Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order
type: Item
instance of: scholarly article
title: Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order
zbMATH Open document ID: 1274.34076
published in: Applications of Mathematics
publication date: 2013-05-13T00:00:00Z
review text: A nonlinear, fourth-order boundary value problem (BVP) \[ u^{(4)}(t)=\lambda h(t) f(t,u(t),u'(t),u''(t)),\quad u(0)=u'(1)=u''(0)=u'''(1)=0 \eqno(P1) \] is considered for \(t\in (0,1)\); the prime stands for the derivative with respect to \(t\). The continuity and nonnegativity of both \(h\) and \(f\) are assumed, but a singularity of \(h(t)f(t,x,y,z)\) is allowed at \(t=0\), \(t=1\), \(x=0\), \(y=0\), and \(z=0\). Further assumptions comprise both \(t\)-dependent bounds put on the value of \(f\) and a condition constraining the values of \(h(t)\). Besides (P1), a simplified problem (P2), where \(f\equiv f(t,u(t))\), is also introduced. By employing the Green function for a simpler BVP, by defining a cone of functions and introducing completely continuous integral operators, and by applying the Guo-Krasnosel'skii fixed point theorem, the author formulates and proves two main results. Roughly speaking, if \(\lambda \) is bounded from below and from above by particular expressions, then (P1) has at least one increasing positive solution. If \(\lambda \) complies with even trickier inequalities, then (P1) has at least two strictly increasing positive solutions. Parallel statements are also formulated for (P2). Moreover, it is shown that the obtained results generalize the existence theorem by \textit{J. R. Graef} and \textit{B. Yang} [Appl. Anal. 74, 201--214 (2000; Zbl 1031.34025)]. Finally, the theory is applied to a nonlinear fourth-order BVP.
reviewed by: Jan Chleboun
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34B16
Mathematics Subject Classification ID: 34B27
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34B09
zbMATH DE Number: 6162814
zbMATH Keywords: nonlinear ordinary differential equation
zbMATH Keywords: singularity
zbMATH Keywords: positive solution
zbMATH Keywords: eigenvalue interval
author: Qingliu Yao
MaRDI profile type: Publication
full work available at URL: s10492-013-0004-8
OpenAlex ID: W2021982853
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DOI: 10.1007/S10492-013-0004-8
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description: scientific article; zbMATH DE number 6162814