Existence of zeros for a class of operators

Abstract

Let \(E\) be a real Banach space. In this note, we establish conditions guaranteeing the existence of zeros of an affine operator \(A : E \to E\). We then extend these results to a class of operators that can be approximated by affine operators. Furthermore, in the finite-dimensional setting, we provide sufficient conditions for the existence of zeros of continuously differentiable operators, without appealing to any min--max theorem. Our approach relies on the introduction of a functional parameter associated with convex subsets of both the Banach space and its dual.