Extended Generalized Contractions, Metric Completeness, and Applications to Solving Nonlinear Diophantine Equations

In this paper, we introduce a generalized (ε-δ)-type contractive condition and develop a unified framework that extends a wide spectrum of classical fixed-point results for self-mappings in complete metric spaces, with or without continuity assumptions. The framework accommodates both contractive and nonexpansive mappings and unifies a broad class of classical fixed-point theorems, including those of Banach, Kannan, Chatterjea, Bianchini, Matkowski, Boyd–Wong, Meir–Keeler, Reich, Ćirić, Ćirić–Jachymski–Matkowski, rational contractions, and Pant, within a multiple-fixed-point setting. One of the established theorems yields a characterization of metric completeness that is different from that of Subrahmanyam in his research in 1975, which covers mappings that are discontinuous at their fixed points as well as mappings admitting multiple fixed points. The theoretical developments are illustrated through several nontrivial examples, including a geometric example in the complex plane whose fixed points are the Nth roots of unity. Following the initial 2024 resolution by Pant and collaborators, this paper provides an alternative constructive solution to the strong version of Rhoades’ 1988 open problem. An example on the integer solutions of a nonlinear Diophantine equation illustrates the results in a discrete arithmetic setting.