AN EXTENSION OF HADAMARD'S THEOREM TO SET-VALUED MAPPINGS AND ITS APPLICATION TO SUBDIFFERENTIALS

Hadamard [3] presented a zero-point theorem, equivalent to Brouwer's fixed point theorem. It states that a continuous mapping g: B -> R-n has a zero point if it satisfies a boundary condition, where B denotes the Euclidean unit ball with center 0. By applying Hadamard's theorem to the gradient vector del f (x) of a smooth function, an existence theorem of a stationary point is obtained. The main purpose of this paper is to extend Hadamard's theorem to set -valued mappings. We show that our zero-point theorem (Theorem 2.2) is equivalent to Kakutani's fixed point theorem. Further, we apply it to the subdifferential of a convex function and the Clarke subdifferential of a locally Lipschitz function, and present sufficient conditions for the existence of a zero subgradient.