Fixed point-based stability analysis of climate and Langevin models
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PLOS ONE OPEN ACCESS Citation: Shah SK, Ahmed WE, Alabdi I, Alahmade A, Aldwoah K, Hassan EI (2025) Fixed point-based stability analysis of climate and Langevin models. PLoS One 20(7): e0327488. https://doi.org/10.1371/journal.pone.0327488 Editor: Pradip Debnath, Tezpur University, INDIA Received: April 16, 2025 Accepted: June 16, 2025 Published: July 3, 2025 Copyright: © 2025 Shah et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data availability statement: This study is purely theoretical and does not involve any empirical, observational, or simulated data. Therefore, no datasets were generated or analyzed during the current study. Funding: The author(s) received no specific funding for this work. Competing interests: The authors have declared that no competing interests exist. RESEARCH ARTICLE Fixed point-based stability analysis of climate and Langevin models Syed Khayyam Shah1, Waleed Eltayeb Ahmed 2, Ishraq Alabdi 3∗, Ayman Alahmade4, Khaled Aldwoah 5∗, Eltigani I. Hassan2 1 Department of Sustainable Environment and Energy Systems (SEES), Middle East Technical University, Northern Cyprus Campus, Kalkanli, Guzelyurt, Mersin 10, Turkey, 2 Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia, 3 Department of Mathematics, Hajjah University, Hajjah, Yemen, 4 Department of Mathematics, College of Science, Taibah
University, Medina, Saudi Arabia, 5 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah, Saudi Arabia ∗ iqalabdi@gmail.com (IA) ; aldwoah@iu.edu.sa (KA) Abstract In this manuscript, the existence and uniqueness of solutions to equations associated with climate change are discussed. For this purpose, we utilize some results from the existing literature to investigate the behavior of these equations. Additionally, the role of fixed point theory in emphasizing the importance of proving the stability and consistency of the models is explored. Several definitions and results, such as the F -contraction, 𝛼-F -contraction, rational type 𝜓, 𝜙-contraction, and Geraghty type contraction, are recalled from the existing literature to illustrate their theoretical foundations and practical applications. 1 Introduction Mathematics serves as a highly efficient and potent tool for understanding the world and addressing intricate challenges across diverse scientific fields, engineering, and technology [1–3]. In this context, the crucial role of mathematics cannot be overstated in addressing planetary issues, particularly the connection between society and nature [4,5]. Humans engage in various activities through interactions with their environment. Humans influence the environment by altering its properties, while environmental conditions, in turn, impact human health and survival capabilities. The present climate change, in contrast to other instances, is anthropogenic and marked by an unparalleled acceleration in the global mean
