Fixed point-based stability analysis of climate and Langevin models

PLOS One Fixed point-based stability analysis of climate and Langevin models Now, utilizing (15) and the above presumptions (m) and (n), |Θ𝜈1ˆt – Θ𝜈2ˆt| V𝜈0  S t 0 È𝜈1, 𝛼1, 𝔣1 dt – 𝜈0 – S t 0 È𝜈2, 𝛼2, 𝔣2 dtV WS t 0 ŒΓˆ𝜈1, 𝛼1, 𝔣1 – È𝜈2, 𝛼2, 𝔣2‘ dtW ≤ 1 ˆ1  t S t 0 ŒWÈ𝜈1, 𝛼1, 𝔣1 – È𝜈2, 𝛼2, 𝔣2W‘ dt ≤ Mbˆ𝜈1, 𝜈2 – 𝜑ˆMbˆ𝜈1, 𝜈2 ˆ1  t S t 0 dt ≤ Mbˆ𝜈1, 𝜈2 – 𝜑ˆMbˆ𝜈1, 𝜈2. Further, Θ𝜈1ˆt – Θ𝜈2ˆt| ≤ Mbˆ𝜈1, 𝜈2 – 𝜑ˆMbˆ𝜈1, 𝜈2. Hence, dmˆΘ𝜈1, Θ𝜈2 ≤ Mbˆ𝜈1, 𝜈2 – 𝜑ˆMbˆ𝜈1, 𝜈2. Define 𝜓  0, ∞ 0, ∞ by 𝜓ˆ𝔓 𝔓. 𝜓ˆdmˆΘ𝜈1, Θ𝜈2 ≤ 𝜓 ˆMbˆ𝜈1, 𝜈2 – 𝜑ˆMbˆ𝜈1, 𝜈2 . Thus, all the necessities of Theorem 2.5 hold true. So, the Eq (7) will have a Uni-Sol. j Example 3.1. Consider the climate model described by the integral equation 𝜈ˆt 𝜈0  S t 0 ˆ𝛼𝜈ˆ𝜏 cosˆ𝜈ˆ𝜏  𝛽e–𝜏 d𝜏, t ∈ 0, T, where 𝜈ˆt represents the temperature anomaly, 𝛼 0.3 is the climate feedback parameter, 𝛽 0.1 models external forcing, and 𝜈0 0 is the initial condition. Let 𝔐 Ct0, 𝛾 represent set of all real-valued functions with no discontinuity on [0,T], endowed with the metric dmˆu, y sup t∈0,T |uˆt – yˆt|. Define the operator Θ  𝔐 𝔐 by ˆΘ𝜈ˆt S t 0 ˆ0.3𝜈ˆ𝜏 cosˆ𝜈ˆ𝜏  0.1e–𝜏 d𝜏. To verify the conditions of Theorem 3.1, note that for all u, y ∈ 𝔐, |Èu – Èy| 0.3|u cosˆu – y cosˆy|. Using the inequality |u cosˆu – y cosˆy| ≤ 2|u – y|, it follows that |Èu – Èy| ≤ 0.6|u – y|. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 10/ 18