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 Now, utilizing (26), (25) and the above presumptions (m1) - (m2) |Bx1t – Bx2t| Vx0t S t 0 gtKs, tds – x0t – S t 0 gsKs, tdsV VS t 0 Kt, sgt – gsdsV ≤ Mbs, t – 𝜑Mbs, t t – s S t 0 Kt, sds ≤ Mbs, t – 𝜑Mbs, t t – s t – s. Further, |Bx1t – Bx2t| ≤ Mbs, t – 𝜑Mbs, t. Hence, dmBx1t, Bx2t ≤ Mbs, t – 𝜑Mbs, t. (27) Define 𝜓 0, ∞ 0, ∞ by 𝜓𝔓 𝔓. 𝜓dmBx1t, Bx2t ≤ 𝜓Mbs, t – 𝜑Mbs, t Therefore, all the necessities of Theorem 2.5 hold true. So, the Eq (18) will have a Uni-Sol and consequently will be stable. j Example 4.1. Let the integral equation: xt S t 0 Kt, sgs ds x0, where the Green’s function K(t,s) for 𝛽 2 is defined as Kt, s ¢¨¨ ¦ ¨¨¤ e–2t–s, t ≥ s, 0, t @ s. Let gs sins, and x0 ∈ ℝ. Then the equation becomes: xt S t 0 e–2t–s sins ds x0. Define 𝔐 C0, T, ℝ, the space of real-valued functions with no discontinuity on [0,T], endowed with the metric: dmu, v sup t∈0,T |ut – vt|. Define the operator B 𝔐 𝔐 by Bxt S t 0 e–2t–s sins ds x0. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 16/ 18