Fuzzy Metric Research Articles

Fixed Point Results for Almost Contraction Mappings in Fuzzy Metric Space

في بعض المسائل الرياضية والحاسوبية والاقتصادية والنمذجة، يكون وجود حل لمشكلة نظرية أو مشكلة في العالم الحقيقي مرادفًا لوجود نقطة صامدة (Fp) لدالة مناسبة. وبالتالي، فأن نظرية النقطة الصامدة Fp تلعب دورًا أساسيًا في مجموعة واسعة من السياقات الرياضية والعلمية. تعتبر نظرية النقطة الصامدة Fp في حد ذاتها مزيجًا مذهلاً مكون من التحليل (التحليل الصرف والتحليل التطبيقي)، والهندسة، والطوبولوجيا. لقد أظهرت السنوات الأخيرة أن نظرية النقطة الصامدة Fp هي أداة قوية ومفيدة للغاية في دراسة الحالات غير الخطية. تهتم نظريات النقطة الصامدة Fp بدالة f من المجموعة X الى المجموعة X نفسها والتي في ظل ظروف معينة، فأنها تسمح بوجود نقطة صامدة Fp ، بعبارة اخرى انه بمعنى لكل نقطة x موجودة في المجموعة X (x∈X) بحيث أن f(x)=x . يقدم هذا العمل ويثبت نظرية النقطة الصامدة Fp لأنواع مختلفة من دوال الانكماش في الفضاء المتري الضبابي (FM-space) والتي تسمى بدالة الانكماش القريبة ودالة الانكماش الضعيفة القريبة ( Ψ̃ ,Φ̃). في البداية، تم التذكير بمفهوم الفضاء المتري الضبابي(FM-space) وبعض المصطلحات المستخدمة في الإطار الضبابي. ثم بعد ذلك تم اعطاء مفهوم دالة المحاكاة. يتم استخدام مفهوم دالة المحاكاة هذا لتقديم تعريف دالة الانكماش Z̃ القريبة في إطار الفضاء المتري الضبابي. بالإضافة إلى ذلك، لقد تم استخدام هذا المفهوم(دالة الانكماش Z̃ القريبة) لبرهان وجود النقطة الصامدة ووحدانية النقطة الصامدة لهذا النوع من الدوال.ثم بعد ذلك تم تقديم فكرة دالة الانكماش الضعيفة القريبة ( Ψ̃ ,Φ̃) في إطار الفضاء المتري الضبابي(FM-space) ، بالإضافة إلى تقديم نظرية النقطة الصامدة Fp لهذا النوع من من الدوال. وفي نهاية البحث تم تقديم بعض الأمثلة لدعم النتائج.

Common Tripled Fixed Point Theorem on M- Fuzzy Metric Space for Occasionally Weakly Compatible Mappings

The fixed point theorems, which are primarily existential in nature, serve as a fundamental topological toolkit for the qualitative analysis of solutions to both linear and nonlinear equations in various branches of mathematics. Many authors have extended and generalized these results in different ways, particularly in the context of fuzzy metric spaces and fuzzy mappings. Numerous researchers have also proved common fixed point theorems under the condition of compatible mappings for fizzy metric spaces. Coupled common fixed point theorems for fuzzy metric spaces with the condition of weakly compatible mappings were attempted to be proved by many authors. Tripled fixed points have emerged as a significant area of research within fixed point theory. Berinde and Borcut introduced the concept of a tripled fixed point for nonlinear mappings in partially ordered metric spaces. They also established a common fixed point theorem for contractive type mappings in M-fuzzy metric spaces. Later, other authors extended these results for common tripled fixed point theorems in fuzzy metric spaces. In this paper we introduce a new technique for proving some new common tripled fixed point theorems for Occasionally Weakly Compatible Mappings in M-fuzzy metric spaces, a method which is not previously utilized by authors in this field. Additionally, we provide illustrative example to support our findings, which represent an improvement over recent results found in the literature.

Some New Results on Partial Fuzzy Metric Spaces

In this work, we introduce a different interpretation of the notion of a partial fuzzy metric, which we refer to as a partial fuzzy co-metric. We define a partial fuzzy co-metric from a t-conorm and compare it with partial fuzzy metric, in contrast to the conventional approach to the theory of partial fuzzy metric spaces, which is based on the use of a t-norm. Here, we limit the scope of our analysis to Sedghi's definition of partial fuzzy metrics. Additionally, we proposed and compared the ideas of strong partial fuzzy co-metric spaces and strong partial fuzzy metric spaces. We also presented a few examples of these novel ideas.

