Global structure of positive solutions for superlinear 2 m th-boundary value problems

Ruyun Ma; Yulian An

Czechoslovak Mathematical Journal (2010)

  • Volume: 60, Issue: 1, page 161-172
  • ISSN: 0011-4642

Abstract

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We consider boundary value problems for nonlinear 2 m th-order eigenvalue problem ( - 1 ) m u ( 2 m ) ( t ) = λ a ( t ) f ( u ( t ) ) , 0 < t < 1 , u ( 2 i ) ( 0 ) = u ( 2 i ) ( 1 ) = 0 , i = 0 , 1 , 2 , ⋯ , m - 1 . where a ∈ C ( [ 0 , 1 ] , [ 0 , ∞ ) ) and a ( t 0 ) > 0 for some t 0 ∈ [ 0 , 1 ] , f ∈ C ( [ 0 , ∞ ) , [ 0 , ∞ ) ) and f ( s ) > 0 for s > 0 , and f 0 = ∞ , where f 0 = lim s → 0 + f ( s ) / s . We investigate the global structure of positive solutions by using Rabinowitz’s global bifurcation theorem.

How to cite

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Ma, Ruyun, and An, Yulian. "Global structure of positive solutions for superlinear $2m$th-boundary value problems." Czechoslovak Mathematical Journal 60.1 (2010): 161-172. <http://eudml.org/doc/37998>.

@article{Ma2010,
abstract = {We consider boundary value problems for nonlinear $2m$th-order eigenvalue problem \[ \begin\{aligned\} (-1)^mu^\{(2m)\}(t)&=\lambda a(t)f(u(t)),\ \ \ \ \ 0<t<1, \\ u^\{(2i)\}(0)&=u^\{(2i)\}(1)=0,\ \ \ \ i=0,1,2,\cdots ,m-1 . \end\{aligned\} \] where $a\in C([0,1], [0,\infty ))$ and $a(t_0)>0$ for some $t_0\in [0,1]$, $f\in C([0,\infty ),[0,\infty ))$ and $f(s)>0$ for $s>0$, and $f_0=\infty $, where $f_0=\lim _\{s\rightarrow 0^+\}f(s)/s$. We investigate the global structure of positive solutions by using Rabinowitz’s global bifurcation theorem.},
author = {Ma, Ruyun, An, Yulian},
journal = {Czechoslovak Mathematical Journal},
keywords = {multiplicity results; Lidstone boundary value problem; eigenvalues; bifurcation methods; positive solutions; multiplicity; Lidstone boundary value problem; eigenvalue; bifurcation method; positive solution},
language = {eng},
number = {1},
pages = {161-172},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Global structure of positive solutions for superlinear $2m$th-boundary value problems},
url = {http://eudml.org/doc/37998},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Ma, Ruyun
AU - An, Yulian
TI - Global structure of positive solutions for superlinear $2m$th-boundary value problems
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 1
SP - 161
EP - 172
AB - We consider boundary value problems for nonlinear $2m$th-order eigenvalue problem \[ \begin{aligned} (-1)^mu^{(2m)}(t)&=\lambda a(t)f(u(t)),\ \ \ \ \ 0<t<1, \\ u^{(2i)}(0)&=u^{(2i)}(1)=0,\ \ \ \ i=0,1,2,\cdots ,m-1 . \end{aligned} \] where $a\in C([0,1], [0,\infty ))$ and $a(t_0)>0$ for some $t_0\in [0,1]$, $f\in C([0,\infty ),[0,\infty ))$ and $f(s)>0$ for $s>0$, and $f_0=\infty $, where $f_0=\lim _{s\rightarrow 0^+}f(s)/s$. We investigate the global structure of positive solutions by using Rabinowitz’s global bifurcation theorem.
LA - eng
KW - multiplicity results; Lidstone boundary value problem; eigenvalues; bifurcation methods; positive solutions; multiplicity; Lidstone boundary value problem; eigenvalue; bifurcation method; positive solution
UR - http://eudml.org/doc/37998
ER -

References

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