Existence of Zeros for Operators Taking Their Values in the Dual of a Banach Space

Throughout the sequel, E denotes a reflexive real Banach space and E∗ its topological dual. We also assume that E is locally uniformly convex. This means that for each x ∈ E, with ‖x‖ = 1, and each > 0, there exists δ > 0 such that, for every y ∈ E satisfying ‖y‖ = 1 and ‖x− y‖ ≥ , one has ‖x + y‖ ≤ 2(1 − δ). Recall that any reflexive Banach space admits an equivalent norm with which it is locally uniformly convex [1, page 289]. For r > 0, we set Br = {x ∈ E : ‖x‖ ≤ r}. Moreover, we fix a topology τ on E, weaker than the strong topology and stronger than the weak topology, such that (E,τ) is a Hausdorff locally convex topological vector space with the property that the τ-closed convex hull of any τ-compact subset of E is still τcompact and the relativization of τ to B1 is metrizable by a complete metric. In practice, the most usual choice of τ is either the strong topology or the weak topology provided E is also separable. The aim of this short paper is to establish the following result and present some of its consequences. Theorem 1. Let X be a paracompact topological space and A : X → E∗ a weakly continuous operator. Assume that there exist a number r > 0, a continuous function α : X → R satisfying ∣∣α(x)∣∣≤ r∥∥A(x)∥∥E∗ (1)