Existence of zeros for operators taking their values in the dual of a Banach space
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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 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)
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) Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:3 (2004) 187–194 2000 Mathematics Subject Classification: 47H10, 54C65, 54C60, 58K05 URL: http://dx.doi.org/10.1155/S1687182004310028
(2)
188
Existence of zeros for dual-valued operators
for all x ∈ C, in such a way that, for every τ-continuous function ψ : X → Br satisfying ψ|C = g, there exists x0 ∈ X such that
A x0 ψ x0
= α x0 .
(3)
Then, there exists x∗ ∈ X such that A(x∗ ) = 0. For the reader’s convenience, we recall that a multifunction F : S → 2V , between topological spaces, is said to be lower semicontinuous at s0 ∈ S if, for every open set Ω ⊆ V meeting F(s0 ), there is a neighborhood U of s0 such that F(s) ∩ Ω = ∅ for all s ∈ U. F is said to be lower semicontinuous if it is so at each point of S. The following well-known results will be our main tools. Theorem 2 [3]. Let X be a paracompact topological space and F : X → 2B1 a τ-lower semicontinuous multifunction with nonempty τ-closed convex values. Then, for each closed set C ⊂ X and each τ-continuous function g : C → B1 satisfying g(x) ∈ F(x) for all x ∈ C, there exists a τ-continuous function ψ : X → B1 such that ψ|C = g and ψ(x) ∈ F(x) for all x ∈ X. Theorem 3 [4]. Let X, Y be two topological spaces, with Y connected and locally connected, and let f : X × Y → R be a function satisfying the following conditions: (a) for each x ∈ X, the function f (x, ·) is continuous, changes sign in Y ,
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