Abstract
In this paper we prove the Belitskii–Lyubich conjecture for triangular and gradient mappings. These results resemble those obtained for the discrete Markus–Yamabe conjecture. However, the proofs are quite different, which sheds new light on the subject. We complete the picture by showing that the general versions of the two conjectures can be turned into theorems at little cost, simply by relaxing their spectral condition.
A Appendix
For completeness we also give the proofs of the following two lemmas, which seem to belong to mathematical folklore. Let
Lemma 3.
For all
Proof (by induction on p).
It is true for p = 1. Suppose it holds for p. Then
Thus it holds for
Lemma 4.
For all p,
Proof.
We have
as desired. ∎
Acknowledgements
I thank the referee for suggesting changes that greatly improved the readability of the paper.
References
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