surface temperature (GMST) [6,7]. In [8], it is indicated that from 1880, the GMST has risen upto 0.07 C by an average per decade. In the 21st century, notably in the first two decades, the growth rate of the GMST was 0.17 C for a decade, which is more than twice the average rate. A notable indicator of human activities is the considerable rise in atmospheric carbon diox- ide (CO2) concentrations. This greenhouse gas, a by-product of the combustion of fuels, turns out to be the primary culprit for global warming. This greenhouse gas, a by-product of fuel combustion, is considered the primary driver of global warming. Global warming has impacted climate across nearly all regions of our planet; however, the extent of this effect varies considerably by region [6,7]. Global and regional climate changes PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 1/ 18
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 Γ𝜈, 𝛼, ℱ 𝜈 ∈ Rn, 𝜈0 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 gt, (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 𝜓, 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
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
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
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
PLOS One Fixed point-based stability analysis of climate and Langevin models (c) ∀ 𝛼, 𝜈, 𝔣 ∈ t0, 𝛾 and for every 𝔲, y ∈ 𝔐, |Γ𝜈1, 𝛼1, 𝔣1 – Γ𝜈2, 𝛼2, 𝔣2 | ≤ M𝔲, y e𝜏1 t , where M𝔲, y is defined in Definition 2.6. Then, (7) will have a Uni-Sol. Proof : Define a map Θ 𝔐 𝔐 Θ𝜈t 𝜈0 S t 0 Γ𝜈, 𝛼, 𝔣 d℘ t ≥ 0. (9) The existence of a FP of Θ in (9) will be evidently the Uni-Sol of the Eq (7). Now utilizing (8) and the above presumptions (b) to (c). |Θ𝜈1t – Θ𝜈2t| V𝜈0 S t 0 Γ𝜈1, 𝛼1, 𝔣1 d℘ – 𝜈0 – S t 0 Γ𝜈2, 𝛼2, 𝔣2 d℘V VS t 0 Γ𝜈1, 𝛼1, 𝔣1 – Γ𝜈2, 𝛼2, 𝔣2 d℘V ≤ S t 0 SΓ𝜈1, 𝛼1, 𝔣1 – Γ𝜈2, 𝛼2, 𝔣2S d℘ ≤ M𝔲, y e𝜏1 t S t 0 d℘ M𝔲, y e𝜏1 t ⋅ t ≤ e–𝜏M𝔲, y. This means that dmΘ𝜈1, Θ𝜈2 ≤ e–𝜏M𝔲, y. Define 𝟋 ℝ ℝ by 𝟋𝔚 ln𝔚 for 𝔚 A 0. 𝟋dmΘ𝜈1, Θ𝜈2 ≤ 𝟋e–𝜏M𝔲, y lnM𝔲, y – 𝜏 ln e. Implies that, 𝜏 𝟋dmΘ𝜈1, Θ𝜈2 ≤ 𝟋M𝔲, y. Thus, all the assumptions of Theorem 2.1 are true. Hence, the Eq (7) will have a Uni-Sol. j For our next new result let 𝔐 t0, 𝛾, R represent all the real valued continuous functions on t0, 𝛾. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 6/ 18
PLOS One Fixed point-based stability analysis of climate and Langevin models Let us define dm 𝔐 × 𝔐 ℝ as follows dm𝔲, y |𝔲t – yt|, for all 𝔲, y ∈ 𝔐. (10) Evidently, 𝔐, dm represent a complete metric space. Theorem 3.2. Suppose the following presumptions are fulfilled: (r) The function Γ t0, 𝛾 × t0, 𝛾 × ℝ ℝ is continuous on t0, 𝛾 with t0, 𝛾2, and t0, 𝛾2 × ℝ, correspondingly. (s) ∀ 𝛼, 𝜈, 𝔣 ∈ t0, 𝛾 and for every 𝔲, y ∈ 𝔐, |Γ𝜈1, 𝛼1, 𝔣1 – Γ𝜈2, 𝛼2, 𝔣2 | ≤ Ma𝔲, y e𝜏1 t , where Ma𝔲, y is defined in Definition 2.7. Then, (7) will have a Uni-Sol. Proof : Define a map Θ 𝔐 𝔐 Θ𝜈t 𝜈0 S t 0 Γ𝜈, 𝛼, 𝔣 d℘ t ≥ 0. (11) The existence of a FP of Θ in (11) will be evidently the Uni-Sol of the Eq (7). Now utilizing (11) and the above presumptions (r) to (s), |Θ𝜈1t – Θ𝜈2t| V𝜈0 S t 0 Γ𝜈1, 𝛼1, 𝔣1 dt – 𝜈0 – S t 0 Γ𝜈2, 𝛼2, 𝔣2 dtV WS t 0 Γ𝜈1, 𝛼1, 𝔣1 – Γ𝜈2, 𝛼2, 𝔣2 dtW ≤ S t 0 SΓ𝜈1, 𝛼1, 𝔣1 – Γ𝜈2, 𝛼2, 𝔣2S dt ≤ Ma𝔲, y e𝜏1 t S t 0 dt Ma𝔲, y e𝜏1 t ⋅ t ≤ e–𝜏Ma𝔲, y. As a result, dmΘ𝜈1, Θ𝜈2 ≤ e–𝜏Ma𝜔1, 𝜔2 Define 𝟋 ℝ ℝ by 𝟋𝔚 ln𝔚, for 𝔚 A 0. 