The method of upper and lower solutions for a Lidstone boundary value problem

Yanping Guo; Ying Gao

Czechoslovak Mathematical Journal (2005)

  • Volume: 55, Issue: 3, page 639-652
  • ISSN: 0011-4642

Abstract

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In this paper we develop the monotone method in the presence of upper and lower solutions for the 2 nd order Lidstone boundary value problem u ( 2 n ) ( t ) = f ( t , u ( t ) , u ' ' ( t ) , ⋯ , u ( 2 ( n - 1 ) ) ( t ) ) , 0 < t < 1 , u ( 2 i ) ( 0 ) = u ( 2 i ) ( 1 ) = 0 , 0 ≤ i ≤ n - 1 , where f [ 0 , 1 ] × ℝ n → ℝ is continuous. We obtain sufficient conditions on f to guarantee the existence of solutions between a lower solution and an upper solution for the higher order boundary value problem.

How to cite

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Guo, Yanping, and Gao, Ying. "The method of upper and lower solutions for a Lidstone boundary value problem." Czechoslovak Mathematical Journal 55.3 (2005): 639-652. <http://eudml.org/doc/30974>.

@article{Guo2005,
abstract = {In this paper we develop the monotone method in the presence of upper and lower solutions for the $2$nd order Lidstone boundary value problem \[ u^\{(2n)\}(t)=f(t,u(t),u^\{\prime \prime \}(t),\dots ,u^\{(2(n-1))\}(t)),\quad 0<t<1, u^\{(2i)\}(0)=u^\{(2i)\}(1)=0,\quad 0\le i\le n-1, \] where $f\:[0,1]\times \mathbb \{R\}^\{n\}\rightarrow \mathbb \{R\}$ is continuous. We obtain sufficient conditions on $f$ to guarantee the existence of solutions between a lower solution and an upper solution for the higher order boundary value problem.},
author = {Guo, Yanping, Gao, Ying},
journal = {Czechoslovak Mathematical Journal},
keywords = {$n$-parameter eigenvalue problem; Lidstone boundary value problem; lower solution; upper solution; -parameter eigenvalue problem; Lidstone boundary value problem; lower solution; upper solution},
language = {eng},
number = {3},
pages = {639-652},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The method of upper and lower solutions for a Lidstone boundary value problem},
url = {http://eudml.org/doc/30974},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Guo, Yanping
AU - Gao, Ying
TI - The method of upper and lower solutions for a Lidstone boundary value problem
JO - Czechoslovak Mathematical Journal
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 3
SP - 639
EP - 652
AB - In this paper we develop the monotone method in the presence of upper and lower solutions for the $2$nd order Lidstone boundary value problem \[ u^{(2n)}(t)=f(t,u(t),u^{\prime \prime }(t),\dots ,u^{(2(n-1))}(t)),\quad 0<t<1, u^{(2i)}(0)=u^{(2i)}(1)=0,\quad 0\le i\le n-1, \] where $f\:[0,1]\times \mathbb {R}^{n}\rightarrow \mathbb {R}$ is continuous. We obtain sufficient conditions on $f$ to guarantee the existence of solutions between a lower solution and an upper solution for the higher order boundary value problem.
LA - eng
KW - $n$-parameter eigenvalue problem; Lidstone boundary value problem; lower solution; upper solution; -parameter eigenvalue problem; Lidstone boundary value problem; lower solution; upper solution
UR - http://eudml.org/doc/30974
ER -

References

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