[Paper Review] Examples in Cone Metric Spaces: A Survey

Mehdi Asadi, Hossein Soleimani|arXiv (Cornell University)|Feb 23, 2011

Fixed Point Theorems Analysis13 references3 citations

TL;DR

This survey presents a comprehensive collection of examples illustrating properties of cone metric spaces, focusing on counterexamples to classical real analysis theorems such as the sandwich and comparison tests. It demonstrates that these theorems do not hold in general cone metric spaces, even with normal cones, highlighting fundamental differences from standard metric spaces.

ABSTRACT

In this survey, at first we review to many examples which have been made on cone metric spaces to verify some properties of cones on real Banach spaces and cone metrics and second, in continue like as examples that sandwich theorem doesn't hold and we shall present an other example that comparison test doesn't hold with an example for normal cones.

Motivation & Objective

• To compile and analyze examples that illustrate the behavior of cones and cone metrics in real Banach spaces.
• To investigate the failure of classical theorems—such as the sandwich and comparison tests—within cone metric spaces.
• To clarify the limitations of normal cones in preserving standard convergence properties found in real-valued metrics.
• To provide a systematic reference for researchers studying cone metric spaces and their deviations from standard metric space behavior.

Proposed method

• Systematic review and classification of existing examples in the literature on cone metric spaces.
• Construction of specific counterexamples to show that the sandwich theorem does not hold in cone metric spaces.
• Development of a counterexample demonstrating the failure of the comparison test in cone metric spaces.
• Analysis of normal cones to show that even under normality, classical comparison principles may not apply.
• Use of real Banach space structures to define and test cone metric properties.
• Logical comparison of cone metric space behavior with standard metric space theorems to isolate non-applicability conditions.

Experimental results

Research questions

• RQ1 Under what conditions does the sandwich theorem fail in cone metric spaces?
• RQ2 Can the comparison test be invalidated in cone metric spaces even when the cone is normal?
• RQ3 What structural properties of cones and metrics lead to the breakdown of classical convergence theorems?
• RQ4 How do normal cones affect the validity of comparison-based convergence arguments in cone metric spaces?
• RQ5 What examples illustrate the fundamental differences between cone metric spaces and standard metric spaces in terms of convergence behavior?

Key findings

• The sandwich theorem does not hold in general cone metric spaces, as demonstrated by constructed counterexamples.
• Even with normal cones, the comparison test can fail in cone metric spaces, indicating that normality alone does not restore classical convergence properties.
• Specific examples show that convergence in cone metric spaces cannot always be inferred from comparison with a dominating sequence.
• The behavior of cones in real Banach spaces significantly influences the validity of classical analysis theorems in cone metric frameworks.
• The survey establishes that cone metric spaces require distinct analytical tools, as standard real analysis results do not universally extend.
• The failure of these theorems underscores the need for specialized techniques in fixed-point theory and convergence analysis within cone metric settings.