A. Ponosov
Agricultural University of Norway,
Dept. of Mathematical Science and Technology,
N-1430 Es-NLH, NORWAY,
A. Shindiapin
Eduardo Mondlane University,
Department of Mathematics and Informatics,
C.P. 257 Maputo, MOZAMBIQUE,
We propose a general method of reducing differential equations
with a distributed delay function to finite or infinite systems of
ordinary differential equations. The idea of the method goes back
to the so-called "W-method" developed by N. V. Azbelev and his
students. As particular cases we obtain the famous "linear chain
trick", on one hand, and the Krasovskii - Hale form of delay
equations, on the other. We also study some general properties of
the proposed version of the W-transform and use these results to
justify the linear chain trick and to show how it can be applied
in the stability theory. Finally, we use this method to
investigate singularly perturbed systems of delay equations by
applying to a certain problem coming from regulatory biology.
More specifically, we consider the following delay differential
equation:
x'(t)=f(t, (Rx)(t)), t>0
with a delay operator R. The W-transform W comes from an
auxiliary ordinary differential equation of the form
z'(t)=A(t)z(t)+y(t), t >0,
where A(t): B -> B is a family of bounded linear operators in
a certain Banach space B, depending on the operator R. Then,
the Cauchy representation
z(t)=Wt (y,z
produces Azbelev's W-transform, which we apply, in a certain
manner, to the original delay equation. This reduces the delay
equation to a system of ordinary differential equations in a
Banach space B.
This work was partially supported by NUFU -
Norwegian Council for Higher Education's Programme for Development
Research and Education, grant no. PRO 06/02.