Local inversion of a class of piecewise regular maps

Abstract

This paper provides sufficient conditions for any map L, that is strongly piecewise linear relatively to a decomposition of R k in admissible cones, to be invertible. Namely, via a degree theory argument, we show that when there are at most four convex pieces (or three pieces with at most a non convex one), the map is invertible. Examples show that the result cannot be plainly extended to a greater number of pieces. Our result is obtained by studying the structure of strongly piecewise linear maps. We then extend the results to the P C 1 case.

Introduction to: Topological degree and fixed point theories in differential and difference equations

“…[73,74] and the many references therein) or contributions to the study of existence and stability of equations (e.g. [75][76][77]; see also the papers [78,79] that have direct applications to optimal control theory [80,81]). However, in spite of the restrictions that we have to operate under, given the strong relations mentioned above between all topological methods we feel that this theme issue offers a valuable service to all mathematicians working in the field of topological methods in nonlinear analysis.…”