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 (c) ∀ 𝛼, 𝜈, 𝔣 ∈ t0, 𝛾 and for every 𝔲, y ∈ 𝔐, |Γ𝜈1, 𝛼1, 𝔣1 – Γ𝜈2, 𝛼2, 𝔣2 | ≤ M𝔲, y e𝜏1 t , where M𝔲, y is defined in Definition 2.6. 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. (9) The existence of a FP of Θ in (9) will be evidently the Uni-Sol of the Eq (7). Now utilizing (8) and the above presumptions (b) to (c). |Θ𝜈1t – Θ𝜈2t| V𝜈0 S t 0 Γ𝜈1, 𝛼1, 𝔣1 d℘ – 𝜈0 – S t 0 Γ𝜈2, 𝛼2, 𝔣2 d℘V VS t 0 Γ𝜈1, 𝛼1, 𝔣1 – Γ𝜈2, 𝛼2, 𝔣2 d℘V ≤ S t 0 SΓ𝜈1, 𝛼1, 𝔣1 – Γ𝜈2, 𝛼2, 𝔣2S d℘ ≤ M𝔲, y e𝜏1 t S t 0 d℘ M𝔲, y e𝜏1 t ⋅ t ≤ e–𝜏M𝔲, y. This means that dmΘ𝜈1, Θ𝜈2 ≤ e–𝜏M𝔲, y. Define 𝟋 ℝ ℝ by 𝟋𝔚 ln𝔚 for 𝔚 A 0. 𝟋dmΘ𝜈1, Θ𝜈2 ≤ 𝟋e–𝜏M𝔲, y lnM𝔲, y – 𝜏 ln e. Implies that, 𝜏 𝟋dmΘ𝜈1, Θ𝜈2 ≤ 𝟋M𝔲, y. Thus, all the assumptions of Theorem 2.1 are true. Hence, the Eq (7) will have a Uni-Sol. j For our next new result let 𝔐 t0, 𝛾, R represent all the real valued continuous functions on t0, 𝛾. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 6/ 18