Fixed and periodic points of general mappings with applications

In this paper, we establish new results on the existence, uniqueness, and convergence of general mappings in two cases: mappings that admit only a fixed point, and mappings that admit both fixed and periodic points. Our findings apply to both contractive and non-expansive mappings, thereby broadening their scope of applicability. We demonstrate that the established fixed-point theorems have significant number-theoretic consequences. The analysis of these classes of mappings leads directly to a generating equation for Pythagorean triples and, ultimately, to the classical negative Pell equation U2 - 2Y2 = -1.