Fixed point-based stability analysis of climate and Langevin models

PLOS One Fixed point-based stability analysis of climate and Langevin models It implies that, 𝜏  𝟋ˆdmˆBx1, Bx2 ≤ 𝛼ˆs, t𝟋ˆMaˆs, t. Thus, all the necessities of Theorem 2.2 hold true. Hence, the Eq (18) will have a Uni-Sol. j Let 𝔐b Ct0, 𝛾 represent set of all real-valued functions with no discontinuity and define bm  𝔐b × 𝔐b ℝ as bmˆ𝔲, y |𝔲ˆt – yˆt|2, for all 𝔲, y ∈ 𝔐b. (22) Evidently, ˆ𝔐b, bm, 𝔟 is a complete b-metric space. Theorem 4.3. Assume the below presumptions fulfill: (c1) The function gˆt  s, t × s, t × ℝ ℝ defined to be continuous on Ct0, 𝛾, Ct0, 𝛾2, and Ct0, 𝛾2 × ℝ, correspondingly. (c2) For all s, t ∈ Ct0, 𝛾 |gˆt – gˆs| ≤ »lnˆ1  Mcˆs, t º𝔟ˆt – s . (c3) ∀ s, t ∈ Ct0, 𝛾, S t 0 Kˆt, sds ≤ ˆt – s, where Mcˆs, t is defined in Theorem 2.4. Then, (18) will have a Uni-Sol and consequently will be stable. Proof : Define a map B  𝔐b 𝔐b Bxˆt S t 0 Kˆt, sgˆsds  x0ˆt. (23) The existence of a FP of B in (23) ensure a Uni-Sol of the integral Eq (18). Now, utilizing (22) and the above presumptions (c1) to (c3) |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 ≤ »lnˆ1  Mcˆs, t º𝔟ˆt – s S t 0 Kˆt, sds »lnˆ1  Mcˆs, t º𝔟ˆt – s ˆt – s. Further, since 𝔟 2, |Bx1ˆt – Bx2ˆt|2 ≤ 1 2 lnˆ1  Mcˆs, t. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 14/ 18