𝟋dmΘ𝜈1, Θ𝜈2 ≤ 𝟋e–𝜏Ma𝔲, y lnMa𝔲, y – 𝜏 ln e. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 7/ 18
PLOS One Fixed point-based stability analysis of climate and Langevin models Next define 𝛼 𝔐 × 𝔐 –∞ ∪ 0, ∞ by 𝛼𝜈1, 𝜈2 ¢¨¨ ¦ ¨¨¤ 1, if dm𝜈1, 𝜈2 ≥ 0, –∞, else. Accordingly, 𝜏 𝟋dmΘ𝜈1, Θ𝜈2 ≤ 𝛼𝜈1, 𝜈2𝟋Ma𝔲, y. Thus, all the presumptions of Theorem 2.2 hold true. So, the equation (7) will have a Uni-Sol. j Following the same flow, taking important results of Geraghty-type contractions in the framework of b-metric space, we can have novel existence and uniqueness results. Let 𝔐b t0, 𝛾, R represent all the real valued continuous functions on 𝔐. Let us define bm 𝔐b × 𝔐b ℝ as follows bm𝔲, y |𝔲 – y|2, for all 𝔲, y ∈ 𝔐b, bm. (12) Evidently, 𝔐b, bm, 𝔟 is a complete b-metric space with 𝔟 ≥ 1. Theorem 3.3. Suppose the following presumptions are fulfilled: (e) The function Γ t0, 𝛾 × t0, 𝛾 × ℝ ℝ is continuous on t0, 𝛾 with t0, 𝛾2, and t0, 𝛾2 × ℝ, correspondingly. (f) ∀ 𝜈, 𝛼, 𝔣 ∈ t0, 𝛾 and for every 𝔲, y ∈ 𝔐, |Γ𝜈1, 𝛼1, 𝔣1 – Γ1𝜈2, 𝛼2, 𝔣2 | ≤ ¿ Á Á À ln1 Mc𝜈1, 𝜈2 1 t𝔟 , where Mc𝜈1, 𝜈2 is defined in Theorem 2.4. Then, (7) will have a Uni-Sol. Proof : Define a map Θ 𝔐 𝔐 Θ𝜈t 𝜈0 S t 0 Γ𝜈, 𝛼, 𝔣 d℘ t ≥ 0. (13) The existence of a FP of Θ in (13) will be evidently the Uni-Sol of the Eq (7). Now, utilizing (12), (13) and the above presumptions (e) and (f) |Θ𝜈1t – Θ𝜈2t|2 V𝜈0 S t 0 Γ𝜈1, 𝛼1, 𝔣1 dt – 𝜈0 – S t 0 Γ𝜈2, 𝛼2, 𝔣2 dtV 2 WS t 0 Γ𝜈1, 𝛼1, 𝔣1 – Γ𝜈2, 𝛼2, 𝔣2 dtW 2 ≤ S t 0 SΓ𝜈1, 𝛼1, 𝔣1 – Γ𝜈2, 𝛼2, 𝔣2S2 dt ≤ 1
𝔟1 t ln1 Mc𝜈1, 𝜈2 S t 0 dt ≤ 1 𝔟 ln1 Mc𝜈1, 𝜈2. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 8/ 18
PLOS One Fixed point-based stability analysis of climate and Langevin models Further, since 𝔟 2 |Θ𝜈1t – Θ𝜈2t|2 ≤ 1 2 ln1 Mc𝜈1, 𝜈2. It implies that, bmΘ𝜈1, Θ𝜈2 ≤ 1 2 ln1 Mc𝜈1, 𝜈2. (14) Define 𝛽 ℝ 0, 1 2 by 𝛽𝔓 ln1 𝔓 2𝔓 , for 𝔓 A 0 and 𝛽0 ∈ 0, 1 2 . Thus, (14) becomes, bmΘ𝜈1, Θ𝜈2 ≤ ln 1 Mc𝜈1, 𝜈2 2Mc𝜈1, 𝜈2 Mc𝜈1, 𝜈2 𝛽Mc𝜈1, 𝜈2Mc𝜈1, 𝜈2. Thus, all the necessities of Theorem 2.4 are true. So, the Eq (7) will have a Uni-Sol. j Let us define dm 𝔐b × 𝔐b ℝ as follows dm𝔲, y |𝔲t – yt|, for all 𝔲, y ∈ 𝔐. (15) Evidently, 𝔐, dm represent a complete metric space. Theorem 3.4. Suppose the following presumptions are fulfilled: (m) The function Γ t0, 𝛾 × t0, 𝛾 × ℝ ℝ is continuous on t0, 𝛾, t0, 𝛾2, and t0, 𝛾2 × ℝ, correspondingly. (n) For all 𝜈, 𝛼, 𝔣 ∈ t0, 𝛾 and for all 𝔲, y ∈ 𝔐, |Γ𝜈1, 𝛼1, 𝔣1 – Γ1𝜈2, 𝛼2, 𝔣2 | ≤ MbΘ𝜈1, Θ𝜈2 – 𝜑MbΘ𝜈1, Θ𝜈2 1 t , where Mb𝜈1, 𝜈2 and 𝜑 are defined in Theorem 2.5. Then, (7) will have a Uni-Sol. Proof : Define a map Θ 𝔐 𝔐 Θ𝜈t 𝜈0 S t 0 Γ𝜈, 𝛼, 𝔣 d℘ t ≥ 0. (16) The existence of a FP of Θ in (16) will ensure a Uni-Sol of the integral Eq (7). PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 9/ 18
