Existence and Uniqueness of Fixed-Point Results in Non-Solid C⋆-Algebra-Valued Bipolar b-Metric Spaces

In this monograph, motivated by the work of Aphane, Gaba, and Xu, we explore fixed-point theory within the framework of C⋆-algebra-valued bipolar b-metric spaces, characterized by a non-solid positive cone. We define and analyze (FH−GH)-contractions, utilizing positive monotone functions to extend classical contraction principles. Key contributions include the existence and uniqueness of fixed points for mappings satisfying generalized contraction conditions. The interplay between the non-solidness of the cone, the C⋆-algebra structure, and the completeness of the space is central to our results. We apply our results to find uniqueness of solutions to Fredholm integral equations and differential equations, and we extend the Ulam–Hyers stability problem to non-solid cones. This work advances the theory of metric spaces over Banach algebras, providing foundational insights with applications in operator theory and quantum mechanics.