Let X be a normed linear space, let C⊆ X be nonempty, and let T:C→ C. Is it true that if there exists b≥ 0 such that |b(x-y)+Tx-Ty|^2≤ (b+1)^2|x-y|^2-|(x-y)-(Tx-Ty)|^2 for all x,y∈ C, then there exist beta≥ 0 and theta∈[0,beta+1) such that |beta(x-y)+Tx-Ty|≤theta|x-y| for all x,y∈ C?

Altun–Hancer–Ateş question on enriched P-contractions

Altun, Hancer, and Ateş posed the question in their 2024 study of enriched $P$ -contractions, which left one converse implication unresolved. The question asks whether that remaining implication between two contraction classes is valid.

Known results

Altun, Hancer, and Ateş (2024) proved that an enriched $P$ -contraction on a nonempty, closed, convex subset of a Banach space has a unique fixed point.
Their corresponding Krasnoselʹskiĭ iteration converges to that fixed point.
Their examples showed that several other converse implications fail, including examples on a finite set and on $[0,1]$ .

August 2026 counterexamples

The manuscript On The Existence of $P$ -Contractions that are not Enriched Contractions claims two explicit counterexamples: one on a finite subset and one given by a continuous operator on an entire normed linear space. It therefore claims that the remaining converse is false; the manuscript is unrefereed, so this resolution remains unverified.

Current status (as of August 2026): The 2024 theorem and earlier separations are established, while the 2026 manuscript claims a counterexample to the remaining converse; that claim has not been independently verified.

Altun–Hancer–Ateş question on enriched P-contractions

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Additional references

Progress summary

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