English

English

A Revised Fuzzy Differential Equations Using Weakly Compatible Self-Mappings in Revised Fuzzy Metric Spaces

In this paper, we demonstrated certain coincidence point and Common Fixed Point [briefly, CFP] findings under the rational type weakly compatible revised fuzzy-contraction conditions in full Revised Fuzzy Metric spaces [briefly, RFMSs] utilizing the "triangular property of revised fuzzy metric" with examples in this work. We also provided Revised fuzzy differential equations [briefly, RFDEs] as an application and demonstrated that the solution of the RFDEs has a unique CFP of the integral operators B,C and g. This new path of weakly-compatible Revised fuzzy-contraction with the use of RFDEs in RFM space will be crucial. With different sorts of weakly-compatible Revised fuzzy contraction conditions for self-mappings and different forms of differential equations in the context of RFMSs, this approach may be expanded and developed in diverse directions.

A Revised Fuzzy Differential Equations Using Weakly Compatible Self-Mappings in Revised Fuzzy Metric Spaces

In this paper, we demonstrated certain coincidence point and Common Fixed Point [briefly, CFP] findings under the rational type weakly compatible revised fuzzy-contraction conditions in full Revised Fuzzy Metric spaces [briefly, RFMSs] utilizing the "triangular property of revised fuzzy metric" with examples in this work. We also provided Revised fuzzy differential equations [briefly, RFDEs] as an application and demonstrated that the solution of the RFDEs has a unique CFP of the integral operators and . This new path of weakly-compatible Revised fuzzy-contraction with the use of RFDEs in RFM space will be crucial. With different sorts of weakly-compatible Revised fuzzy contraction conditions for self-mappings and different forms of differential equations in the context of RFMSs, this approach may be expanded and developed in diverse directions.

Fuzzy H-Quasi-Contraction and Fixed Point Theorems in Tripled Fuzzy Metric Spaces

Fuzzy H-Quasi-Contraction and Fixed Point Theorems in Tripled Fuzzy Metric Spaces

Fixed Point Theorems in Partial Fuzzy Metric Spaces

The aim of this research article is basically to give some fixed point theorems in partial fuzzy metric space using a distance function.

Another Aspect of Generalized Fuzzy Metric Space

Another Aspect of Generalized Fuzzy Metric Space

Note on Intuitionistic Fuzzy Metric-like Spaces with Application in Image Processing

Recently, the fixed-point theorem for fuzzy contractive mappings has been investigated within the framework of intuitionistic fuzzy metric-like spaces. This interesting topic was explored through the utilization of G-Cauchy sequences as defined by Grabiec. The aim of this study is to enhance the aforementioned results in a few aspects. Initially, the proof of the fixed-point theorem is simplified and condensed, allowing for potential generalization to papers focusing on similar fixed-point analyses. Furthermore, instead of G-Cauchy sequences, the classical Cauchy sequences proposed by George and Veeramani are examined, incorporating an additional condition on the fuzzy metric. Within this context, a solution to an old unresolved question posed by Gregory and Sapena is provided. The findings are reinforced by relevant examples. Finally, the introduced fuzzy metrics are applied to the field of image processing.

Fixed Point Theorems in Revised Fuzzy Metric Space via R_F-Contraction

Fixed Point Theorems in Revised Fuzzy Metric Space via R_F-Contraction

Some Novel Coincidence and Fixed Point Results Based on Z Family of Functions in Fuzzy Metrics

Some Novel Coincidence and Fixed Point Results Based on Z Family of Functions in Fuzzy Metrics

Integrating fuzzy metrics and negation operator in FCM algorithm via genetic algorithm for MRI image segmentation

In this study, we redefine FCM algorithm by integrating fuzzy set theory, fuzzy metrics, and Sugeno negation principles. This innovative approach overcomes the limitations inherent in conventional machine learning models, especially in situations characterized by uncertainty, noise, and ambiguity. Our model utilizes the membership degrees from fuzzy set theory, and transforms the concept of proximity defined by fuzzy metrics into a minimization problem. This transformation is achieved using a linguistic negation operator, which is crucial for optimizing FCM algorithm's objective function. A significant innovation in our research is the use of GA for optimizing parameters within the contexts of fuzzy metrics and Sugeno negation. The precise optimization capabilities of GA greatly enhance the sensitivity and adaptability of FCM algorithm, thereby improving overall performance. By leveraging the meticulous parameter adjustments provided by GA, our approach has shown superior results in practical applications, such as brain MRI image segmentation, surpassing traditional methods. Experimental results highlight the considerable enhancements our proposed FCM algorithms bring over existing methods across various performance metrics. In conclusion, this study makes a valuable addition to the field of fuzzy-based machine learning methodologies. It combines the optimization strength of GA with the flexible classification capabilities of fuzzy logic. The integration of Sugeno negation and fuzzy metrics not only improves the accuracy and precision of FCM algorithm but also provides significant benefits in handling complex and ambiguous datasets. This research signifies a major advance in machine learning and fuzzy logic, setting the stage for future applications and studies.

