Fixed point-based stability analysis of climate and Langevin models

PLOS One Fixed point-based stability analysis of climate and Langevin models Next define 𝛼  𝔐 × 𝔐 ˜–∞ ∪ 0, ∞ by 𝛼ˆ𝜈1, 𝜈2 ¢¨¨ ¦ ¨¨¤ 1, if dmˆ𝜈1, 𝜈2 ≥ 0, –∞, else. Accordingly, 𝜏  𝟋ˆdmˆΘ𝜈1, Θ𝜈2 ≤ 𝛼ˆ𝜈1, 𝜈2𝟋ˆMaˆ𝔲, y. Thus, all the presumptions of Theorem 2.2 hold true. So, the equation (7) will have a Uni-Sol. j Following the same flow, taking important results of Geraghty-type contractions in the framework of b-metric space, we can have novel existence and uniqueness results. Let 𝔐b t0, 𝛾, R represent all the real valued continuous functions on 𝔐. Let us define bm  𝔐b × 𝔐b ℝ as follows bmˆ𝔲, y |𝔲 – y|2, for all 𝔲, y ∈ ˆ𝔐b, bm. (12) Evidently, ˆ𝔐b, bm, 𝔟 is a complete b-metric space with 𝔟 ≥ 1. Theorem 3.3. Suppose the following presumptions are fulfilled: (e) The function Γ  t0, 𝛾 × t0, 𝛾 × ℝ ℝ is continuous on t0, 𝛾 with t0, 𝛾2, and t0, 𝛾2 × ℝ, correspondingly. (f) ∀ 𝜈, 𝛼, 𝔣 ∈ t0, 𝛾 and for every 𝔲, y ∈ 𝔐, |È𝜈1, 𝛼1, 𝔣1 – Γ1ˆ𝜈2, 𝛼2, 𝔣2 | ≤ ¿ Á Á À lnˆ1  Mcˆ𝜈1, 𝜈2 ˆ1  t𝔟 , where Mcˆ𝜈1, 𝜈2 is defined in Theorem 2.4. Then, (7) will have a Uni-Sol. Proof : Define a map Θ  𝔐 𝔐 Θ𝜈ˆt 𝜈0  S t 0 È𝜈, 𝛼, 𝔣 d℘ t ≥ 0. (13) The existence of a FP of Θ in (13) will be evidently the Uni-Sol of the Eq (7). Now, utilizing (12), (13) and the above presumptions (e) and (f) |Θ𝜈1ˆt – Θ𝜈2ˆt|2 V𝜈0  S t 0 È𝜈1, 𝛼1, 𝔣1 dt – 𝜈0 – S t 0 È𝜈2, 𝛼2, 𝔣2 dtV 2 WS t 0 ŒΓˆ𝜈1, 𝛼1, 𝔣1 – È𝜈2, 𝛼2, 𝔣2‘ dtW 2 ≤ S t 0 SÈ𝜈1, 𝛼1, 𝔣1 – È𝜈2, 𝛼2, 𝔣2S2 dt ≤ 1

𝔟ˆ1  t lnˆ1  Mcˆ𝜈1, 𝜈2 S t 0 dt ≤ 1 𝔟 lnˆ1  Mcˆ𝜈1, 𝜈2. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 8/ 18