Existence and Uniqueness of a Solution of a Boundary Value Problem Used in Chemical Sciences via a Fixed Point Approach

In this paper, we present Proinov-type fixed point theorems in the setting of bi-polar metric spaces and fuzzy bi-polar metric spaces. Fuzzy bi-polar metric spaces with symmetric property extend classical metric spaces to address dual structures and uncertainty, ensuring consistency and balance. We provide different concrete conditions on the real-valued functions Ω,Π:0,∞→R for the existence of fixed points via the (Ω,Π)-contraction in bi-polar metric spaces. Further, we define real-valued functions Ω,Π:(0,1]→R to obtain fixed point theorems in fuzzy bi-polar metric spaces. We apply Ω,Π fuzzy bi-polar version of a Banach fixed point theorem to show the existence of solutions. Furthermore, we provide some non-trivial examples to show the validity of our results. In the end, we find the existence and uniqueness of a solution of integral equations and boundary value problem used in chemical sciences by applying main results.