This paper presents a novel concept of $G$-Hardy-Rogers functional operators on metric spaces endowed with a graph. It investigates sufficient circumstances under which such a mapping becomes a Picard operator. As applications of the principal idea discussed herein, a few important corresponding fixed point results in ordered metric spaces and cyclic operators are pointed out and analyzed. For upcoming research papers in this field, comparative graphical illustrations are created to highlight the pre-eminence of proposed notions with respect to the existing ones.

In this article, a Green's function for a fractional boundary value problem in connection with modified analytic kernel has been constructed to study the existence of multiple solutions of a type of characteristic fractional boundary value problems. It is done here by using a well-known result: Krasnoselskii fixed point theorem. Moreover, a practical example is created to understand the importance of main results regarding the existence of solution of a boundary value fractional differential problem with homogeneous conditions. This example analytically and graphically, explains circumstances under which the Green's functions with different types of differential operator are compatible.

In this note, by using basic properties of the recently introduced concepts of generalized metric spaces, new conditions for the existence of a fixed point for weakly type contractive operator which sends a closed subset into the ambient space under consideration are examined. Our obtained result extends and unifies its corresponding ideas in metric and modular spaces. A comparative non-trivial example is provided to show the novelty and preeminence of our proposed notion.

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 $C^{\ast }$-algebra. Inspired by this, the notion of $C^{\ast }$-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.