Exploring Fixed Points and Common Fixed Points of Contractive Mappings in Complex-Valued Intuitionistic Fuzzy Metric Spaces

The current manuscript aims to introduce complex-valued intuitionisitic fuzzy metric spaces as a fresh perspective on complex-valued fuzzy metric spaces and intuitionistic fuzzy metric spaces. Existence together with distinctiveness of the fixed points within maps with diverse contractive criteria in this novel space are established. Additionally, our work yields a few common fixed-point findings for intuitionistic fuzzy Banach contraction on this newly introduced space. The outcomes presented in this study go beyond the existing literature, adding to the growing body of knowledge in this field. Our research outcomes are exemplified through examples that are included in this paper to help readers better grasp our findings. Our paper concludes with a discussion of how our findings can be applied to the problem of determining the presence of an exclusive solution for Fredholm integral equations.

On Neutrosophic Fuzzy Metric Space and Its Topological Properties

The present research introduces a novel concept termed “neutrosophic fuzzy metric space”, which extends the traditional metric space framework by incorporating the notion of neutrosophic fuzzy sets. A thorough investigation of various structural and topological properties within this newly proposed generalization of metric space has been conducted. Additionally, counterparts of well-known theorems such as the Uniform Convergence Theorem and the Baire Category Theorem have been established for this generalized metric space. Through rigorous analysis, a detailed understanding of its fundamental characteristics has been attained, illuminating its potential applications and theoretical significance.

Type-2 Fuzzy Metric Spaces

In this paper, an attempt is made to define a Type-2 fuzzy metric on a nonempty set [Formula: see text] by allowing it to take fuzzy numbers as values of distance of a pair of points under a membership grade which is also a fuzzy number. Its type-1 counterpart is a fuzzy metric mostly similar to those defined by Kramosil and Michalek[Formula: see text] and also by George and Veeramani.6 It is shown that the topology induced by this fuzzy metric is Hausdorff in nature. In this type of fuzzy metric space, Banach’s fixed point theorem and Edelstein’s fixed point theorems are extended. Finally decomposition theorems for this fuzzy metric are proved from which a justification of type-2 behavior of this fuzzy metric can also be interpreted.

Vector-Valued Fuzzy Metric Spaces and Fixed Point Theorems

The purpose of this paper is to generalize the concept of classical fuzzy set to vector-valued fuzzy set which can attend values not only in the real interval [0, 1], but in an ordered interval of a Banach algebra as well. This notion allows us to introduce the concept of vector-valued fuzzy metric space which generalizes, extends and unifies the notion of classical fuzzy metric space and complex-valued fuzzy metric space and permits us to consider the fuzzy sets and metrics in a larger domain. Some topological properties of such spaces are discussed and some fixed point results in this new setting are proved. Multifarious examples are presented which clarify and justify our claims and results.

APPLICATION OF FUZZY METRICS IN CLUSTERING PROBLEMS OF AGRICULTURAL CROP VARIETIES

The problem of image-based detection of the variety of beans, using artificial intelligence, is currently dealt with by scientists of various profiles. The idea of this paper is to show the possibility of applying different types of distances, primarily those that are fuzzy metrics, in clustering models in order to improve existing models and obtain more accurate results. The paper presents the method of variable neighborhood search, which uses both standard and fuzzy t-metrics and dual fuzzy s-metrics characterized by appropriate parameters. By varying those parameters of the fuzzy metric as well as the parameters of the metaheuristic used, we have shown how it is possible to improve the clustering results. The obtained results were compared with existing ones from the literature. The criterion function used in clustering is a fuzzy metric, which is proven in the paper.

Fuzzy Metric Spaces: Optimizing Coincidence and Proximity Points

Our manuscript puts forward two novel fuzzy proximal contractive conditions. First, we present two variants of fuzzy α-proximal quasi-H-contractions and establish optimal coincidence point outcomes for these contractions in fuzzy metric space. This manuscript’s second part proposes the fuzzy ψ-contraction for a multivalued mapping equipped with fuzzy weak P-property and achieves the best proximity point outcome in strong fuzzy metric space. The findings of this study broaden and generalize some existing research results.

On the Completeness and Compactness with Eldestein Fixed Point Theorem in Standard Intuitionistic Fuzzy Metric Spaces

In recent years, Kider and Hussain introduced the concept of standard fuzzy metric space and proved some of its key properties. In this paper, we extend this notion to the intuitionistic fuzzy setting. Furthermore, a number of properties related to this notion are studied such as compactness and completeness properties defined in standard intuitionistic fuzzy metric space. Moreover, the Edelstein contraction theorem on this space is established.