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

Bryan P. Rynne

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

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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 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.

Original language	English
Pages (from-to)	1-16
Number of pages	16
Journal	Electronic Journal of Differential Equations
Volume	2004
Publication status	Published - 3 Mar 2004

Keywords

Nonlinear boundary value problems
Ordinary differential equations

Cite this

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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

ER -

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

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Keywords

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