Conjecture on the Schauder fixed point theorem and the degree of non-densifiability
Let be a nonempty closed, bounded, convex subset of a Banach space, let be continuous, and suppose that is compact for some integer . Then has a fixed point; that is, there exists such that .
References
Primary source
Additional references
- A conjecture on the Schauder fixed point theorem and the degree of non-densifiability — Archiv der Mathematik — Gonzalo García
Progress summary
A new paper proposes a connection between Schauder fixed points and non-densifiability, but its mathematical conclusions cannot yet be assessed.
The entry concerns a conjectural relationship between the Schauder fixed point theorem and a degree-like notion of non-densifiability. Gonzalo García is named as author, but the retrieved record gives no abstract or result details, so it does not show whether the conjecture was proved, disproved, or merely formulated.
Known results
For the broader Schauder conjecture, Cauty reported proofs in 2005 and 2010, but earlier arguments were challenged for gaps. A 2021 paper by Ennassik and Taoudi claimed an alternative proof, while a 2023 article reportedly identified an incorrect recent proof and supplied a counterexample to an intermediate lemma.
September 2026 paper
The newly published article formulates and investigates the proposed relationship, but the available record contains no theorem statement, proof, counterexample, or conclusion. Its contribution is therefore a claimed advance, not a verified resolution.
Current status (as of September 2026): The specific non-densifiability conjecture remains unverified; a new paper has been published, but whether it proves or refutes the conjecture is unknown.
Solutions 0
No solutions have been posted yet.