C*-algebra-valued perturbed modular metric spaces and existence results of fourth-order boundary value problems

The primary objective of perturbed metric spaces was to set route for the advancement of problems involving fixed point findings for a distraught structure, where errors inevitably affected the measurement of distance between two points. This objective developed into a broadly applicable context when the range set of the distance function was a $ C^{*} $-algebra. Inspired by this, the notion of $ C^{*} $-algebra-valued perturbed modular metric space was introduced in this paper. Thereafter, some fixed point theorems in the new space were studied, using paired contractive mappings. In order to demonstrate the novelty of the concepts presented herein and to generalize some significant related discoveries in the literature, contrasting examples were given. Under this comparative illustration, numerical and graphical approaches were adopted to study the rate of convergence of Picard-type and paired contractive operators. As an application, new conditions for the existence and uniqueness of a solution to fourth-order boundary value problems were obtained.