Introducing Fixed-Point Theorems and Applications in Fuzzy Bipolar <i>b</i>-Metric Spaces with <i>ψ_α</i>- and <i>ϝ_η</i>-Contractive Maps

Abstract

In this study, we introduce novel concepts within the framework of fuzzy bipolar b-metric spaces, focusing on various mappings such as ψα-contractive and ϝη-contractive mappings, which are essential for quantifying distances between dissimilar elements. We establish fixed-point theorems for these mappings, demonstrating the existence of invariant points under certain conditions. To enhance the credibility and applicability of our findings, we provide illustrative examples that support these theorems and expand the existing knowledge in this field. Furthermore, we explore practical applications of our research, particularly in solving integral equations and fractional differential equations, showcasing the robustness and utility of our theoretical advancements. Symmetry, both in its traditional sense and within the fuzzy context, is fundamental to our study of fuzzy bipolar b-metric spaces. The introduced contractive mappings and fixed-point theorems expand the theoretical framework and offer robust tools for addressing practical problems where symmetry is significant.

Keywords

• fixed points
• fuzzy <i>b</i>-metric space
• fuzzy bipolar <i>b</i>-metric space
• <i>b</i>-triangular property
• integral equations
• fractional differential equation