Fixed point-based stability analysis of climate and Langevin models
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PLOS One Fixed point-based stability analysis of climate and Langevin models Let us define dm 𝔐 × 𝔐 ℝ as follows dm𝔲, y |𝔲t – yt|, for all 𝔲, y ∈ 𝔐. (10) Evidently, 𝔐, dm represent a complete metric space. Theorem 3.2. Suppose the following presumptions are fulfilled: (r) The function Γ t0, 𝛾 × t0, 𝛾 × ℝ ℝ is continuous on t0, 𝛾 with t0, 𝛾2, and t0, 𝛾2 × ℝ, correspondingly. (s) ∀ 𝛼, 𝜈, 𝔣 ∈ t0, 𝛾 and for every 𝔲, y ∈ 𝔐, |Γ𝜈1, 𝛼1, 𝔣1 – Γ𝜈2, 𝛼2, 𝔣2 | ≤ Ma𝔲, y e𝜏1 t , where Ma𝔲, y is defined in Definition 2.7. Then, (7) will have a Uni-Sol. Proof : Define a map Θ 𝔐 𝔐 Θ𝜈t 𝜈0 S t 0 Γ𝜈, 𝛼, 𝔣 d℘ t ≥ 0. (11) The existence of a FP of Θ in (11) will be evidently the Uni-Sol of the Eq (7). Now utilizing (11) and the above presumptions (r) to (s), |Θ𝜈1t – Θ𝜈2t| V𝜈0 S t 0 Γ𝜈1, 𝛼1, 𝔣1 dt – 𝜈0 – S t 0 Γ𝜈2, 𝛼2, 𝔣2 dtV WS t 0 Γ𝜈1, 𝛼1, 𝔣1 – Γ𝜈2, 𝛼2, 𝔣2 dtW ≤ S t 0 SΓ𝜈1, 𝛼1, 𝔣1 – Γ𝜈2, 𝛼2, 𝔣2S dt ≤ Ma𝔲, y e𝜏1 t S t 0 dt Ma𝔲, y e𝜏1 t ⋅ t ≤ e–𝜏Ma𝔲, y. As a result, dmΘ𝜈1, Θ𝜈2 ≤ e–𝜏Ma𝜔1, 𝜔2 Define 𝟋 ℝ ℝ by 𝟋𝔚 ln𝔚, for 𝔚 A 0. 𝟋dmΘ𝜈1, Θ𝜈2 ≤ 𝟋e–𝜏Ma𝔲, y lnMa𝔲, y – 𝜏 ln e. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 7/ 18