Remarks on the infinite‐dimensional counterparts of the Darboux theorem

Abstract

The Darboux theorem, one of the fundamental results in analysis, states that the derivative of a real (not necessarily continuously) differentiable function defined on a compact interval has the intermediate value property, that is, attains each value between the derivatives at the endpoints. The Bolzano intermediate value theorem, which implies Darboux's theorem when the derivative is continuous, states that a continuous real‐valued function defined on satisfying and , has a zero, that is, for at least one number . It has numerous counterparts in multivariate calculus as well as in the infinite‐dimensional setting. The present paper is devoted to the discussion of some infinite‐dimensional variants of the Darboux theorem, which does not seem to be sufficiently deeply discussed. The study relies on different notions of nonsmooth differentiability of real functions and some appropriate compactness conditions. Problems involving functionals bounded below, monotone operators, and some general questions concerning the existence of the so‐called generalized equilibria are discussed.