PLOS One Fixed point-based stability analysis of climate and Langevin models Now, utilizing (15) and the above presumptions (m) and (n), |Θ𝜈1t – Θ𝜈2t| V𝜈0 S t 0 Γ𝜈1, 𝛼1, 𝔣1 dt – 𝜈0 – S t 0 Γ𝜈2, 𝛼2, 𝔣2 dtV WS t 0 Γ𝜈1, 𝛼1, 𝔣1 – Γ𝜈2, 𝛼2, 𝔣2 dtW ≤ 1 1 t S t 0 WΓ𝜈1, 𝛼1, 𝔣1 – Γ𝜈2, 𝛼2, 𝔣2W dt ≤ Mb𝜈1, 𝜈2 – 𝜑Mb𝜈1, 𝜈2 1 t S t 0 dt ≤ Mb𝜈1, 𝜈2 – 𝜑Mb𝜈1, 𝜈2. Further, Θ𝜈1t – Θ𝜈2t| ≤ Mb𝜈1, 𝜈2 – 𝜑Mb𝜈1, 𝜈2. Hence, dmΘ𝜈1, Θ𝜈2 ≤ Mb𝜈1, 𝜈2 – 𝜑Mb𝜈1, 𝜈2. Define 𝜓 0, ∞ 0, ∞ by 𝜓𝔓 𝔓. 𝜓dmΘ𝜈1, Θ𝜈2 ≤ 𝜓 Mb𝜈1, 𝜈2 – 𝜑Mb𝜈1, 𝜈2 . Thus, all the necessities of Theorem 2.5 hold true. So, the Eq (7) will have a Uni-Sol. j Example 3.1. Consider the climate model described by the integral equation 𝜈t 𝜈0 S t 0 𝛼𝜈𝜏 cos𝜈𝜏 𝛽e–𝜏 d𝜏, t ∈ 0, T, where 𝜈t represents the temperature anomaly, 𝛼 0.3 is the climate feedback parameter, 𝛽 0.1 models external forcing, and 𝜈0 0 is the initial condition. Let 𝔐 Ct0, 𝛾 represent set of all real-valued functions with no discontinuity on [0,T], endowed with the metric dmu, y sup t∈0,T |ut – yt|. Define the operator Θ 𝔐 𝔐 by Θ𝜈t S t 0 0.3𝜈𝜏 cos𝜈𝜏 0.1e–𝜏 d𝜏. To verify the conditions of Theorem 3.1, note that for all u, y ∈ 𝔐, |Γu – Γy| 0.3|u cosu – y cosy|. Using the inequality |u cosu – y cosy| ≤ 2|u – y|, it follows that |Γu – Γy| ≤ 0.6|u – y|. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 10/ 18
PLOS One Fixed point-based stability analysis of climate and Langevin models Let 𝜏 – ln0.6 A 0 and define FP ln𝔚. Then, dmΘu, Θy ≤ 0.6 dmu, y ⇒ FdmΘu, Θy ≤ Fdmu, y – 𝜏, which satisfies the Ćirić-type contraction condition. Moreover, the function Γ𝜈, 𝛼, F 0.3𝜈 cos𝜈 0.1e–t, is continuous in both 𝜈 and t. Additionally, the growth condition |Γ𝜈1 – Γ𝜈2| ≤ 0.6|𝜈1 – 𝜈2| ≤ Mu, y e𝜏1 t , holds with Mu, y dmu, y, since 0.6 @ e–𝜏 for 𝜏 – ln0.6. Therefore, all the conditions of Theorem 3.1 are true, and thus integral equation admits a unique solution 𝜈 ∈ 𝔐. 4 Existence results for Langevin equation In this section, the same approach is utilized to investigate the stability of the Langevin equation. dx dt 𝛽x gt, (17) where, g(t) is the external force, 𝛽 is a positive constant (damping coefficient or stabilizing term). Using the technique of green function, the Eq (17) can be transformed to xt S t 0 Kt, sgsds x0t, (18) where the Green function that describes how the system responds to the external forcing is K ¢¨¨ ¦ ¨¨¤ e–𝛽t–s, if t ≥ s, 0, if t @ s. Let 𝔐 Ct0, 𝛾 represent set of all real-valued functions with no discontinuity. Let us define dm 𝔐 × 𝔐 ℝ as dm𝔲, y |𝔲t – yt|, for all 𝔲, y ∈ 𝔐. (19) Evidently, 𝔐, dm represents a complete 𝕄-space. Theorem 4.1. Assume the fulfillment of the following hypotheses: (a2) The function gt s, t × s, t × ℝ ℝ is continuous on Ct0, 𝛾, Ct0, 𝛾2, and Ct0, 𝛾2 × ℝ, correspondingly.
PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 11/ 18
PLOS One Fixed point-based stability analysis of climate and Langevin models (a3) ∀ s, t ∈ Ct0, 𝛾, |gt – gs| ≤ e𝜏 Ms, t t – s . (a4) ∀ s, t ∈ Ct0, 𝛾, S t 0 Kt, sds ≤ t – s e2𝜏 . Where M(s,t) is defined in Definition 2.6 and 𝜏 A 0. Then the Eq (18) will have a unique solution. Proof : Define a map B 𝔐 𝔐 Bxt S t 0 Kt, sgsds x0t. (20) The existence of a FP of B in (20) will guarantee a Uni-Sol of the integral Eq (18). Now utilizing (20) and the above presumptions (a2) to (a3), |Bx1t – Bx2t| Vx0t S t 0 gtKs, tds – x0t – S t 0 gsKs, tdsV VS t 0 Kt, sgt – gsdsV ≤ e𝜏 Ms, t t – s S t 0 Kt, sds ≤ We𝜏 t – s e2𝜏 Ms, t t – s W e–𝜏Ms, t. This means that dmBx1, Bx2 ≤ e–𝜏Ms, t. Define 𝟋 ℝ ℝ by 𝟋𝔚 ln𝔚 for 𝔚 A 0. 𝟋dmBx1, Bx2 ≤ 𝟋e–𝜏Ms, t lnMs, t – 𝜏 ln e. Implies that, 𝜏 𝟋dmBx1, Bx2 ≤ 𝟋Ms, t Thus, all the necessities of Theorem 2.1 hold true. So, the Eq (18) will have a Uni-Sol. j Theorem 4.2. Consider that the subsequent assumptions are met: (b2) The function gt s, t × s, t × ℝ ℝ is continuous on Ct0, 𝛾, Ct0, 𝛾2, and Ct0, 𝛾2 × ℝ, correspondingly. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 12/ 18
PLOS One Fixed point-based stability analysis of climate and Langevin models (b3) ∀ s, t ∈ Ct0, 𝛾, |gt – gs| ≤ e𝜏 Mas, t t – s . (b4) ∀ s, t ∈ Ct0, 𝛾, S t 0 Kt, sds ≤ t – s e2𝜏 , where Mas, t is defined in Definition 2.7. Then, (18) will have a Uni-Sol. Proof : Define a map B 𝔐 𝔐 Bxt S t 0 Kt, sgsds x0t. (21) The existence of a FP of B in (21) ensures a Uni-Sol of the integral Eq (18). Now utilizing (21) and the above presumptions (b2) to (b3), |Bx1t – Bx2t| Vx0t S t 0 gtKs, tds – x0t – S t 0 gsKs, tdsV VS t 0 Kt, sgt – gsdsV ≤ e𝜏 Ms, t t – s S t 0 Kt, sds ≤ e𝜏 t – s e2𝜏 Ms, t t – s e–𝜏Ms, t. Hence, dmBx1, Bx2 ≤ e–𝜏Mas, t. Define 𝟋 ℝ ℝ by 𝟋𝔚 ln𝔚, for 𝔚 A 0. 𝟋dmBx1, Bx2 ≤ 𝟋e–𝜏Mas, t lnMas, t – 𝜏 ln e. Therefore, 𝜏 𝟋dmBx1, Bx2 ≤ 𝟋Mas, t. Next define 𝛼 𝔐 × 𝔐 –∞ ∪ 0, ∞ by 𝛼s, t ¢¨¨ ¦ ¨¨¤ 1, if dmBx1, Bx2 ≥ 0, –∞, else. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 13/ 18
PLOS One Fixed point-based stability analysis of climate and Langevin models It implies that, 𝜏 𝟋dmBx1, Bx2 ≤ 𝛼s, t𝟋Mas, t. Thus, all the necessities of Theorem 2.2 hold true. Hence, the Eq (18) will have a Uni-Sol. j Let 𝔐b Ct0, 𝛾 represent set of all real-valued functions with no discontinuity and define bm 𝔐b × 𝔐b ℝ as bm𝔲, y |𝔲t – yt|2, for all 𝔲, y ∈ 𝔐b. (22) Evidently, 𝔐b, bm, 𝔟 is a complete b-metric space. Theorem 4.3. Assume the below presumptions fulfill: (c1) The function gt s, t × s, t × ℝ ℝ defined to be continuous on Ct0, 𝛾, Ct0, 𝛾2, and Ct0, 𝛾2 × ℝ, correspondingly. (c2) For all s, t ∈ Ct0, 𝛾 |gt – gs| ≤ »ln1 Mcs, t º𝔟t – s . (c3) ∀ s, t ∈ Ct0, 𝛾, S t 0 Kt, sds ≤ t – s, where Mcs, t is defined in Theorem 2.4. Then, (18) will have a Uni-Sol and consequently will be stable. Proof : Define a map B 𝔐b 𝔐b Bxt S t 0 Kt, sgsds x0t. (23) The existence of a FP of B in (23) ensure a Uni-Sol of the integral Eq (18). Now, utilizing (22) and the above presumptions (c1) to (c3) |Bx1t – Bx2t| Vx0t S t 0 gtKs, tds – x0t – S t 0 gsKs, tdsV VS t 0 Kt, sgt – gsdsV ≤ »ln1 Mcs, t º𝔟t – s S t 0 Kt, sds »ln1 Mcs, t º𝔟t – s t – s. Further, since 𝔟 2, |Bx1t – Bx2t|2 ≤ 1 2 ln1 Mcs, t. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 14/ 18
