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The fixed point property for nonexpansive maps in reflexive Banach spaces
Theorem 1.1: if is a real reflexive Banach space and is nonempty, closed, bounded and convex, every nonexpansive has a fixed point, with no uniform convexity, normal structure or separability assumed and the norm unchanged. Corollary: for any commuting family of nonexpansive self-maps of such , the common fixed set is nonempty and a nonexpansive retract (via Bruck 1974). The proof argues by contradiction from a minimal invariant set, using the Goebel-Karlovitz lemma, an adaptive anchor, a tree of approximate fixed point images and a weighted switching estimate. It says nothing about nonreflexive spaces with the weak fixed point property, and gives no constructive iteration.
Exact FrontierDelta
Prior state unknown→proved
Scope and record
Occurred: Sep 24, 2026
Delta type: SOURCE CLAIM
Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.
Canonical aliases: The fixed point property for nonexpansive maps in reflexive Banach spaces · Nonexpansive fixed points in reflexive spaces
Confidence: Not scored
Registry verification: lean checked · announcement · candidate
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VibeMathed
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Unreleased internal OpenAI model
model · ai model contributor · OpenAI
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This event attributed to Unreleased internal OpenAI model
VibeMathed record: The fixed point property for nonexpansive maps in reflexive Banach spaces evidence for this event
Theorem 1.1: if is a real reflexive Banach space and is nonempty, closed, bounded and convex, every nonexpansive has a fixed point, with no uniform convexity, normal structure or separability assumed and the norm unchanged. Corollary: for any commuting family of nonexpansive self-maps of such , the common fixed set is nonempty and a nonexpansive retract (via Bruck 1974). The proof argues by contradiction from a minimal invariant set, using the Goebel-Karlovitz lemma, an adaptive anchor, a tree of approximate fixed point images and a weighted switching estimate. It says nothing about nonreflexive spaces with the weak fixed point property, and gives no constructive iteration. parent of this event
The fixed point property for nonexpansive maps in reflexive Banach spaces parent of this event
The fixed point property for nonexpansive maps in reflexive Banach spaces evidence for this event
Lean: OAI/Analysis/Nonexpansive/Main.lean (exists_fixedPoint_of_canonicallyReflexive) evidence for this event
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