In the first part of this paper, we prove some generalized versions of the result of Matthews in (Matthews, 1994) using different types of conditions in partially ordered partial metric spaces for dominated self-mappings or in partial metric spaces for self-mappings. In the second part, using our results, we deduce a characterization of partial metric 0-completeness in terms of fixed point theory. This result extends the Subrahmanyam characterization of metric completeness. 1. Introduction In the mathematical field of domain theory, attempts were made in order to equip semantics domain with a notion of distance. In particular, Matthews [1] introduced the notion of a partial metric space as a part of the study of denotational semantics of data for networks, showing that the contraction mapping principle can be generalized to the partial metric context for applications in program verification. Moreover, the existence of several connections between partial metrics and topological aspects of domain theory has been lately pointed by other authors as O'Neill [2], Bukatin and Scott [3], Bukatin and Shorina [4], Romaguera and Schellekens [5], and others (see also [6–14] and the references therein). After the result of Matthews [1], the interest for fixed point theory developments in partial metric spaces has been constantly growing, and many authors presented significant contributions in the directions of establishing partial metric versions of well-known fixed point theorems for the existence of fixed points, common fixed points, and coupled fixed points in classical metric spaces (see e.g., [15, 16]). Obviously, we cannot cite all these papers but we give only a partial list [17–49]. Recently, Romaguera [50] proved that a partial metric space is 0-complete if and only if every -Caristi mapping on has a fixed point. In particular, the result of Romaguera extended Kirk’s [51] characterization of metric completeness to a kind of complete partial metric spaces. Successively, Karapinar in [36] extended the result of Caristi and Kirk [52] to partial metric spaces. In the first part of this paper, following this research direction, we prove some generalized versions of the result of Matthews by using different types of conditions in ordered partial metric spaces for dominated self-mappings or in partial metric spaces for self-mappings. The notion of dominated mapping of economics, finance, trade, and industry is also applied to approximate the unique solution of nonlinear functional equations. In the second part, using the results obtained in the first part, we
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