Chaos for Linear Operators and Abstract Differential Equations

$230.00

Marko Kostić
Faculty of Technical Sciences, Novi Sad, Serbia

Series: Mathematics Research Developments
BISAC: MAT027000

The theory of linear topological dynamics is a rapidly growing field of research over the last three decades or so. This book presents a survey of recent results of the author obtained in this field during the period 2016-2019.

Without any doubt, this is the first research monograph concerning the topological dynamics of multivalued operators and binary relations, especially, multivalued linear woperators, simple graphs, digraphs and tournaments (we feel duty bound to say that multivalued topological dynamics is still a very undeveloped field of investigation, full of open problems and possible for further expansion). Asiede from that, the main purpose of this monograph is to consider topologically dynamical properties of linear single-valued operators in Frechet spaces and abstract fractional differential equations in Frechet spaces, which could be degenerate or non-degenerate in time variable. In this monograph, we use only two types of fractional derivatives, namely the Caputo time-fractional derivatives and Weyl time-fractional derivatives. However, most results on dynamics of differential equations are given to the abstract differential equations with integer order derivatives, especially those of first and second order in time.

The monograph is consistsed of two chapters; the first chapter is further broken down into nine sections, while the second chapter is broken down into seven sections. It is not of introductory character to linear topological dynamics and it is not written in a traditional manner. As in all my previously published monographs, the numbering of definitions, theorems, propositions, remarks, lemmas, corollaries, definitions, etc., are by chapter and section; the bibliography is by author in alphabetic order.

Concerning target audience, wWe deeply believe that the book could be of invaluable help to experts in linear topological dynamics, researchers in abstract partial differential equations but and also to PhD students and advanced graduate students in mathematics as well. A potential reader should be familiar with backgrounds including elementary functional analysis, measure and integration theory as well as the basic theory of abstract (degenerate) Volterra integro-differential equations. At some places, the knowledge of graph theory is preferable but not demandedable. This monograph is not intended to be a comprehensive review of current trends; albeit includes several recent results from the field of linear topological dynamics and has more than 450 titles, our reference list is far from being exhaustively complete.
(Imprint: Nova)

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Table of Contents

Preface

Introduction

Chapter 1. Dynamical Properties of Linear Operators and Abstract Differential Equations

Chapter 2. Disjoint Dynamical Properties of Linear Operators and Abstract Differential Equations

Bibliography


Reviews

“Linear dynamics is a trending topic of intensive research in the last years. It is quite common that new contributions in this area require a combination of tools from functional analysis, operator theory, topology, geometry, and number theory. In recent years, there is a growing interest to provide more accurate descriptions of how reiterative the orbits visit different parts of the domain space. This excellent book contributes to the development of this part of the theory, providing recent advances of the author in this line. The reader will also find the extension of linear dynamics of operators to multivalued linear operators (MLO’s) and what the impact of the results is for the case of fractional differential operators.”  Jose Alberto Conejero, University Institute of Pure and Applied Mathematics, Polytechnic University of Valencia, Valencia, Spain

“This book is a helpful and readable guide full of important results and examples for those interested readers in linear topological dynamics in connection with chaos and abstract differential equations.”  Marina Murillo-Arcila, Department of Mathematics, Universitat Jaume I, Campus de Ríu Sec, Castelló de la Plana, Spain

“In this interesting monograph, dynamics and disjoint dynamics for linear operators and abstract differential equations are investigated systemically. It does not cover only the preliminary and background of linear dynamics, but also the newest results in this theme. Therefore, this book is excellent for researchers and graduate students who wish to get in or keep up with this active field.”  Chung-Chuan Chen, Department of Mathematics Education, National Taichung University of Education, Taichung, Taiwan

“The author continues with his research analysis of integro-differential equations in abstract environment with a highly sophisticated technic. This monograph is a first one related to the topological dynamics of multivalued operators and binary relations, especially, multivalued linear operators, simple graphs, digraphs and tournaments. Based on the topologically dynamical properties of linear single-valued operators in Frechet spaces and abstract multi-term fractional differential equations in Frechet spaces, the book goes ahead with up-to-date results in the field. The author shoes brilliant knowledge of the existing literature. His monograph provides a large number of data about what has been done and what can be done in the field. It is very useful for experts in linear topological dynamics and in abstract partial differential equations. Moreover, it is an excellent literature for the PhD students.”  StevanPilipović, Faculty of Sciences, Department of Mathematics and Informatics, University of Novi Sad, Serbia

“An interesting approximation to the study of different types of hypercyclicity and chaos with applications to abstract differential equations.”  Antonio Bonilla Ramírez, Department of Mathematical Analysis, Universidad de la Laguna, La Laguna (Tenerife), Spain


Audience

Concerning target audience, we deeply believe that the book could be of invaluable help to experts in linear topological dynamics, researchers in abstract partial differential equations but also to PhD students and advanced graduate students in mathematics.


Keywords

Linear topological dynamics, chaos, abstract differential equations, abstract fractional differential equations, graph theory

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