In this article, we establish common $\alpha$ -fuzzy fixed point theorems for Chatterjea type contractions involving rational expression in complex valued extended $b$-metric space. Our results generalize and extend some familiar results in the literature. Some common fixed point results for multivalued and single valued mappings are derived for complex valued extended $b$-metric space, complex valued $b$-metric space and complex valued metric space as consequences of our leading results. As an application, we investigate the solution of Fredholm integral inclusion.

The purpose of this article is to investigate the existence of solutions for Urysohn integral equations. To achieve our objectives, we take advantage of common fixed point results for self-mappings satisfying a generalized contraction involving control functions of two variables in the context of complex valued metric spaces. We also supply a non-trivial example to show the validity of obtained results.

This study explores the concept of complex-valued suprametric spaces, a recent and generalized framework in fixed point theory. This concept extends both complex-valued spaces and classical metric spaces. The article aims to establish common fixed point results for rational contractions governed by control functions dependent on two variables within the setting of complex-valued suprametric spaces. These findings generalize several well-known results in the field. An illustrative example is provided to validate the novelty of the key result. Furthermore, the study derives common fixed point theorems for rational contractions with control functions of a single variable under the same framework. As a practical application, the findings are applied to study the solution of a nonlinear mixed Volterra–Fredholm integral equation within the context of predator-prey dynamics.

The objective of this research is to propose a new concept known as rational ( $\alpha \eta$- $\psi )$-contractions in the framework of $\mathcal{F}$-metric spaces and to establish several fixed point theorems. These theorems help to generalize and unify various established fixed point results from the existing literature. To demonstrate the practical effectiveness of our approach, we provide a significant example that confirms our findings. In addition, we introduce a generalized multivalued ( $\alpha$- $\psi )$-contraction concept in $\mathcal{F}$-metric spaces and use it to prove fixed point theorems specifically designed for multivalued mappings. To demonstrate the practical utility of our findings, we apply our main results to the solution of synaptic delay differential equations in neural networks.