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The purpose of this note is to give an impression of some of the research in the project operator theory and applications which focuses on operator theoretical methods for analyzing problems arising from concrete classes of integral, differential and difference equations.

In the study of predator-prey models and viscoelasticity, Volterra [8] formulated some general differential equations incorporating the past states of the system:


In this model, it is assumed that the solution tex2html_wrap_inline3984 is given on the interval [-1,0] and satisfies equation (1) for tex2html_wrap_inline3988 .

This system is an example of a retarded functional differential equation. Systems of this type appear in a number of interesting applications, i.e., biology, economics, feedback control and number theory.

To understand the qualitative behaviour of the solutions of eq. (1) near an equilibrium, a thorough understanding of the corresponding linearized equation is needed, that is, an equation


where tex2html_wrap_inline3992 is an tex2html_wrap_inline3994 -matrix valued function of bounded variation (in the specific example n=2). The initial data is given by tex2html_wrap_inline3998 for tex2html_wrap_inline4000 , where tex2html_wrap_inline4002 is a given continuous function on [-1,0] with values in tex2html_wrap_inline4006 . The solution with initial data tex2html_wrap_inline4002 we shall denote by tex2html_wrap_inline4010 , tex2html_wrap_inline4012 , where C denotes the space of tex2html_wrap_inline4006 -valued continuous functions on the interval [-1,0] provided with the sup-norm.

Fri Mar 20 16:01:06 MET 1998