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 Further, since 𝔟 2 |Θ𝜈1t – Θ𝜈2t|2 ≤ 1 2 ln1 Mc𝜈1, 𝜈2. It implies that, bmΘ𝜈1, Θ𝜈2 ≤ 1 2 ln1 Mc𝜈1, 𝜈2. (14) Define 𝛽 ℝ 0, 1 2 by 𝛽𝔓 ln1 𝔓 2𝔓 , for 𝔓 A 0 and 𝛽0 ∈ 0, 1 2 . Thus, (14) becomes, bmΘ𝜈1, Θ𝜈2 ≤ ln 1 Mc𝜈1, 𝜈2 2Mc𝜈1, 𝜈2 Mc𝜈1, 𝜈2 𝛽Mc𝜈1, 𝜈2Mc𝜈1, 𝜈2. Thus, all the necessities of Theorem 2.4 are true. So, the Eq (7) will have a Uni-Sol. j Let us define dm 𝔐b × 𝔐b ℝ as follows dm𝔲, y |𝔲t – yt|, for all 𝔲, y ∈ 𝔐. (15) Evidently, 𝔐, dm represent a complete metric space. Theorem 3.4. Suppose the following presumptions are fulfilled: (m) The function Γ t0, 𝛾 × t0, 𝛾 × ℝ ℝ is continuous on t0, 𝛾, t0, 𝛾2, and t0, 𝛾2 × ℝ, correspondingly. (n) For all 𝜈, 𝛼, 𝔣 ∈ t0, 𝛾 and for all 𝔲, y ∈ 𝔐, |Γ𝜈1, 𝛼1, 𝔣1 – Γ1𝜈2, 𝛼2, 𝔣2 | ≤ MbΘ𝜈1, Θ𝜈2 – 𝜑MbΘ𝜈1, Θ𝜈2 1 t , where Mb𝜈1, 𝜈2 and 𝜑 are defined in Theorem 2.5. 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. (16) The existence of a FP of Θ in (16) will ensure a Uni-Sol of the integral Eq (7). PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 9/ 18