Two-point higher-order BVPs with singularities in phase variables
type: Item
instance of: scholarly article
title: Two-point higher-order BVPs with singularities in phase variables
zbMATH Open document ID: 1057.34005
DOI: 10.1016/S0898-1221(03)90238-0
author: Ravi P. Agarwal
author: Donal O'Regan
author: Irena Rachůnková
author: Svatoslav Staněk
publication date: 2004-08-20T00:00:00Z
review text: This paper provides results on the existence of positive solutions to two kinds of singular boundary value problems. The first one is the singular Lidstone boundary value problem \((-1)^n x^{(2n)}(t)=f(t,x(t), \dots, x^{(2n-2)}(t)), x^{(2j)}(0)=0= x^{(2j)}(T), 0\leq j \leq n-1\). It is assumed that the function \(f\) satisfies suitable Carathéodory conditions; moreover, the function \(f(t,x_0,\dots, x_{n-2})\) may be singular at the points \(x_i=0, 0\leq i \leq 2n-2\). The second one is the singular \((n,p)\)-boundary value problem \(-x^{(n)}(t)=f(t,x(t), \dots, x^{(n-1)}(t)), x^{(i)}(0)=0, 0\leq i \leq n-2, x^{(p)}(T)=0\), with \(p \in {\mathbb N}, 0\leq p \leq n-1\). It is assumed that the function \(f\) satisfies suitable Carathéodory conditions; moreover, the function \(f(t,x_0,\dots, x_{n-1})\) may be singular at the points \(x_i=0, 0\leq i \leq n-2\). For the Lidstone problem, it is assumed that, on a suitable subset of the space variables, the function \(f\) satisfies a condition of the form \(f(t,x_0,\dots, x_{n-2}) \leq \phi(t) + \sum_{j=0}^{2n-2} q_j(t) \omega_j(| x_j| )+\sum_{j=0}^{2n-2} h_j(t) | x_j| \), where \(\phi, h_j,q_j\) satisfy suitable technical assumptions and \(\omega_j\) is nonincreasing. A similar condition is assumed for the singular \((n,p)\)-boundary value problem. The proof is performed by applying the Leray-Schauder continuation theorem. Several related results can be found in the book by \textit{R. P. Agarwal, D. O'Regan} and \textit{P. J. Y. Wong} [Positive solutions of differential, difference and integral equations. Dordrecht: Kluwer Academic Publishers, (1999; Zbl 1157.34301)]. Other related results for Dirichlet second-order BVPs can be found, e.g., in the paper by \textit{C. De Coster, M. R. Grossinho} and \textit{P. Habets} [Appl. Anal. 59, No. 1--4, 241--256 (1995; Zbl 0847.34022)]; see also \textit{C. De Coster} and \textit{P. Habets} [F. Zanolin (ed.), Nonlinear analysis and boundary value problems for ordinary differential e…
reviewed by: Anna Capietto
Mathematics Subject Classification ID: 34B16
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 2093588
zbMATH Keywords: singular higher-order boundary value problem
zbMATH Keywords: topological degree
zbMATH Keywords: Lidstone
MaRDI profile type: Publication
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full work available at URL: s0898-1221(03)90238-0
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published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 2093588