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EIGENFUNCTIONS ARISING FROM A FIRST-ORDER FUNCTIONAL DIFFERENTIAL EQUATION IN A CELL GROWTH MODEL

Part of: Functional-differential and differential-difference equations

Published online by Cambridge University Press:  19 April 2011

BRUCE VAN BRUNT  and
M. VLIEG-HULSTMAN
BRUCE VAN BRUNT*
Affiliation:
Institute of Fundamental Sciences, Mathematics, Massey University, Palmerston North, New Zealand (email: b.vanbrunt@massey.ac.nz)
M. VLIEG-HULSTMAN
Affiliation:
Institute of Fundamental Sciences, Mathematics, Massey University, Palmerston North, New Zealand (email: b.vanbrunt@massey.ac.nz)
*
✉For correspondence; e-mail: b.vanbrunt@massey.ac.nz
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Abstract

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A boundary-value problem for cell growth leads to an eigenvalue problem. In this paper some properties of the eigenfunctions are studied. The first eigenfunction is a probability density function and is of importance in the cell growth model. We sharpen an earlier uniqueness result and show that the distribution is unimodal. We then show that the higher eigenfunctions have nested zeros. We show that the eigenfunctions are not mutually orthogonal, but that there are certain orthogonality relations that effectively partition the set of eigenfunctions into two sets.

Keywords

pantograph equationcell growth modelnonlocal eigenvalue problem

MSC classification

Secondary: 34K06: Linear functional-differential equations
Type
Research Article
Information
The ANZIAM Journal , Volume 52 , Issue 1 , July 2010 , pp. 46 - 58
Copyright
Copyright © Australian Mathematical Society 2011

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