The theory of fixed points occupies a central position in modern mathematical analysis because of its deep interconnections with nonlinear functional analysis, topology, computer science, and applied modeling. Classical results such as the Banach contraction principle established a powerful paradigm for guaranteeing the existence and uniqueness of fixed points in complete metric spaces. However, the increasing complexity of modern mathematical structures, particularly those emerging from computer science, fuzzy logic, optimization, and dynamical systems, has required the extension of fixed-point theory far beyond standard metric spaces and simple contraction conditions. Partial metric spaces, symmetric spaces, fuzzy metric spaces, and dislocated or quasi-metric structures represent some of the most important generalizations in which distance-like functions allow nonzero self-distances, asymmetry, or uncertainty. At the same time, the classical notion of contraction has been weakened into various forms of weak contractions, generalized contractions, and implicit relational contractions, enabling the treatment of a much broader class of nonlinear operators.

This article develops a comprehensive, integrated, and theoretically coherent study of common fixed-point theory for weakly contractive and compatible mappings defined on partial metric spaces, symmetric spaces, and related generalized structures. Drawing strictly on the conceptual and methodological foundations provided by the referenced works, including Jungck’s theory of compatible mappings, the developments of Aamri and El Moutawakil on common fixed points in symmetric spaces, Matthews’ foundational topology of partial metric spaces, and the modern formulations of weak contractions in ordered and partial metric contexts by Aydi, Ciric, Samet, and their collaborators, the present work unifies these strands into a single interpretative framework.

The study emphasizes the role of compatibility, weak compatibility, and contractive-type implicit relations in ensuring the convergence of iterative sequences and the existence of unique or nonunique common fixed points. Particular attention is given to the subtle interplay between order structures, partial metrics, and weak contractive conditions, which allows one to overcome the absence of classical continuity, self-mapping, or strict contraction assumptions. Theoretical implications for the completeness of generalized spaces, the stability of fixed points, and the robustness of iterative processes are analyzed in depth.

By elaborating the logical structure underlying existing theorems and comparing their assumptions and consequences, this paper clarifies how diverse fixed-point results in partial metric spaces, symmetric spaces, fuzzy metric spaces, and ordered sets can be viewed as manifestations of a unified principle: the balance between generalized distance, compatibility, and weak contraction. The findings underscore the enduring relevance of fixed-point theory as a foundational tool in abstract mathematics and its growing importance in applications where classical metric assumptions are no longer adequate.

Partial metric spaces, compatible mappings, weak contractions

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