DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

FAST CONVERGENT ITERATIVE METHODS FOR SOME
PROBLEMS OF MATHEMATICAL BIOLOGY
HENRYK LESZCZYŃSKI
We investigate how fast some iterative methods converge to the exact solution of a differential-functional
von Foerster-type equation which describes a single population dependent
on its past time, state densities, and on its total size.
Copyright © 2006 Henryk Leszczyński. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Von Foerster and Volterra-Lotka equations arise in biology, medicine and chemistry, see
[1, 9]. The independent variables x j and the unknown function u stand for certain features
and densities, respectively. It follows from this natural interpretation that x j ≥ 0
and u ≥ 0. We are interested in von Foerster-type models, which are essentially nonlocal,
because there are included also the total sizes of population ∫ u(t,x)dx.
Existence results for certain von Foerster-type problems have been established by
means of the Banach contraction principle, the Schauder fixed point theorem or iterative
method, see [2–5]. These theorems are closely related to direct iterations for a natural
integral fixed point operator. Because of nonlocal terms, these methods demand very
thorough calculations and a proper choice of subspaces of continuous and integrable
functions. Sometimes, it may cost some simplifications of the real model. On the other
hand, there is a very consistent theory of first-order partial differential-functional equations
in [7], based on properties of bicharacteristics and on fixed point techniques with
respect to the uniform norms. Our research group has also obtained some convergence
results for the direct iterative method under nonlinear comparison conditions.
In the present paper, we find natural conditions which guarantee L ∞ ∩ L 1 -convergence
of iterative methods of Newton type. These conditions are preceded by analogous (slightly
weaker) conditions for direct iterations, however, we do not formulate any convergence
results for them (this will be stated in another paper).
Let τ = (τ 1 ,...,τ n ) ∈ R n +, τ 0 > 0, where R + := [0,+∞). Define B = [−τ 0 ,0]× [−τ,τ],
where [−τ,τ] = [−τ 1 ,τ 1 ] ×···×[−τ n ,τ n ]andE 0 = [−τ 0 ,0]× R n , E = [0,a] × R n , a>0.
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 661–666