Fuzzy b-Metric Spaces

Metric spaces and their various generalizations occur frequently in computer science applications. This is the reason why, in this paper, we introduced and studied the concept of fuzzy b-metric space, generalizing, in this way, both the notion of fuzzy metric space introduced by I. Kramosil and J. Michálek and the concept of b-metric space. On the other hand, we introduced the concept of fuzzy quasi-bmetric space, extending the notion of fuzzy quasi metric space recently introduced by V. Gregori and S. Romaguera. Finally, a decomposition theorem for a fuzzy quasipseudo- b-metric into an ascending family of quasi-pseudo-b-metrics is established. The use of fuzzy b-metric spaces and fuzzy quasi-b-metric spaces in the study of denotational semantics and their applications in control theory will be an important next step

Fuzzy Euclidean Normed Spaces for Data Mining Applications

The aim of this paper is to introduce some special fuzzy norms on Kn and to obtain, in this way, fuzzy Euclidean normed spaces. In order to introduce this concept we have proved that the cartesian product of a finite family of fuzzy normed linear spaces is a fuzzy normed linear space. Thus any fuzzy norm on K generates a fuzzy norm on Kn. Finally, we prove that each fuzzy Euclidean normed space is complete. Fuzzy Euclidean normed spaces can be proven to be a suitable tool for data mining. The method is based on embedding the data in fuzzy Euclidean normed spaces and to carry out data analysis in these spaces

Fuzzy Logic and Soft Computing—Dedicated to the Centenary of the Birth of Lotfi A. Zadeh (1921–2017)

In 1965, Lotfi A. Zadeh published “Fuzzy Sets”, his pioneering and controversialpaper, which has now reached over 115,000 citations [...

Fixed Point Theory in Fuzzy Normed Linear Spaces: A General View

In this paper we have presented, firstly, an evolution of the concept of fuzzy normed linear spaces, different definitions, approaches as well as generalizations. A special section is dedicated to fuzzy Banach spaces. In the case of fuzzy normed linear spaces, researchers have been working, until now, with a definition of completeness inspired by M. Grabiec’s work in the context of fuzzy metric spaces. We propose another definition and we prove that it is much more adequate, inspired by the work of A.George and P. Veeramani. Finally, some important results in fuzzy fixed point theory were highlighted

Reinstatement of the Extension Principle in Approaching Mathematical Programming with Fuzzy Numbers

Optimization problems in the fuzzy environment are widely studied in the literature. We restrict our attention to mathematical programming problems with coefficients and/or decision variables expressed by fuzzy numbers. Since the review of the recent literature on mathematical programming in the fuzzy environment shows that the extension principle is widely present through the fuzzy arithmetic but much less involved in the foundations of the solution concepts, we believe that efforts to rehabilitate the idea of following the extension principle when deriving relevant fuzzy descriptions to optimal solutions are highly needed. This paper identifies the current position and role of the extension principle in solving mathematical programming problems that involve fuzzy numbers in their models, highlighting the indispensability of the extension principle in approaching this class of problems. After presenting the basic ideas in fuzzy optimization, underlying the advantages and disadvantages of different solution approaches, we review the main methodologies yielding solutions that elude the extension principle, and then compare them to those that follow it. We also suggest research directions focusing on using the extension principle in all stages of the optimization process

A Study of Boundedness in Fuzzy Normed Linear Spaces

In the present paper some different types of boundedness in fuzzy normed linear spaces of type ( X , N , ∗ ) , where ∗ is an arbitrary t-norm, are considered. These boundedness concepts are very general and some of them have no correspondent in the classical topological metrizable linear spaces. Properties of such bounded sets are given and we make a comparative study among these types of boundedness. Among them there are various concepts concerning symmetrical properties of the studied objects arisen from the classical setting appropriate for this journal topics. We establish the implications between them and illustrate by examples that these concepts are not similar

Fixed-Point Theorems in Fuzzy Normed Linear Spaces for Contractive Mappings with Applications to Dynamic-Programming

The aim of this paper is to provide new ways of dealing with dynamic programming using a context of newly proven results about fixed-point problems in linear spaces endowed with a fuzzy norm. In our paper, the general framework is set to fuzzy normed linear spaces as they are defined by Nădăban and Dzitac. When completeness is required, we will use the George and Veeramani (G-V) setup, which, for our purposes, we consider to be more suitable than Grabiec-completeness. As an important result of our work, we give an original proof for a version of Banach’s fixed-point principle on this particular setup of fuzzy normed spaces, a variant of Jungck’s fixed-point theorem in the same setup, and they are proved in G-V-complete fuzzy normed spaces, paving the way for future developments in various fields within this framework, where our application of dynamic programming makes a proper example. As the uniqueness of almost every dynamic programming problem is necessary, the fixed-point theorems represent an important tool in achieving that goal

Fixed-Point Theorems in Fuzzy Normed Linear Spaces for Contractive Mappings with Applications to Dynamic-Programming

The aim of this paper is to provide new ways of dealing with dynamic programming using a context of newly proven results about fixed-point problems in linear spaces endowed with a fuzzy norm. In our paper, the general framework is set to fuzzy normed linear spaces as they are defined by Nădăban and Dzitac. When completeness is required, we will use the George and Veeramani (G-V) setup, which, for our purposes, we consider to be more suitable than Grabiec-completeness. As an important result of our work, we give an original proof for a version of Banach’s fixed-point principle on this particular setup of fuzzy normed spaces, a variant of Jungck’s fixed-point theorem in the same setup, and they are proved in G-V-complete fuzzy normed spaces, paving the way for future developments in various fields within this framework, where our application of dynamic programming makes a proper example. As the uniqueness of almost every dynamic programming problem is necessary, the fixed-point theorems represent an important tool in achieving that goal