Abstract
In analogy to the Eisenfeld-Lakshmikantham measure of nonconvexity and the Hausdorff measure of noncompactness, we introduce two mutually equivalent measures of noncircularity for Banach spaces satisfying a Cantor type property, and apply them to establish a fixed point theorem of Darbo type for multifunctions. Namely, we prove that every multifunction with closed values, defined on a closed set and contractive with respect to any one of these measures, has the origin as a fixed point.
Highlights
Let X, · be a Banach space over the field K ∈ {R, C}
Around 1955, Darbo 1 ensured the existence of fixed points for so-called condensing operators on Banach spaces, a result which generalizes both Schauder fixed point theorem and Banach contractive mapping principle
Darbo proved that if M ∈ b X is closed and convex, κ is a measure of noncompactness, and f : M → M is continuous and κ-contractive, that is, κ f A ≤ rκ A A ∈ b M for some r ∈ 0, 1, f has a fixed point
Summary
Let X, · be a Banach space over the field K ∈ {R, C}. In what follows, we write BX {x ∈ X : x ≤ 1} for the closed unit ball of X. Darbo proved that if M ∈ b X is closed and convex, κ is a measure of noncompactness, and f : M → M is continuous and κ-contractive, that is, κ f A ≤ rκ A A ∈ b M for some r ∈ 0, 1 , f has a fixed point. We recall the axiomatic definition of a regular measure of noncompactness on X; we refer to 2 for details. A regular measure of noncompactness κ possesses the following properties:. In R the only bounded balanced sets are the open or closed intervals centered at the origin. Denoting by γ either one of the two measures introduced, in Section 4 we prove a result of Darbo type for γ-contractive multimaps see Section 4 for precise definitions. It is shown that the origin is a fixed point of every γ-contractive multimap F with closed values defined on a closed set M ∈ b X such that F M ⊂ M
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