Overview
This research introduces two novel classifications of cyclic-type mappings: co-cyclic mappings and sub-cyclic mappings. These mappings are defined on the union of two subsets within a metric space. The study aims to establish unique common fixed point theorems for these newly defined mappings within the context of complete metric spaces. The methodology relies on specific Jungck-type contractive conditions and compatibility assumptions.
Research Context
The field of fixed point theory is fundamental in various areas of mathematics. This paper contributes to this field by expanding the types of mappings considered. Specifically, it focuses on cyclic-type mappings, which typically involve elements moving between subsets in a structured manner. The introduction of 'co-cyclic' and 'sub-cyclic' mappings represents an extension of this established framework.
Approach
The researchers defined co-cyclic mappings and sub-cyclic mappings. Both are cyclic-type mappings and are characterized by their operation on the union of two subsets of a metric space. The core of the approach involved applying Jungck-type contractive conditions. These conditions are known in fixed point theory for establishing the existence and uniqueness of fixed points under certain contractual behaviors of mappings. Additionally, compatibility assumptions between the mappings were integrated into the framework. The proofs for the unique common fixed point theorems were constructed based on three key elements:
- The construction of appropriate sequences.
- The associated contractive conditions (Jungck-type).
- The completeness property of the underlying metric space.
To demonstrate the utility and validity of the proposed concepts and the main results, the study included illustrative examples.
Findings
The primary finding is the establishment of unique common fixed point theorems for both co-cyclic mappings and sub-cyclic mappings. These theorems are valid in complete metric spaces. The conditions for these theorems include the adherence to suitable Jungck-type contractive conditions and compatibility assumptions. The proofs for these theorems were successfully derived through the construction of specific sequences, the application of defined contractive conditions, and leveraging the completeness property of the metric space. Examples were provided to illustrate the introduced concepts and demonstrate the applicability of the main results.
Why This Matters
The introduction of co-cyclic and sub-cyclic mappings expands the theoretical framework of cyclic-type mappings within fixed point theory. The establishment of unique common fixed point theorems for these new classes provides new mathematical tools. Illustrative examples accompanying the results demonstrate the applicability of these newly defined concepts.
Potential Applications
While specific real-world applications are not detailed, the paper states that examples are provided to demonstrate the applicability of the main results, suggesting the theorems have practical utility within mathematical contexts.
Key Limitations Mentioned by Researchers
The paper concludes by presenting some open questions and possible directions for further research concerning co-cyclic and sub-cyclic mappings, indicating areas where the current work could be extended or deepened.