Locally Convex Valued Rectangular Metric Spaces and The Kannans Fixed Point Theorem

Rectangular TVS-cone metric spaces are introduced and Kannan’s fixed point theorem is proved in these spaces. Two approaches are followed for the proof. At first we prove the theorem by a direct method using the structure of the space itself. Secondly, we use the nonlinear scalarization used recently by Wei-Shih Du in [A note on cone metric fixed point theory and its equivalence, {Nonlinear Analysis},72(5),2259-2261 (2010).] to prove the equivalence of the Banach contraction principle in cone metric spaces and usual metric spaces. The proof is done without any normality assumption on the cone of the locally convex topological vector space, and hence generalizing several previously obtained results.

💡 Research Summary

The paper introduces a new class of metric‑type spaces called rectangular TVS‑cone metric spaces (R‑TVS‑CMS). These spaces are built on a real Hausdorff locally convex topological vector space (E, S) equipped with a closed, solid cone P ⊂ E. A vector‑valued distance p : X × X → E is required to satisfy four conditions: non‑negativity and definiteness, symmetry, and a rectangular inequality p(x,z) ≤ p(x,y)+p(y,w)+p(w,z) for any four distinct points x, y, z, w. This rectangular inequality replaces the usual triangle inequality and allows a broader class of spaces; every TVS‑cone metric space (TVS‑CMS) is an R‑TVS‑CMS, but the converse need not hold, as illustrated by a concrete example with E = ℝ² and P = ℝ²₊.

The authors first review the standard cone theory, emphasizing the normality condition (which links the cone order to the topology). Normality is traditionally required to translate cone‑valued inequalities into real‑valued ones, but the present work deliberately avoids any normality assumption. Instead, they employ the nonlinear scalarization map ξₑ associated with a fixed interior point e ∈ int P, defined by ξₑ(y)=inf{t∈ℝ | y∈t e−P}. This map is positively homogeneous, continuous, and sub‑additive, and it satisfies ξₑ(y) ≤ t ⇔ y ∈ t e−P, among other useful properties.

By composing ξₑ with the vector‑valued distance p, the authors obtain a real‑valued distance dₚ = ξₑ∘p. Lemma 1.6 (citing Du) guarantees that dₚ is a genuine metric on X, and Theorem 3.1 shows that dₚ also satisfies the rectangular inequality, thus turning (X,dₚ) into a rectangular metric space (R‑MS). Lemma 3.2 establishes the equivalence of convergence, Cauchy sequences, and completeness between (X,p) and (X,dₚ). Consequently, any fixed‑point result proved in the classical rectangular metric setting can be transferred to the cone‑valued setting without invoking normality.

The main fixed‑point result is a Kannan‑type theorem. For a self‑map T : X→X satisfying the cone‑valued Kannan contractive condition

 p(Tx,Ty) ≤ β