Fixed point-based stability analysis of climate and Langevin models

PLOS One Fixed point-based stability analysis of climate and Langevin models For all s, t ∈ 0, T, |gˆt – gˆs| | sinˆt – sinˆs| ≤ |t – s| ≤ Mˆs, t e𝜏ˆ1  t , where M(s,t) is defined in Theorem 4.1 and we may take 𝜏 1. S t 0 e–2ˆt–sds 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 ˆ𝜓, 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.

Writing – review & editing: Syed Khayyam Shah, W. Eltayeb Ahmed, Ishraq Alabdi, Ayman Alahmade, Khaled Aldwoah, E.I. Hassan. References 1. Stroud K. Essential mathematics for science and technology: a self-learning guide. South Norwalk, CT, USA: Industrial Press; 2009. 2. Riley KF, Hobson MP. Student solution manual for mathematical methods for physics and engineering. 3rd ed. Cambridge, UK: Cambridge University Press; 2019. 3. Aldwoah K, Shah SK, Hussain S, Almalahi MA, Arko YAS, Hleili M. Investigating fractal fractional PDEs, electric circuits, and integral inclusions via (𝜓,𝜙)-rational type contractions. Sci Rep. 2024;14(1):23546. https://doi.org/10.1038/s41598-024-74046-8 PMID: 39384570 4. Yang X-S. Introductory mathematics for earth scientists. Edinburgh, UK: Dunedin Academic Press; 2009. PLOS One https://doi.org/10.1371/journal.pone.0327488 July 3, 2025 17/ 18