Fixed point based analysis of climate change model via rational type modified contractions

This paper introduces a novel class of rational-type $\alpha -\mathcal{F}_{c}$-contractions within the framework of extended b-metric spaces. The generality offered by the auxiliary function $\mathcal{F}_{c}$ facilitates a noteworthy extension of the classical contraction principle, enabling the derivation of unique fixed-point results in broader and more flexible analytical settings. These results are particularly valuable for tackling complex problems arising in the study of nonlinear differential and integral equations. In pursuit of practical relevance, we apply the developed theory to examine the dynamics of a climate change model, demonstrating the theoretical findings in a real-world context. Carefully constructed examples are included to validate the main results and highlight their robustness. The extended b-metric space serves as a powerful analytical framework, particularly in situations where the traditional b-metric structure fails to provide adequate applicability.