Fixed point-based stability analysis of climate and Langevin models
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PLOS One Fixed point-based stability analysis of climate and Langevin models Accordingly, bmBx1t – Bx2t ≤ 1 2 ln1 Mcs, t. (24) Define 𝛽 ℝ 0, 1 2 by 𝛽𝔓 ln1 𝔓 2𝔓 for 𝔓 A 0 and 𝛽0 ∈ 0, 1 2 . Thus, (24) becomes, bmBx1t – Bx2t ≤ ln 1 Mcs, t 2Mcs, t Mcs, t. 𝛽Mcs, tMcs, t. Thus, all the necessities of Theorem 2.2 are true. So, the Eq (18) will have a Uni-Sol and consequently will be stable. j Let 𝔐 Ct0, 𝛾 represent set of all real-valued functions with no discontinuity and define dm 𝔐 × 𝔐 ℝ as dm𝔲, y |𝔲t – yt|, for all 𝔲, y ∈ 𝔐. (25) Evidently, 𝔐, dm represent a complete metric space. Theorem 4.4. Assume the fulfillment of the following hypotheses: (m1) The function gt s, t × s, t × ℝ ℝ is continuous on Ct0, 𝛾, Ct0, 𝛾2, and Ct0, 𝛾2 × ℝ, correspondingly. (m2) For all s, t ∈ Ct0, 𝛾 |gt – gs| ≤ Mbs, t – 𝜑Mbs, t t – s . (m3) ∀ s, t ∈ Ct0, 𝛾, S t 0 Kt, sds ≤ t – s. Then, (18) will have a Uni-Sol. Proof : Define a map B 𝔐 𝔐 by Bxt S t 0 Kt, sgsds x0t. (26) The existence of a Uni-Sol of the integral Eq (18) is equivalent to the existence of a FP of B in (26). PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 15/ 18