Fixed point-based stability analysis of climate and Langevin models

PLOS One Fixed point-based stability analysis of climate and Langevin models (a3) ∀ s, t ∈ Ct0, 𝛾, |gˆt – gˆs| ≤ e𝜏 Mˆs, t t – s . (a4) ∀ s, t ∈ Ct0, 𝛾, S t 0 Kˆt, sds ≤ t – s e2𝜏 . Where M(s,t) is defined in Definition 2.6 and 𝜏 A 0. Then the Eq (18) will have a unique solution. Proof : Define a map B  𝔐 𝔐 Bxˆt S t 0 Kˆt, sgˆsds  x0ˆt. (20) The existence of a FP of B in (20) will guarantee a Uni-Sol of the integral Eq (18). Now utilizing (20) and the above presumptions (a2) to (a3), |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 ≤ We𝜏 ˆt – s e2𝜏 Mˆs, t ˆt – s W e–𝜏Mˆs, t. This means that dmˆBx1, Bx2 ≤ e–𝜏Mˆs, t. Define 𝟋  ℝ ℝ by 𝟋ˆ𝔚 lnˆ𝔚 for 𝔚 A 0. 𝟋ˆdmˆBx1, Bx2 ≤ 𝟋ˆe–𝜏Mˆs, t lnˆMˆs, t – 𝜏 ln e. Implies that, 𝜏  𝟋ˆdmˆBx1, Bx2 ≤ 𝟋ˆMˆˆs, t Thus, all the necessities of Theorem 2.1 hold true. So, the Eq (18) will have a Uni-Sol. j Theorem 4.2. Consider that the subsequent assumptions are met: (b2) The function gˆt  s, t × s, t × ℝ ℝ is continuous on Ct0, 𝛾, Ct0, 𝛾2, and Ct0, 𝛾2 × ℝ, correspondingly. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 12/ 18