PLOS One Fixed point-based stability analysis of climate and Langevin models Accordingly, bmBx1t – Bx2t ≤ 1 2 ln1 Mcs, t. (24) Define 𝛽 ℝ 0, 1 2 by 𝛽𝔓 ln1 𝔓 2𝔓 for 𝔓 A 0 and 𝛽0 ∈ 0, 1 2 . Thus, (24) becomes, bmBx1t – Bx2t ≤ ln 1 Mcs, t 2Mcs, t Mcs, t. 𝛽Mcs, tMcs, t. Thus, all the necessities of Theorem 2.2 are true. So, the Eq (18) will have a Uni-Sol and consequently will be stable. j Let 𝔐 Ct0, 𝛾 represent set of all real-valued functions with no discontinuity and define dm 𝔐 × 𝔐 ℝ as dm𝔲, y |𝔲t – yt|, for all 𝔲, y ∈ 𝔐. (25) Evidently, 𝔐, dm represent a complete metric space. Theorem 4.4. Assume the fulfillment of the following hypotheses: (m1) The function gt s, t × s, t × ℝ ℝ is continuous on Ct0, 𝛾, Ct0, 𝛾2, and Ct0, 𝛾2 × ℝ, correspondingly. (m2) For all s, t ∈ Ct0, 𝛾 |gt – gs| ≤ Mbs, t – 𝜑Mbs, t t – s . (m3) ∀ s, t ∈ Ct0, 𝛾, S t 0 Kt, sds ≤ t – s. Then, (18) will have a Uni-Sol. Proof : Define a map B 𝔐 𝔐 by Bxt S t 0 Kt, sgsds x0t. (26) The existence of a Uni-Sol of the integral Eq (18) is equivalent to the existence of a FP of B in (26). PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 15/ 18
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
PLOS One Fixed point-based stability analysis of climate and Langevin models For all s, t ∈ 0, T, |gt – gs| | sint – sins| ≤ |t – s| ≤ Ms, t e𝜏1 t , where M(s,t) is defined in Theorem 4.1 and we may take 𝜏 1. S t 0 e–2t–sds 1 – e–2t 2 ≤ t 2 ≤ t e–2 te2𝜏, for 𝜏 1, so the kernel satisfies the integral growth condition. All conditions of Theorem 4.1 turn true. Therefore, the equation admits a unique solution x ∈ 𝔐. 5 Conclusion This paper investigates the stability of some climate systems models. In this direction, some important contractions such as F-contraction, 𝛼-F-contraction, rational type 𝜓, 𝜙-contraction, and Geraghty type contraction have been utilized to formulate some novel existence results to ensure the stability of these models. 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 thorough analysis and applying fixed-point theorems from the existing literature. The uniqueness and existence of these mod- els are significant for making sure that the models are accurate and reliable, which in turn is necessary for making smart decisions about climate change and disasters. Acknowledgments This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2502). Author contributions Methodology: Syed Khayyam Shah. Supervision: Khaled Aldwoah. Validation: Ishraq Alabdi. Writing – original draft: Syed Khayyam Shah.
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