Existence of zeros for operators taking their values in the dual of a Banach space

Existence of zeros for operators taking their values in the dual of a Banach space BIAGIO RICCERI 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 ǫ > 0 there exists δ > 0 such that, for every y ∈ E satisfying y = 1 and x − y ≥ ǫ, 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], p. 289). For r > 0, we set B r = {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 B 1 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 satsfying |α(x)| ≤ rA(x) E * for all x ∈ X, a closed set C ⊂ X, and a τ-continuous function g : C → B r satisfying A(x)(g(x)) = α(x) for all x ∈ C, in such a way that, for every τ-continuous function ψ : X → B r satisfying ψ |C = g, there exists x 0 ∈ X such that A(x 0)(ψ(x 0)) = α(x 0). Then, there exists x * ∈ X such that A(x *) = 0. For the reader's convenience, we recall that a multifunction F : S → 2 V , between topological spaces, is said to be lower semicontinuous at s 0 ∈ S if, for every open set Ω ⊆ V meeting F (s 0), there is a neighbourhood U of s 0 such that F …