EXISTENCE OF ZEROS FOR OPERATORS TAKING THEIR VALUES IN THE DUAL OF A BANACH SPACE

Using continuous selections, we establish some existence results about the zeros of weakly continuous operators from a paracompact topological space into the dual of a reflexive real 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 τ-compact 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) for all x ∈ X, a (possibly empty) closed set C ⊂ X, and a τ-continuous function g: C → Br satisfying A(x) g(x)) = α(x) (2