Local and Global Dynamics of Contractiv...

This thesis covers the following topics:

We formulate initial value problems for delay difference equations in Banach spaces as
 fixed-point problems in sequence spaces. By choosing appropriate sequence spaces various types of attractivity can be described. This allows us to establish global attractivity by means of fixed-point results. Finally, we provide an application to delay integrodifference equations in the space of continuous functions over a compact domain.

The construction of attractors of a dissipative difference equation is usually based on compactness assumptions. We replace them with contractivity assumptions under which the pullback and forward attractors are identical. As a consequence, attractors degenerate to unique bounded entire solutions. As an application, we investigate attractors of integrodifference equations which are popular models in theoretical ecology.

Integrodifference equations are infinite-dimensional discrete dynamical systems. Numerical illustrations require spatial discretizations. Using piecewise linear collocation and the Nyström method we show that the discretized systems have also a periodic solution. Moreover, a convergence rate maintaining the order of the method is established.