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## Über dieses Buch

This book may be used as reference for graduate students interested in fuzzy differential equations and researchers working in fuzzy sets and systems, dynamical systems, uncertainty analysis, and applications of uncertain dynamical systems. Beginning with a historical overview and introduction to fundamental notions of fuzzy sets, including different possibilities of fuzzy differentiation and metric spaces, this book moves on to an overview of fuzzy calculus thorough exposition and comparison of different approaches. Innovative theories of fuzzy calculus and fuzzy differential equations using fuzzy bunches of functions are introduced and explored. Launching with a brief review of essential theories, this book investigates both well-known and novel approaches in this field; such as the Hukuhara differentiability and its generalizations as well as differential inclusions and Zadeh’s extension. Through a unique analysis, results of all these theories are examined and compared.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
This chapter provides a brief overview on the most known approaches for fuzzy initial value problems, namely via Hukuhara and strongly generalized derivatives, Zadeh’s extension of the classical (or crisp) solution and fuzzy differential inclusions. It also situates among them the recent theory of fuzzy differential equations via extension of the derivative operator. It points out important conceptual differences among all the mentioned theories and provides an historical overview on the subject.
Luciana Takata Gomes, Laécio Carvalho de Barros, Barnabas Bede

### Chapter 2. Basic Concepts

Abstract
Basic definitions such as fuzzy subsets, fuzzy numbers, Zadeh’s extension and operations of fuzzy numbers are presented. This chapter also introduces arithmetic for fuzzy numbers and the various definitions for difference of fuzzy numbers. The last section of this chapter is devoted to the understanding of two kinds of fuzzy functions (fuzzy-set-valued functions and fuzzy bunches of functions) since it is necessary to the development of the fuzzy calculus and the different approaches of fuzzy differential equations.
Luciana Takata Gomes, Laécio Carvalho de Barros, Barnabas Bede

### Chapter 3. Fuzzy Calculus

Abstract
This chapter develops fuzzy calculus under two different perspectives. One uses fuzzy-set-valued functions and the other one is for fuzzy bunches of functions. Concepts such as Hukuhara derivative and its generalizations, fuzzy Aumann, Henstock and Riemann integrals and derivative and integral via Zadeh’s extension are introduced, explored and compared.
Luciana Takata Gomes, Laécio Carvalho de Barros, Barnabas Bede

### Chapter 4. Fuzzy Differential Equations

Abstract
The following approaches of fuzzy differential equations are depicted in this chapter: via Hukuhara and strongly generalized derivatives, Zadeh’s extension of the classical (or crisp) solution, fuzzy differential inclusions and extension of the derivative operator. Theorems assuring existence of solutions to fuzzy initial value problems are provided to all theories. Comparisons are carried out and conditions assure equivalence of results under different approaches.
Luciana Takata Gomes, Laécio Carvalho de Barros, Barnabas Bede

### Backmatter

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