Fixed Point Techniques in Analog Systems

Analog computation is concerned with continuous rather than discrete spaces. Most of the physical processes arising in nature are modeled by differential equations, either ordinary (example: spring/mass/damper system) or partial (example: heat diffusion). In analog computability, the existence of an effective way to obtain solutions (either exact or approximate) of these systems is essential. We develop a framework in which the solutions can be seen as fixed points of certain operators on continuous data streams, using the framework of Fréchet spaces. We apply a fixed point construction to retrieve these solutions and present sufficient conditions on the operators and initial inputs to ensure existence and uniqueness of these corresponding fixed points.