Fixed Points in Complex-Valued Extended Fuzzy b-Metric Spaces with Application to Integral Equations

This paper introduces and develops the framework of complex-valued extended fuzzy b-metric spaces, thereby broadening the scope of classical fuzzy metric and fuzzy b-metric spaces into a richer and more versatile setting. We first establish the fundamental structural properties of these spaces and then investigate the existence of fixed points within this generalized framework, thus advancing the theoretical underpinnings of fuzzy analysis. In particular, we derive fixed point theorems tailored to complex-valued extended fuzzy b-metric spaces, laying a rigorous foundation for subsequent research. An illustrative example is provided to elucidate the applicability of the proposed results, while a concrete application to integral equations demonstrates the practical relevance of the theory. By extending classical principles, most notably the Banach Contraction Principle, our findings encompass more intricate spaces that integrate both real and imaginary components. This generalization not only enriches the theory of fixed points but also situates it within a broader perspective, offering new avenues for exploration in fuzzy spaces and related mathematical disciplines.