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Differential Equations are very important tools in Mathematical
Analysis. They are widely found in mathematics itself and in its
applications to statistics, computing, electrical circuit analysis,
dynamical systems, economics, biology, and so on. Recently there
has been an increasing interest in and widely-extended use of
differential equations and systems of fractional order (that is, of
arbitrary order) as better models of phenomena in various physics,
engineering, automatization, biology and biomedicine, chemistry,
earth science, economics, nature, and so on. Now, new unified
presentation and extensive development of special functions
associated with fractional calculus are necessary tools, being
related to the theory of differentiation and integration of
arbitrary order (i.e., fractional calculus) and to the fractional
order (or multi-order) differential and integral equations. This
book provides learners with the opportunity to develop an
understanding of advancements of special functions and the skills
needed to apply advanced mathematical techniques to solve complex
differential equations and Partial Differential Equations (PDEs).
Subject matters should be strongly related to special functions
involving mathematical analysis and its numerous applications. The
main objective of this book is to highlight the importance of
fundamental results and techniques of the theory of complex
analysis for differential equations and PDEs and emphasizes
articles devoted to the mathematical treatment of questions arising
in physics, chemistry, biology, and engineering, particularly those
that stress analytical aspects and novel problems and their
solutions. Specific topics include but are not limited to Partial
differential equations Least squares on first-order system Sequence
and series in functional analysis Special functions related to
fractional (non-integer) order control systems and equations
Various special functions related to generalized fractional
calculus Operational method in fractional calculus Functional
analysis and operator theory Mathematical physics Applications of
numerical analysis and applied mathematics Computational
mathematics Mathematical modeling This book provides the recent
developments in special functions and differential equations and
publishes high-quality, peer-reviewed book chapters in the area of
nonlinear analysis, ordinary differential equations, partial
differential equations, and related applications.
Originally published in 1987. This study evaluates micro- and
macroeconomic determinants for the export performance of European
suppliers to the markets of the Association of Southeast Asian
Nations (ASEAN). By comparing marketing strategies and the
respective economic environment of suppliers from ElH"ope , Japan ,
and the U S - the major exporter
A study of difference equations and inequalities. This second
edition offers real-world examples and uses of difference equations
in probability theory, queuing and statistical problems, stochastic
time series, combinatorial analysis, number theory, geometry,
electrical networks, quanta in radiation, genetics, economics,
psychology, sociology, and other disciplines. It features 200 new
problems, 400 additional references, and a new chapter on the
qualitative properties of solutions of neutral difference
equations.
Interest in the mathematical analysis of multi-functions has
increased rapidly over the past thirty years, partly because of its
applications in fields such as biology, control theory and
optimization, economics, game theory, and physics. Set Valued
Mappings with Applications to Nonlinear Analysis contains 29
research articles from leading mathematicians in this area. The
contributors were invited to submit papers on topics such as
integral inclusion, ordinary and partial differential inclusions,
fixed point theorems, boundary value problems, and optimal control.
This collection will be of interest to researchers in analysis and
will pave the way for the creation of new mathematics in the
future.
This collection of 24 papers, which encompasses the construction
and the qualitative as well as quantitative properties of solutions
of Volterra, Fredholm, delay, impulse integral and
integro-differential equations in various spaces on bounded as well
as unbounded intervals, will conduce and spur further research in
this direction.
This book summarizes the qualitative theory of differential
equations with or without delays, collecting recent oscillation
studies important to applications and further developments in
mathematics, physics, engineering, and biology. The authors address
oscillatory and nonoscillatory properties of first-order delay and
neutral delay differential equations, second-order delay and
ordinary differential equations, higher-order delay differential
equations, and systems of nonlinear differential equations. The
final chapter explores key aspects of the oscillation of dynamic
equations on time scales-a new and innovative theory that
accomodates differential and difference equations simultaneously.
