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where s is a positive fixed constant and h is a positive, bounded, decreasing function satisfying that for some η > 0, lim r → ∞ r n + η h ( r s ) = 0. (1.5).
where T λ n + 1 x n = T x n − λ n + 1 μ F ( T x n ) for all x n ∈ H, x 0 ∈ H is an initial point, F : H → H is an η-strongly monotone and k-Lipschitzian mapping, μ is a positive fixed constant.
Suppose u ≠ u ∗ is a positive fixed point of A. By Lemma 2.6, we can get u ∈ Q.
Let (E_{2}) be a positive fixed point of (3).
Suppose that (H1)–(holdhold and that the operator has a positive fixed point in at.
which shows that has a positive fixed point at The proof is complete.
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Compared with single layer a-SiN x :H, a lower positive fixed charge density was revealed by SHG measurements, while field-effect passivation was absent for a reference stack comprising thermally grown SiO2.
(as the solution A < 0 is not physical, it is ignored, leaving a single positive fixed point for positive h), and the eigenvalue of this fixed point is given by λ = − h 2 + 4 h.
We algebraically show that system (1.2) undergoes a bifurcation (flip or Neimark-Sacker) at a unique positive fixed point if r varies around the sets (F_{B}) and (H_{B}).
Hence has a unique positive fixed point.
To conclude, F has a unique positive fixed point (y^) and (F' y^)<1).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com