Fixed point results for four-point Kannan mappings with applications to fractional BVPs and Diophantine equations

Abstract

In this paper, we introduce and study Kannan-type aggregate pairwise distance mappings defined on four points, formulated through mapping conditions involving the sum of all pairwise distances of a quadrilateral configuration. We analyze their structural properties and clarify their connections with classical Kannan mappings, generalized Kannan-type mappings, as well as with mappings involving the total pairwise distance of a quadrilateral. We further extend the theory by introducing functional generalizations, namely the G-Kannan-type and B-Kannan-type aggregate pairwise distance mappings on four points, in which the constant parameter is replaced by appropriate control functions. Under the assumptions of asymptotic regularity or approximating fixed point sequence, with or without weaker continuity hypotheses, we obtain functional extensions that improve the admissible range of the associated parameters. Finally, applications are provided to the existence of solutions for fractional differential equations with boundary conditions and to a nonlinear Diophantine equation.