Description

• Title:
  Positive solutions for a $2n$th-order boundary value problem involving all derivatives of odd orders
• Creator:
  Yang, Zhilin;
  O'Regan, Donal
• Description:
  We are concerned with the existence, multiplicity and uniqueness of positive solutions for the $2n$-order boundary value problem $$ \cases (-1)^nu^{(2n)}=f(t,u,u',-u''',\ldots, \\ (-1)^{i-1}u^{(2i-1)},\ldots, (-1)^{n-1}u^{(2n-1)}), \\ u^{(2i)}(0)=u^{(2i+1)}(1)=0, \quad i=0,\ldots,n-1. \endcases $$ where $n\geq 2$ and $f\in C([0,1]\times \mathbb{R}_+^{n+1},\mathbb{R}_+)$ $(\mathbb{R}_+:=[0,\infty))$ depends on $u$ and all derivatives of odd orders. Our main hypotheses on $f$ are formulated in terms of the linear function $g(x):=x_1+2\sum_{i=2}^{n+1}x_i$. We use fixed point index theory to establish our main results, based on a priori estimates achieved by utilizing some integral identities and an integral inequality. Finally, we apply our main results to establish the existence, multiplicity and uniqueness of positive symmetric solutions for a Lidostone problem involving an open question posed by P. W. Eloe in 2000.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2011.04.23
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 37, No 1 (March 2011); 87 - 101;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2011.004/8394
• Subject:
  Positive solution;
  integro-differential equation;
  fixed point index;
  a priori estimate;
  symmetric positive solution;
  34B18;
  45J05;
  47H07;
  47H11
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2011.004
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other