Fixed point-based stability analysis of climate and Langevin models

PLOS One Fixed point-based stability analysis of climate and Langevin models Let 𝜏 – lnˆ0.6 A 0 and define FˆP lnˆ𝔚. Then, dmˆΘu, Θy ≤ 0.6 dmˆu, y ⇒ FˆdmˆΘu, Θy ≤ Fˆdmˆu, y – 𝜏, which satisfies the Ćirić-type contraction condition. Moreover, the function È𝜈, 𝛼, F 0.3𝜈 cosˆ𝜈  0.1e–t, is continuous in both 𝜈 and t. Additionally, the growth condition |È𝜈1 – È𝜈2| ≤ 0.6|𝜈1 – 𝜈2| ≤ Mˆu, y e𝜏ˆ1  t , holds with Mˆu, y dmˆu, y, since 0.6 @ e–𝜏 for 𝜏 – lnˆ0.6. Therefore, all the conditions of Theorem 3.1 are true, and thus integral equation admits a unique solution 𝜈‡ ∈ 𝔐. 4 Existence results for Langevin equation In this section, the same approach is utilized to investigate the stability of the Langevin equation. dx dt  𝛽x gˆt, (17) where, g(t) is the external force, 𝛽 is a positive constant (damping coefficient or stabilizing term). Using the technique of green function, the Eq (17) can be transformed to xˆt S t 0 Kˆt, sgˆsds  x0ˆt, (18) where the Green function that describes how the system responds to the external forcing is K ¢¨¨ ¦ ¨¨¤ e–𝛽ˆt–s, if t ≥ s, 0, if t @ s. Let 𝔐 Ct0, 𝛾 represent set of all real-valued functions with no discontinuity. Let us define dm  𝔐 × 𝔐 ℝ as dmˆ𝔲, y |𝔲ˆt – yˆt|, for all 𝔲, y ∈ 𝔐. (19) Evidently, ˆ𝔐, dm represents a complete 𝕄-space. Theorem 4.1. Assume the fulfillment of the following hypotheses: (a2) 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 11/ 18