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2018 | Buch

Preliminaries Kapitel lesen Erstes Kapitel lesen

Fuzzy Operator Theory in Mathematical Analysis

verfasst von: Yeol Je Cho, Prof. Themistocles M. Rassias, Reza Saadati

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This self-contained monograph presents an overview of fuzzy operator theory in mathematical analysis. Concepts, principles, methods, techniques, and applications of fuzzy operator theory are unified in this book to provide an introduction to graduate students and researchers in mathematics, applied sciences, physics, engineering, optimization, and operations research. New approaches to fuzzy operator theory and fixed point theory with applications to fuzzy metric spaces, fuzzy normed spaces, partially ordered fuzzy metric spaces, fuzzy normed algebras, and non-Archimedean fuzzy metric spaces are presented.

Surveys are provided on: Basic theory of fuzzy metric and normed spaces and its topology, fuzzy normed and Banach spaces, linear operators, fundamental theorems (open mapping and closed graph), applications of contractions and fixed point theory, approximation theory and best proximity theory, fuzzy metric type space, topology and applications.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Preliminaries
Abstract
In this chapter, we recall some definitions and results as triangular norms (co-norm), fuzzy sets, and lattices which will be used later on in this book.
Yeol Je Cho, Themistocles M. Rassias, Reza Saadati
Chapter 2. Fuzzy Normed Spaces and Fuzzy Metric Spaces
Abstract
In this chapter, we define fuzzy normed spaces and show that every fuzzy normed space induces a fuzzy metric space. Then we consider the topology induced by fuzzy normed (metric) spaces and show some important topological properties of them. Next, we study fuzzy inner product spaces and some properties of these spaces.
Yeol Je Cho, Themistocles M. Rassias, Reza Saadati
Chapter 3. Further Properties of Fuzzy Banach Spaces
Abstract
In this chapter, we consider some important properties of fuzzy Banach spaces. In Section 3.1, we discuss about finite dimensional fuzzy Banach spaces and prove some important theorems on linearly independent set. Next, we prove, in a finite dimensional vector space X, every two fuzzy norms are equivalent. Finally, we study some bounded and continuous linear operators in fuzzy normed spaces.
Yeol Je Cho, Themistocles M. Rassias, Reza Saadati
Chapter 4. Fundamental Theorems in Fuzzy Normed Spaces
Abstract
Some important theorems in this chapter are the open mapping theorem and the closed graph theorem. These are the cornerstones of the theory of fuzzy Banach spaces. Open mapping theorem states that a fuzzy bounded linear operator T from a fuzzy Banach space onto a fuzzy Banach space is an open mapping, that is, maps open sets onto open sets. Closed graph theorem gives conditions under which a closed linear operator is fuzzy bounded. Closed linear operators are of importance in physical and other applications.
Yeol Je Cho, Themistocles M. Rassias, Reza Saadati
Chapter 5. Fixed Point Theorems in Fuzzy Metric Spaces
Abstract
In this chapter, we study the fixed point theory in fuzzy metric spaces. This subject is very important in fuzzy nonlinear operator theory. In Section 5.1, we define weak compatible mappings in fuzzy metric spaces and prove some common fixed point theorems for four mappings satisfying some contractions. In Section 5.2, we define R-weakly commuting mappings in intuitionistic fuzzy metric spaces and prove some common fixed point theorems in these spaces. In Section 5.3, we prove some common fixed point theorems for six mappings in three complete fuzzy metric spaces. In Section 5.4, we consider \(\mathcal {L}\)-fuzzy metric spaces and prove a famous theorem, i.e., Jungck’s Theorem in these spaces. In Section 5.5, we study hyper \(\mathcal {L}\)-fuzzy metric spaces and prove some important fixed point theorems in these spaces. Finally, in Section 5.6, we consider the concept of intuitionistic fuzzy quasi-metric spaces and prove a fixed point theorem to obtain the existence of a solution for a recurrence equation associated with the analysis of Quicksort algorithms.
Yeol Je Cho, Themistocles M. Rassias, Reza Saadati
Chapter 6. Generalized Distances and Fixed Point Theorems in Fuzzy Metric Spaces
Abstract
In this chapter, we study the generalized distance in fuzzy metric spaces and prove some fixed point theorems for some contractive mappings in fuzzy metric spaces by using the generalized distances.
Yeol Je Cho, Themistocles M. Rassias, Reza Saadati
Chapter 7. Fixed Point Theorems in Partially Ordered Fuzzy Metric Spaces
Abstract
In this chapter, we study some fixed point theorems for nonlinear mappings satisfying some contractions in fuzzy partially metric spaces.
Yeol Je Cho, Themistocles M. Rassias, Reza Saadati
Chapter 8. Fixed Point Theorems in Fuzzy Normed Spaces
Abstract
In this chapter, we consider some fixed point theorems for nonlinear mappings with some properties in fuzzy normed spaces and fuzzy inner product spaces.
Yeol Je Cho, Themistocles M. Rassias, Reza Saadati
Chapter 9. Approximation Theory in Fuzzy Metric Spaces
Abstract
In this chapter, we study some topics on approximation theory in fuzzy metric spaces.
Yeol Je Cho, Themistocles M. Rassias, Reza Saadati
Chapter 10. Topologies and Fixed Points in Fuzzy Metric-Type Spaces
Abstract
In this chapter, first, we introduce some extensions of metric spaces. Next, we introduce the concept of fuzzy metric-type spaces and consider the topology induced by the fuzzy metric type. Finally, we consider some fixed point theorems for some nonlinear mapping satisfying some conditions in complete fuzzy metric-type spaces.
Yeol Je Cho, Themistocles M. Rassias, Reza Saadati
Chapter 11. Operator Theory and Fixed Points in Fuzzy Normed Algebras and Applications
Abstract
In this chapter, first, we consider the concept of fuzzy Banach algebras and fuzzy compact operators in fuzzy normed spaces. Then we apply some fixed point theorems to solve the operator equation AxBx = x in fuzzy Banach algebras under some nonlinear contraction.
Yeol Je Cho, Themistocles M. Rassias, Reza Saadati
Chapter 12. Fixed Points in Non-Archimedean Fuzzy Metric Spaces
Abstract
Recently, Miheţ enlarged the class of fuzzy contractive mappings of Gregori and Sapena and proved a fuzzy Banach contraction result in complete non-Archimedean fuzzy metric spaces.
Yeol Je Cho, Themistocles M. Rassias, Reza Saadati
Chapter 13. Coincidence Points for Set-Valued Mappings in Fuzzy Metric Spaces
Abstract
In this chapter, we prove some coincidence theorem for set-valued mappings in fuzzy metric spaces with a view to generalizing Downing-Kirk’s fixed point theorem in metric spaces.
Yeol Je Cho, Themistocles M. Rassias, Reza Saadati
Backmatter
Metadaten
Titel
Fuzzy Operator Theory in Mathematical Analysis
verfasst von
Yeol Je Cho
Prof. Themistocles M. Rassias
Reza Saadati
Copyright-Jahr
2018
Electronic ISBN
978-3-319-93501-0
Print ISBN
978-3-319-93499-0
DOI
https://doi.org/10.1007/978-3-319-93501-0