Dioids and Nonlinear Analysis

Most of the problems dealt with in the preceding chapters have concerned finite dimensional or discrete problems. The aim of the present chapter is to show that the structures of dioids lend themselves to defining, in the continuous domain, new branches of nonlinear analysis.

The basic idea is to replace the classical field structure on the reals by a dioid structure. Thus, a new branch of nonlinear analysis will correspond to each type of dioid. This approach was pioneered by Maslov (1987b), Maslov and Samborskii (1992), under the name of idempotent analysis. (The underlying dioid structure considered being $\left( {{\rm R} \cup \left\{ { + \infty } \right\},{\rm Min}, + } \right)$, the so-called MINPLUS dioid).