Fixed point-based stability analysis of climate and Langevin models

PLOS One Fixed point-based stability analysis of climate and Langevin models evidently influence individuals and their activities [9,10]. Evaluating climate hazards is a necessary step in determining how humanity may be affected by changing climate condi- tions [11]. Mathematics and its methodologies are essential in climate study [12,13]. It can be asserted assuredly that mathematics is the sole discipline that enables us to measure and forecast the impacts of external natural and man-made disturbances on the dynamics of the climate system. Quantitative research fundamentally relies on mathematical representations of the systems and objects being studied. In this context, we see that replicating climate and forecasting its alterations due to external influences, unlike traditional physics problems, pos- sesses a fundamental characteristic stemming from the inability to conduct comprehensive direct physical experiments. This highlights the importance of stability of the solutions of the equations associated with climate change models is of great importance. The equation, d𝜈 dt È𝜈, 𝛼, ℱ 𝜈 ∈ Rn, 𝜈ˆ0 𝜈0, t ≥ 0, (1) denotes the mathematical representation of the Earth’s Climate System (ECS) [14–16], encompassing its dynamic development. Here, 𝜈 is a vector of state variables that delineate the system’s state at time t, while 𝜈0 represents its initial state. The parameter vector 𝛼 ∈ ℝm gov- erns the complex traits of the system, and ℱ symbolizes external forcing, such as solar radia-

tion or anthropogenic impacts. This system of three-dimensional nonlinear partial differential equations captures the intended connection between internal climate parameters and external influences, offering a deterministic, semi-dynamic formulation. The Langevin equation draws the interest of researchers, as it possesses significant impor- tance in stochastic dynamics. This equation is commonly utilized to demonstrate the systems affected by deterministic forces and random fluctuations [14]: dx dt  𝛽x gˆt, (2) where x represents the state variable, 𝛽 stands for the damping effect with g(t) means the external influence on the system. Fixed point theory (FPT) is significant for analyzing mathematical models that represent climate change. FPT provides the groundwork for showcasing the stability and consistency of solutions for differential and integral equations associated with these models. In the mod- els of Climate proper simulation, good mathematical modeling is essential for stability and predictability, notably with the help of fixed-point results like Banach’s Fixed Point Theorem, etc. This indicates that, at a fixed point, we may determine if a stable solution exists to which this system will converge under particular conditions. These methods ensure the existence and stability of solutions, which is essential for analysis and informed decision-making. In this direction, this manuscript examines the worth of the FPT in investigating the stability of those equations that are in association with the climate change system. In this

direction, some important contractions such as F-contraction, 𝛼-F-contraction, rational type ˆ𝜓, 𝜙-contraction, and Geraghty type contraction have been recalled and utilized to define the existence and uniqueness criteria for those equations. The objective is to emphasize the fundamental characteristics that guarantee the practical effectiveness of these models in the disciplines of meteorology and climatology by conducting a thorough analysis and applying fixed-point theorems from the existing literature. 2 Preliminaries Onward, we will denote the fixed point as (FP) and the unique solution as (Uni-Sol). PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 2/ 18