This volume encompasses the mathematical analysis of multifunctions and contains twenty-nine research articles from leading mathematicians in this area. Interest in the mathematical analysis of multifunctions has increased rapidly over the past thirty years. This is partly due to the rich and plentiful supply of applications in diverse fields such as biology, control theory and optimization, economics, game theory and physics. The papers within this book were invited and, among others, include topics such as integral inclusion, ordinary and partial differential inclusions, fixed point theorems, boundary value problems, and optimal control. This collection of papers will be of mnterest to researchers and will pave the way for the creation of new mathematics in the future. eBook available with sample pages: 0203216490
A study of difference equations and inequalities. This second
edition offers real-world examples and uses of difference equations
in probability theory, queuing and statistical problems, stochastic
time series, combinatorial analysis, number theory, geometry,
electrical networks, quanta in radiation, genetics, economics,
psychology, sociology, and other disciplines. It features 200 new
problems, 400 additional references, and a new chapter on the
qualitative properties of solutions of neutral difference
equations.
This monograph establishes a theory of classification and
translation closedness of time scales, a topic that was first
studied by S. Hilger in 1988 to unify continuous and discrete
analysis. The authors develop a theory of translation function on
time scales that contains (piecewise) almost periodic functions,
(piecewise) almost automorphic functions and their related
generalization functions (e.g., pseudo almost periodic functions,
weighted pseudo almost automorphic functions, and more). Against
the background of dynamic equations, these function theories on
time scales are applied to study the dynamical behavior of
solutions for various types of dynamic equations on hybrid domains,
including evolution equations, discontinuous equations and
impulsive integro-differential equations. The theory presented
allows many useful applications, such as in the Nicholson`s
blowfiles model; the Lasota-Wazewska model; the Keynesian-Cross
model; in those realistic dynamical models with a more complex
hibrid domain, considered under different types of translation
closedness of time scales; and in dynamic equations on mathematical
models which cover neural networks. This book provides readers with
the theoretical background necessary for accurate mathematical
modeling in physics, chemical technology, population dynamics,
biotechnology and economics, neural networks, and social sciences.
This book provides an extensive survey on Lyapunov-type
inequalities. It summarizes and puts order into a vast literature
available on the subject, and sketches recent developments in this
topic. In an elegant and didactic way, this work presents the
concepts underlying Lyapunov-type inequalities, covering how they
developed and what kind of problems they address. This survey
starts by introducing basic applications of Lyapunov's
inequalities. It then advances towards even-order, odd-order, and
higher-order boundary value problems; Lyapunov and Hartman-type
inequalities; systems of linear, nonlinear, and quasi-linear
differential equations; recent developments in Lyapunov-type
inequalities; partial differential equations; linear difference
equations; and Lyapunov-type inequalities for linear, half-linear,
and nonlinear dynamic equations on time scales, as well as linear
Hamiltonian dynamic systems. Senior undergraduate students and
graduate students of mathematics, engineering, and science will
benefit most from this book, as well as researchers in the areas of
ordinary differential equations, partial differential equations,
difference equations, and dynamic equations. Some background in
calculus, ordinary and partial differential equations, and
difference equations is recommended for full enjoyment of the
content.
This monograph establishes a theory of classification and
translation closedness of time scales, a topic that was first
studied by S. Hilger in 1988 to unify continuous and discrete
analysis. The authors develop a theory of translation function on
time scales that contains (piecewise) almost periodic functions,
(piecewise) almost automorphic functions and their related
generalization functions (e.g., pseudo almost periodic functions,
weighted pseudo almost automorphic functions, and more). Against
the background of dynamic equations, these function theories on
time scales are applied to study the dynamical behavior of
solutions for various types of dynamic equations on hybrid domains,
including evolution equations, discontinuous equations and
impulsive integro-differential equations. The theory presented
allows many useful applications, such as in the Nicholson`s
blowfiles model; the Lasota-Wazewska model; the Keynesian-Cross
model; in those realistic dynamical models with a more complex
hibrid domain, considered under different types of translation
closedness of time scales; and in dynamic equations on mathematical
models which cover neural networks. This book provides readers with
the theoretical background necessary for accurate mathematical
modeling in physics, chemical technology, population dynamics,
biotechnology and economics, neural networks, and social sciences.
The book is devoted to dynamic inequalities of Hardy type and
extensions and generalizations via convexity on a time scale T. In
particular, the book contains the time scale versions of classical
Hardy type inequalities, Hardy and Littlewood type inequalities,
Hardy-Knopp type inequalities via convexity, Copson type
inequalities, Copson-Beesack type inequalities, Liendeler type
inequalities, Levinson type inequalities and Pachpatte type
inequalities, Bennett type inequalities, Chan type inequalities,
and Hardy type inequalities with two different weight functions.
