Fixed point-based stability analysis of climate and Langevin models

PLOS One Fixed point-based stability analysis of climate and Langevin models (b3) ∀ s, t ∈ Ct0, 𝛾, |gˆt – gˆs| ≤ e𝜏 Maˆs, t t – s . (b4) ∀ s, t ∈ Ct0, 𝛾, S t 0 Kˆt, sds ≤ t – s e2𝜏 , where Maˆs, t is defined in Definition 2.7. Then, (18) will have a Uni-Sol. Proof : Define a map B  𝔐 𝔐 Bxˆt S t 0 Kˆt, sgˆsds  x0ˆt. (21) The existence of a FP of B in (21) ensures a Uni-Sol of the integral Eq (18). Now utilizing (21) and the above presumptions (b2) to (b3), |Bx1ˆt – Bx2ˆt| Vx0ˆt  S t 0 gˆtKˆs, tds – x0ˆt – S t 0 gˆsKˆs, tdsV VS t 0 Kˆt, sˆgˆt – gˆsdsV ≤ e𝜏 Mˆs, t t – s S t 0 Kˆt, sds ≤ e𝜏 ˆt – s e2𝜏 Mˆs, t ˆt – s e–𝜏Mˆs, t. Hence, dmˆBx1, Bx2 ≤ e–𝜏Maˆs, t. Define 𝟋  ℝ ℝ by 𝟋ˆ𝔚 lnˆ𝔚, for 𝔚 A 0. 𝟋ˆdmˆBx1, Bx2 ≤ 𝟋ˆe–𝜏Maˆs, t lnˆMaˆs, t – 𝜏 ln e. Therefore, 𝜏  𝟋ˆdmˆBx1, Bx2 ≤ 𝟋ˆMaˆs, t. Next define 𝛼  𝔐 × 𝔐 ˜–∞ ∪ 0, ∞ by 𝛼ˆs, t ¢¨¨ ¦ ¨¨¤ 1, if dmˆBx1, Bx2 ≥ 0, –∞, else. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 13/ 18