential operator (m = 2 is an integer), ? ? [0, 8), and the function f : R ? R is C^{2} and satisfies f(?) > 0, ? ? R. Under various convexity or concavity type assumptions on f we show that this problem has a smooth curve, S_{0}, of solutions (?,u), emanating from (?, u) = (0, 0), and we describe the shape and asymptotes of S_{0}. All the solutions on S_{0} are positive and all solutions for which u is stable lie on S_{0}.

Original language | English |
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Pages (from-to) | 1-16 |

Number of pages | 16 |

Journal | Electronic Journal of Differential Equations |

Volume | 2004 |

Publication status | Published - 3 Mar 2004 |

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### Keywords

- Nonlinear boundary value problems
- Ordinary differential equations

### Cite this

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*Electronic Journal of Differential Equations*, vol. 2004, pp. 1-16.

**Solution curves of 2m-th order boundary-value problems.** / Rynne, Bryan P.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Solution curves of 2m-th order boundary-value problems

AU - Rynne, Bryan P.

PY - 2004/3/3

Y1 - 2004/3/3

N2 - We consider a boundary-value problem of the form Lu = ?f(u), where L is a 2m-th order disconjugate ordinary differential operator (m = 2 is an integer), ? ? [0, 8), and the function f : R ? R is C2 and satisfies f(?) > 0, ? ? R. Under various convexity or concavity type assumptions on f we show that this problem has a smooth curve, S0, of solutions (?,u), emanating from (?, u) = (0, 0), and we describe the shape and asymptotes of S0. All the solutions on S0 are positive and all solutions for which u is stable lie on S0.

AB - We consider a boundary-value problem of the form Lu = ?f(u), where L is a 2m-th order disconjugate ordinary differential operator (m = 2 is an integer), ? ? [0, 8), and the function f : R ? R is C2 and satisfies f(?) > 0, ? ? R. Under various convexity or concavity type assumptions on f we show that this problem has a smooth curve, S0, of solutions (?,u), emanating from (?, u) = (0, 0), and we describe the shape and asymptotes of S0. All the solutions on S0 are positive and all solutions for which u is stable lie on S0.

KW - Nonlinear boundary value problems

KW - Ordinary differential equations

UR - http://www.scopus.com/inward/record.url?scp=3042550347&partnerID=8YFLogxK

M3 - Article

VL - 2004

SP - 1

EP - 16

JO - Electronic Journal of Differential Equations

JF - Electronic Journal of Differential Equations

SN - 1072-6691

ER -