Mutational equations in metric spaces

Abstract

This paper summarizes an extension of differential calculus to a mutational calculus for maps from one metric space to another. The simple idea is to replace half-lines allowing to define difference quotients of maps and their various limits in the case of vector space by ‘transitions’ with which we can also define differential quotients of a map. Their various limits are called ‘mutations’ of a map. Many results of differential calculus and set-valued analysis, including the Inverse Function Theorem, do not really rely on the linear structure and can be adapted to the nonlinear case of metric spaces and exploited. Furthermore, the concept of differential equation can be extended tomutational equation governing the evolution in metric spaces. Basic Theorems as the Nagumo Theorem, the Cauchy-Lipschitz Theorem, the Center Manifold Theorem and the second Lyapunov Method hold true for mutational equations. This work was motivated by evolution equations of ‘tubes’ in ‘visual servoing’ on one hand, mathematical morphology on the other, when the metric spaces are ‘power spaces’. This paper begins by listing some consequences of general theorems concerning ‘mutational equations for tubes’