On the nonhomogeneous fourth-order \(p\)-Laplacian generalized Sturm-Liouville nonlocal boundary value problems
type: Item
instance of: scholarly article
title: On the nonhomogeneous fourth-order \(p\)-Laplacian generalized Sturm-Liouville nonlocal boundary value problems
zbMATH Open document ID: 1264.34046
DOI: 10.1155/2012/831960
author: Jian Liu
author: Zengqin Zhao
published in: Discrete Dynamics in Nature and Society
publication date: 2012-12-18T00:00:00Z
review text: The authors consider the boundary value problem \[ \begin{aligned} (\phi_p(u''(t)))''- k^2\phi_p(u''(t))&= g(t) f(t,u(t)),\quad t\in (0,1),\\ u''(0)= \phi_q(a),&\quad u''(1)= \phi_q(b),\\ \alpha u(0)-\beta u'(0)&= \sum^{n-2}_{i=1} a_i u(\xi_i),\\ \gamma u(1)+\delta u'(1)&= \sum^{n-2}_{i=1} b_i u(\xi_i),\end{aligned}\tag{\(*\)} \] where \(\phi_p\) is the \(p\)-Laplacian operator, that is, \(\phi_p(u)=|u|^{p-2}u\), \(p> 1\), \(\phi^{-1}_p= \phi_q\), \(1/p+1/q= 1\), \(k\neq 0\), \(\alpha,\beta,\gamma,\delta\geq 0\), \(\xi_i\in (0,1)\), \(a,b,a_i,b_i\in (0,\infty)\) \((i= 1,2,\dotsc, n-2)\), \(f\in C([0,1]\times [0,\infty),[0,\infty))\), \(f(t,0)\not\equiv 0\), \(g\in C([0,1],[0,\infty))\). By using the Leray-Schauder nonlinear alternative, they prove the existence of a positive solution of \((*)\). Finally, the existence of three solutions of \((*)\) is established by means of the Leggett-Williams fixed point theorem.
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34B10
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 34B18
zbMATH DE Number: 6116528
reviewed by: Klaus R. Schneider
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2012/831960
OpenAlex ID: W1996784181
Wikidata QID: Q58700789
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description: scientific article; zbMATH DE number 6116528