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 Consider the following two functions, 𝜓 0, ∞ → 0, ∞ a monotone non-decreasing and continuous function with 𝜓𝔤 0 iff 𝔤 0. 𝜑 0, ∞ → 0, ∞ is a lower semi-continuous function in a way that 𝜑𝔤 0 iff 𝔤 0. Theorem 2.5. [23] Suppose 𝔐, dm represent a complete metric space and suppose B 𝔐 𝔐 represent a mapping. Let 𝜓dmB𝔲, By ≤ 𝜓MbB𝔲, By – 𝜑MbB𝔲, By, ∀ 𝔲, y ∈ 𝔐, (4) whereas Mb𝔲, y max dm𝔲, y, dm𝔲, B𝔲, dmy, By, dmy,B𝔲dm𝔲,By 2 , dm𝔲,B𝔲dmy,By 2 , dmy, By 1dm𝔲,B𝔲 1dm𝔲,y , dm𝔲, B𝔲 1dmy,B𝔲 1dm𝔲,y . (5) Then there will be a unique point 𝔲∗ ∈ 𝔐 in a sense that 𝔲∗ B𝔲∗. 3 Existence results for ECS’s equation In this section, various important contractions from the fixed point theory will be utilized to investigate the stability of the equation associated with ECS. d𝜈 dt Γ𝜈, 𝛼, ℱ, 𝜈 ∈ Rn, 𝜈0 𝜈0, t ≥ 0, (6) where 𝜈 represents the vector of state variables that define the system’s state at certain points in time t, 𝜈0 represents a specified initial condition of a system, n denotes the dimension for a dynamical system, 𝛼 ∈ Rm express the m-dimensional parameter vector along ℱ representing external forcing. By transforming the differential equation into the integral form, the model equation will become: 𝜈t 𝜈0 S t 0 Γ𝜈, 𝛼, ℱ d℘, t ≥ 0. (7) In this direction, we aim to establish an existence result for investigating the solution of the
integral equation of the type (7), utilizing Definition 2.6 and Theorem 2.1 from the existing literature. Now, our objective is to demonstrate the existence of a solution to the integral Eq (7) in the given space and also to verify the uniqueness of the solution. Let 𝔐 Ct0, 𝛾 denote the set of all real-valued continuous functions defined on t0, 𝛾. We define dm 𝔐 × 𝔐 ℝ as follows: dm𝔲, y |𝔲t – yt|, for all 𝔲, y ∈ 𝔐. (8) Evidently, 𝔐, dm represent a complete metric space. Here is our first new result. Theorem 3.1. Suppose the following presumptions are fulfilled: (b) The function Γ t0, 𝛾 × t0, 𝛾 × ℝ ℝ is continuous on t0, 𝛾 with t0, 𝛾2, and t0, 𝛾2 × ℝ, correspondingly. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 5/ 18