<p>In this paper, we extend the Bianchini fixed point theorem to encompass both contractive and non-expansive mappings within metric spaces. This extended result allows for the existence of multiple fixed points, with the fixed-point sets and domains of these mappings exhibiting intriguing algebraic, geometric, and dynamical properties. Our theorem offers a significant generalization of many existing results concerning contractive mappings. Our main theorem allows us to obtain the integral solutions of a nonlinear Diophantine equation. These solutions are Pythagorean triples, which represent right-angled triangles, with each integer in the triple belonging to a Fibonacci-type sequence. We also identify additional application areas where our results could be valuable, given their alignment with well-established concepts like the <i>n</i>th roots of unity, which are utilized in various fields of science, mathematics, medicine, and engineering.</p>