Solvability of a nonlinear general third order two-point eigenvalue problem on time scales
type: Item
instance of: scholarly article
title: Solvability of a nonlinear general third order two-point eigenvalue problem on time scales
zbMATH Open document ID: 1212.34293
author: K. Rajendra Prasad
author: Sabbavarapu Nageswara Rao
author: Penugurthi Murali
published in: Differential Equations and Dynamical Systems
publication date: 2010-11-30T00:00:00Z
review text: The third order dynamic equation \[ y^{\triangle \triangle \triangle}(t)+\lambda f(t,y(t),y^{\triangle}(t),y^{\triangle \triangle}(t)) = 0, \quad a < t < b, \] is investigated subject to the boundary conditions \[ \alpha_{11}y(a)-\alpha_{12}y(b) = 0, \] \[ \alpha_{21}y^{\triangle}(a)-\alpha_{22}y^{\triangle}(b) = 0, \] \[ \alpha_{31}y^{\triangle \triangle}(a)-\alpha_{32}y^{\triangle \triangle}(b) = 0, \] where \(f\) is continuous and nonnegative valued and \(\alpha_{i,j} \geq 0\), \(i=1,2,3\), \(j=1,2\). The authors obtain conditions on the constants \(\alpha_{i,j}\) that ensure positivity of the Green function. As an underlying space, the authors choose \((C^3_{rd}[a,b],||\cdot||)\) with \(\|u\| = \max_{t \in [a,\sigma^3(b)]} |u(t)|\), which is claimed to be a Banach space. Since it is not a Banach space, the cone-theoretic method applied in this paper fails to work.
reviewed by: Nickolai Kosmatov
Mathematics Subject Classification ID: 34N05
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 34B27
zbMATH DE Number: 5821534
zbMATH Keywords: time scales
zbMATH Keywords: boundary value problem
zbMATH Keywords: dynamical equation
zbMATH Keywords: eigenvalue interval
zbMATH Keywords: cone
MaRDI profile type: Publication
full work available at URL: s12591-009-0020-z
OpenAlex ID: W1988159366
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DOI: 10.1007/S12591-009-0020-Z
description: scientific article; zbMATH DE number 5821534