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 Theorem 2.1. [20] Suppose 𝔐, dm represent a complete metric space and B 𝔐 → 𝔐 is a Ćirić type generalized 𝟋 – contraction. When B or 𝟋 is continuous, then B admits a unique FP. Definition 2.7. [19] Suppose 𝔐, dm express a metric space, a mapping B 𝔐 → 𝔐 be labeled as modified 𝛼 – 𝟋-contraction. When for functions 𝟋 and 𝛼 𝔐 × 𝔐 –∞, 0 ∪ 0, ∞ some 𝜏 A 0 exists such a way that dmB𝜈, B℘ A 0 and, 𝜏 𝛼𝜈, ℘𝟋dmB𝜈, B℘ ≤ 𝟋Ma𝜈, ℘, where Ma𝜈, ℘ max dm𝜈, ℘, dm𝜈, B𝜈, dm℘, B℘, dm𝜈, B𝜈dm𝜈, B𝜈 dm℘, B𝜈dm℘, B℘ dm℘, B𝜈 dm𝜈, B℘ , dm𝜈, B𝜈dm℘, B℘ dm𝜈, ℘ dm𝜈, B℘ dm℘, B𝜈 ¡ . Theorem 2.2. [19] Suppose 𝔐, dm express a complete metric space, when a continuous mapping B 𝔐 → 𝔐 is 𝛼 – 𝟋 – weak contraction and 𝛼 – admissible, with 𝛼℘0, B℘0 ≥ 1. Then B possesses a unique FP. Definition 2.8. [21] Suppose a family of functions W, as 𝛼c ℝ 0, 1 with the property of: lim n∞ 𝛼czn 1 ⟹ lim n∞ zn 0. Theorem 2.3. [21] Suppose 𝔐, dm represent a complete metric space. Suppose C 𝔐 𝔐 be a self-map fulfilling the below: dmCe, Ck ≤ 𝛼cdme, kdme, k, ∀e, k ∈ 𝔐. where 𝛼c ∈ W. So this will ensure a unique FP for C. Definition 2.9. [22] For b-metric space 𝔐b, bm with 𝔟 ≥ 1, suppose B𝔟 is a set representing all the functions that are defined as 𝛽 0, ∞ 0, 1 𝔟 that satisfy the below lim sup n∞ 𝛽zn 1 𝔟 ⟹ zn 0 as n ∞.
Theorem 2.4. [22] Suppose 𝔐b, bm represents a complete b-metric space along 𝔟 ≥ 1. Moreover, let P 𝔐𝔟 𝔐𝔟 denotes a self-map. If there exists 𝛽 ∈ B𝔟 in a sense that bmP𝔲, Py ≤ 𝛽 Mc𝔲, y Mc𝔲, y, where Mc𝔲, y max bm𝔲, y, bm𝔲, P𝔲, bmy, Py, 1 2𝔟 bm𝔲, Py bmy, P𝔲 , then P ensures a unique FP. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 4/ 18