These dynamic inequalities contain the classical continuous and
discrete inequalities as special cases when T = R and T = N and can
be extended to different types of inequalities on different time
scales such as T = hN, h > 0, T = qN for q > 1, etc.In this
book the authors followed the history and development of these
inequalities. Each section in self-contained and one can see the
relationship between the time scale versions of the inequalities
and the classical ones. To the best of the authors' knowledge this
is the first book devoted to Hardy-typeinequalities and their
extensions on time scales.
This book is based on lectures from a one-year course at the Far
Eastern Federal University (Vladivostok, Russia) as well as on
workshops on optimal control offered to students at various
mathematical departments at the university level. The main themes
of the theory of linear and nonlinear systems are considered,
including the basic problem of establishing the necessary and
sufficient conditions of optimal processes. In the first part of
the course, the theory of linear control systems is constructed on
the basis of the separation theorem and the concept of a
reachability set. The authors prove the closure of a reachability
set in the class of piecewise continuous controls, and the problems
of controllability, observability, identification, performance and
terminal control are also considered. The second part of the course
is devoted to nonlinear control systems. Using the method of
variations and the Lagrange multipliers rule of nonlinear problems,
the authors prove the Pontryagin maximum principle for problems
with mobile ends of trajectories. Further exercises and a large
number of additional tasks are provided for use as practical
training in order for the reader to consolidate the theoretical
material.
Written by a team of leading experts in the field, this volume
presents a self-contained account of the theory, techniques and
results in metric type spaces (in particular in G-metric spaces);
that is, the text approaches this important area of fixed point
analysis beginning from the basic ideas of metric space topology.
The text is structured so that it leads the reader from
preliminaries and historical notes on metric spaces (in particular
G-metric spaces) and on mappings, to Banach type contraction
theorems in metric type spaces, fixed point theory in partially
ordered G-metric spaces, fixed point theory for expansive mappings
in metric type spaces, generalizations, present results and
techniques in a very general abstract setting and framework. Fixed
point theory is one of the major research areas in nonlinear
analysis. This is partly due to the fact that in many real world
problems fixed point theory is the basic mathematical tool used to
establish the existence of solutions to problems which arise
naturally in applications. As a result, fixed point theory is an
important area of study in pure and applied mathematics and it is a
flourishing area of research.
The book is devoted to dynamic inequalities of Hardy type and
extensions and generalizations via convexity on a time scale T. In
particular, the book contains the time scale versions of classical
Hardy type inequalities, Hardy and Littlewood type inequalities,
Hardy-Knopp type inequalities via convexity, Copson type
inequalities, Copson-Beesack type inequalities, Liendeler type
inequalities, Levinson type inequalities and Pachpatte type
inequalities, Bennett type inequalities, Chan type inequalities,
and Hardy type inequalities with two different weight functions.
These dynamic inequalities contain the classical continuous and
discrete inequalities as special cases when T = R and T = N and can
be extended to different types of inequalities on different time
scales such as T = hN, h > 0, T = qN for q > 1, etc.In this
book the authors followed the history and development of these
inequalities. Each section in self-contained and one can see the
relationship between the time scale versions of the inequalities
and the classical ones. To the best of the authors' knowledge this
is the first book devoted to Hardy-typeinequalities and their
extensions on time scales.
Environmental variation plays an important role in many biological
and ecological dynamical systems. This monograph focuses on the
study of oscillation and the stability of delay models occurring in
biology. The book presents recent research results on the
qualitative behavior of mathematical models under different
physical and environmental conditions, covering dynamics including
the distribution and consumption of food. Researchers in the fields
of mathematical modeling, mathematical biology, and population
dynamics will be particularly interested in this material.
This work introduces readers to the topic of maximal regularity for
difference equations. The authors systematically present the method
of maximal regularity, outlining basic linear difference equations
along with relevant results. They address recent advances in the
field, as well as basic semi group and cosine operator theories in
the discrete setting. The authors also identify some open problems
that readers may wish to take up for further research. This book is
intended for graduate students and researchers in the area of
difference equations, particularly those with advance knowledge of
and interest in functional analysis.
Written by a team of leading experts in the field, this volume
presents a self-contained account of the theory, techniques and
results in metric type spaces (in particular in G-metric spaces);
that is, the text approaches this important area of fixed point
analysis beginning from the basic ideas of metric space topology.
