A new approach for fixed point theorems for $C$-class functions in Hilbert $C^{\ast }$-modules

In this paper, we introduced a new contraction principle via altering distance and $C$-class functions with rational forms which extends and generalizes the existing version provided by Hasan Ranjbar et al. [H. Ranjbar, A. Niknam, A fixed point theorem in Hilbert $C^{\ast }$-modules, Korean J. Math., 30 (2022), 297–304]. Specifically, the rational forms involved in the contraction condition we presented involve the $p$-th power of the displacements which can exceed the second power mentioned in Hasan Ranjbar et al.'s paper. Moreover, we also proved a fixed point theorem for this type of contraction in the Hilbert $C^{\ast }$-module. Some adequate examples were provided to support our results. As an application, we applied our result to prove the existence of a unique solution to an integral equation and a second-order $(p,q)$-difference equation with integral boundary value conditions.