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 Some important definitions in this sequel are listed below: Definition 2.1. [17] For a non-empty set 𝔐𝔟, define bm 𝔐𝔟 × 𝔐𝔟 0, ∞ with 𝔟 ≥ 1, if the below hold: 1. bmq, s 0 ⇒ q s, ∀q, s ∈ 𝔐𝔟. 2. bmq, s bms, q, ∀q, s ∈ 𝔐𝔟. 3. bmq, s ≤ 𝔟bmq, t bmt, s, ∀q, s, t ∈ 𝔐𝔟. Then, we labeled 𝔐𝔟, bm as 𝔟-metric space. Definition 2.2. [18] Suppose 𝟋 ℝ → ℝ represent a mapping which fulfill the below conditions: (1) 𝟋 is strictly increasing, i.e., when 𝜎 @𝜖 , for all 𝜎,𝜖 ∈ ℝ implies to 𝟋𝜎 @ 𝟋𝜖 . (2) For each sequence 𝜎𝜂𝜂∈ℕ of positive terms, lim𝜂∞ 𝜎𝜂 0 iff limn∞ 𝟋𝜎𝜂 –∞. (3) There is t ∈ 0, 1 in a sense that lim𝜎0 𝜎t𝟋𝜎 0. Example 2.1. [19] For 𝟋 ℝ ℝ: 1. 𝟋ℬ ln ℬ, ℬ A 0, 2. 𝟋ℬ ln ℬ ℬ, ℬ A 0, 3. 𝟋ℬ –1 ºℬ , ℬ A 0, 4. 𝟋ℬ lnℬ2 ℬ, ℬ A 0, are all the examples of 𝟋. Definition 2.3. [3] Let 𝔐𝔟, bm represent a 𝔟-metric space, and let vn be a sequence in 𝔐𝔟 with e ∈ 𝔐𝔟. Then the sequence vn is referred to be: (a) convergent in 𝔐𝔟, bm and converges to e when, for every 𝜀 A 0, there exists n0 ∈ 𝒩 in a way that bmvn, e @ 𝜀 for all n A n0, and will be expressed as limn∞ vn e or vn e as n ∞. (b) Cauchy if, for every 𝜀 A 0, there is some N N 𝜀 ∈ 𝒩 such that bmvm, vn @ 𝜀 for all m, n ≥ N. Definition 2.4. [3] Let 𝔐𝔟, bm denote a 𝔟-metric space then, it is labeled as complete if every Cauchy Sequence in 𝔐𝔟 converges.
Definition 2.5. [19] For a non-empty set 𝔐, a mapping 𝒲 𝔐 𝔐 will be 𝛼-admissible when there exists a function 𝛼 𝔐 × 𝔐 ℝ in a sense q, y ∈ 𝔐, 𝛼q, y ≥ 1 ⟹ 𝛼𝒲q, 𝒲y ≥ 1. Definition 2.6. [20] Suppose 𝔐, dm express the structure metric space, a mapping B 𝔐 → 𝔐 will be termed Ćirić type generalized 𝟋-contraction when there is 𝜏 A 0 in a sense that, dmB𝜈, B𝜔 A 0 ⇒ 𝜏 𝟋dmB𝜈, B𝜔 ≤ 𝟋M𝜈, 𝜔 , ∀𝜈, 𝜔 ∈ 𝔐, (3) while M𝜈, 𝜔 max dm𝜈, 𝜔, dm𝜈, B𝜈, dm𝜔, B𝜔, 1 2 dm𝜈, B𝜔 dm𝜔, B𝜈 . PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 3/ 18