The text is structured so that it leads the reader from
preliminaries and historical notes on metric spaces (in particular
G-metric spaces) and on mappings, to Banach type contraction
theorems in metric type spaces, fixed point theory in partially
ordered G-metric spaces, fixed point theory for expansive mappings
in metric type spaces, generalizations, present results and
techniques in a very general abstract setting and framework. Fixed
point theory is one of the major research areas in nonlinear
analysis. This is partly due to the fact that in many real world
problems fixed point theory is the basic mathematical tool used to
establish the existence of solutions to problems which arise
naturally in applications. As a result, fixed point theory is an
important area of study in pure and applied mathematics and it is a
flourishing area of research.
The approximation of functions by linear positive operators is an
important research topic in general mathematics and it also
provides powerful tools to application areas such as computer-aided
geometric design, numerical analysis, and solutions of differential
equations. q-Calculus is a generalization of many subjects, such as
hypergeometric series, complex analysis, and particle physics. This
monograph is an introduction to combining approximation theory and
q-Calculus with applications, by using well- known operators. The
presentation is systematic and the authors include a brief summary
of the notations and basic definitions of q-calculus before delving
into more advanced material. The many applications of q-calculus in
the theory of approximation, especially on various operators, which
includes convergence of operators to functions in real and complex
domain forms the gist of the book. This book is suitable for
researchers and students in mathematics, physics and engineering,
and for professionals who would enjoy exploring the host of
mathematical techniques and ideas that are collected and discussed
in the book.
This textbook introduces the subject of complex analysis to
advanced undergraduate and graduate students in a clear and concise
manner. Key features of this textbook: effectively organizes the
subject into easily manageable sections in the form of 50
class-tested lectures, uses detailed examples to drive the
presentation, includes numerous exercise sets that encourage
pursuing extensions of the material, each with an "Answers or
Hints" section, covers an array of advanced topics which allow for
flexibility in developing the subject beyond the basics, provides a
concise history of complex numbers. An Introduction to Complex
Analysis will be valuable to students in mathematics, engineering
and other applied sciences. Prerequisites include a course in
calculus.
This monograph is the first one to systematically present a series
of local and global estimates and inequalities for differential
forms, in particular the ones that satisfy the A-harmonic
equations. The presentation focuses on the Hardy-Littlewood,
Poincare, Cacciooli, imbedded and reverse Holder inequalities.
Integral estimates for operators, such as homotopy operator, the
Laplace-Beltrami operator, and the gradient operator are discussed
next. Additionally, some related topics such as BMO inequalities,
Lipschitz classes, Orlicz spaces and inequalities in Carnot groups
are discussed in the concluding chapter. An abundance of
bibliographical references and historical material supplement the
text throughout. This rigorous presentation requires a familiarity
with topics such as differential forms, topology and Sobolev space
theory. It will serve as an invaluable reference for researchers,
instructors and graduate students in analysis and partial
differential equations and could be used as additional material for
specific courses in these fields.
This work introduces readers to the topic of maximal regularity for
difference equations. The authors systematically present the method
of maximal regularity, outlining basic linear difference equations
along with relevant results. They address recent advances in the
field, as well as basic semi group and cosine operator theories in
the discrete setting. The authors also identify some open problems
that readers may wish to take up for further research. This book is
intended for graduate students and researchers in the area of
difference equations, particularly those with advance knowledge of
and interest in functional analysis.
Environmental variation plays an important role in many biological
and ecological dynamical systems. This monograph focuses on the
study of oscillation and the stability of delay models occurring in
biology. The book presents recent research results on the
qualitative behavior of mathematical models under different
physical and environmental conditions, covering dynamics including
the distribution and consumption of food. Researchers in the fields
of mathematical modeling, mathematical biology, and population
dynamics will be particularly interested in this material.
The study of linear positive operators is an area of mathematical
studies with significant relevance to studies of computer-aided
geometric design, numerical analysis, and differential equations.
This book focuses on the convergence of linear positive operators
in real and complex domains. The theoretical aspects of these
operators have been an active area of research over the past few
decades. In this volume, authors Gupta and Agarwal explore new and
more efficient methods of applying this research to studies in
Optimization and Analysis. The text will be of interest to
upper-level students seeking an introduction to the field and to
researchers developing innovative approaches.
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