A unified theory of cone metric spaces and its applications to the fixed point theory

In this paper we develop a unified theory for cone metric spaces over a solid vector space. As an application of the new theory we present full statements of the iterated contraction principle and the Banach contraction principle in cone metric spaces over a solid vector space.

💡 Research Summary

The paper develops a comprehensive framework for cone metric spaces built upon solid vector spaces and demonstrates how this unified approach streamlines fixed‑point theory. The authors begin by introducing a minimalist notion of convergence on a real vector space Y, defined solely by five axioms (C1–C5) that guarantee linearity and compatibility with limits but deliberately avoid assuming uniqueness of limits. This abstract convergence relation → induces a sequential topology, allowing the definition of open and closed sets in purely sequential terms.

Next, the authors focus on cones. A cone K⊂Y is a non‑empty closed set satisfying positive scalability, additivity, and pointedness. A cone is called solid if its interior K° is non‑empty. Theorem 3.3 provides a precise characterization of K°: it is the unique non‑empty open subset of K that is stable under positive scaling, absorbs K, and does not contain the zero vector. This interior criterion becomes the cornerstone for linking cones with order structures.

Ordered vector spaces are then introduced. A vector ordering ≼ is compatible with the vector operations and the convergence relation (axioms V1–V3). The positive cone Y₊ associated with ≼ is shown to be a cone, and conversely any cone K determines a vector ordering via x≼y ⇔ y−x∈K (Theorem 4.3). The paper distinguishes several important subclasses: solid ordered spaces (where Y₊ is solid), normal spaces (satisfying a sandwich theorem for sequences), and regular spaces (where every bounded monotone sequence converges).

A key contribution is the establishment of a one‑to‑one correspondence between solid cones and strict vector orderings (Theorem 5.2). A strict ordering is defined by x≺y ⇔ x≼y and x≠y. The authors prove that a solid positive cone admits a unique strict ordering, and that the existence of a strict ordering forces the cone to be solid. This result unifies the algebraic, order‑theoretic, and topological aspects of the space.

Section 6 introduces the order topology τ generated by the convergence →. It is proved that τ coincides with the sequential topology defined by →, and that every convergent sequence has a unique limit (Theorem 6.6). Consequently, solid vector spaces behave like ordinary topological vector spaces with respect to continuity and closure.

Using the Minkowski functional associated with the interior of a solid cone, the authors show in Section 7 that the order topology is normable. For a normal solid space Y, there exists a norm ‖·‖ₖ such that xₙ→x iff ‖xₙ−x‖ₖ→0 (Theorem 7.7). Moreover, the classical sandwich theorem holds in this setting (Theorem 7.10), reinforcing the analogy with ℝ‑valued convergence.

With this groundwork, the paper defines cone metric spaces (X,d) where the distance d:X×X→Y takes values in a solid cone K⊂Y. The usual metric axioms are adapted to the cone‑valued setting. The authors prove that every cone metric space over a solid vector space is metrizable (Theorem 9.5) and that the nested‑ball theorem, completeness, and other fundamental results from classical metric space theory carry over (Theorems 9.22, 9.33). They also show that cone normed spaces are normable (Theorem 9.12), extending the classical equivalence between normed and metric structures.

The main applications are presented in Sections 10 and 11. The iterated contraction principle (Theorem 10.5) states that if a mapping T satisfies an m‑step contraction condition with constant α∈(0,1), then the Picard iterates {Tⁿx₀} converge to a unique fixed point in any complete cone metric space over a solid vector space. The theorem includes explicit a‑priori and a‑posteriori error estimates, thereby generalizing earlier partial results by Du (2010), Kadelburg et al. (2011), and others.

The Banach contraction principle (Theorem 11.1) is similarly generalized: a mapping T with a global contraction constant c∈