Fixed Point Theorems Involving Generalized Hypergeometric and Bessel-Type Functions

Abstract

This paper investigates new classes of fixed point theorems generated by mappings constructed from generalized hypergeometric and Bessel-type functions. Motivated by the analytical structure of special functions, we define nonlinear operators of the form T(x) = pF (a;b;λJν(x)) and establish sufficient conditions under which such mappings admit unique fixed points in complete and G-metric spaces. Several contraction-type results, including Banach, Kannan, and Ciric variants, are extended to these function-based operators. The analytical properties of pF and Jν(x)—such as recurrence relations, differentiability, and asymptotic behavior—are employed to demonstrate the existence and stability of the obtained fixed points. Illustrative numerical examples are provided to highlight the rate of convergence of iterative sequences associated with the proposed mappings. Furthermore, potential applications to nonlinear integral and differential equations are discussed, revealing the usefulness of these theorems in the analysis of equations containing special-function kernels. The presented framework not only unifies several existing results in fixed point theory but also bridges the gap between analytic number theory and nonlinear functional analysis.