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Difference Equations in Normed Spaces, Volume 206 - Stability and Oscillations (Hardcover, 206th edition)
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Difference Equations in Normed Spaces, Volume 206 - Stability and Oscillations (Hardcover, 206th edition)
Series: North-Holland Mathematics Studies
Expected to ship within 12 - 19 working days
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Many problems for partial difference and integro-difference
equations can be written as difference equations in a normed space.
This book is devoted to linear and nonlinear difference equations
in a normed space. Our aim in this monograph is to initiate
systematic investigations of the global behavior of solutions of
difference equations in a normed space. Our primary concern is to
study the asymptotic stability of the equilibrium solution. We are
also interested in the existence of periodic and positive
solutions. There are many books dealing with the theory of ordinary
difference equations. However there are no books dealing
systematically with difference equations in a normed space. It is
our hope that this book will stimulate interest among
mathematicians to develop the stability theory of abstract
difference equations.
Note that even for ordinary difference equations, the problem of
stability analysis continues to attract the attention of many
specialists despite its long history. It is still one of the most
burning problems, because of the absence of its complete solution,
but many general results available for ordinary difference
equations
(for example, stability by linear approximation) may be easily
proved for abstract difference equations.
The main methodology presented in this publication is based on a
combined use of recent norm estimates for operator-valued functions
with the following
methods and results:
a) the freezing method;
b) the Liapunov type equation;
c) the method of majorants;
d) the multiplicative representation of solutions.
In addition, we present stability results for abstract Volterra
discrete equations.
The bookconsists of 22 chapters and an appendix. In Chapter 1,
some definitions and preliminary results are collected. They are
systematically used in the next chapters.
In, particular, we recall very briefly some basic notions and
results of the theory of operators in Banach and ordered spaces. In
addition, stability concepts are presented and Liapunov's functions
are introduced. In Chapter 2 we review various classes of linear
operators and their spectral properties. As examples, infinite
matrices are considered. In Chapters 3 and 4, estimates for the
norms of operator-valued and matrix-valued functions are suggested.
In particular, we consider Hilbert-Schmidt, Neumann-Schatten,
quasi-Hermitian and quasiunitary operators. These classes contain
numerous infinite matrices arising in applications. In Chapter 5,
some perturbation results for linear operators in a Hilbert space
are presented. These results are then used in the next chapters to
derive bounds for the spectral radiuses. Chapters 6-14 are devoted
to asymptotic and exponential stabilities, as well as boundedness
of solutions of linear and nonlinear difference equations. In
Chapter 6 we investigate the linear equation with a bounded
constant operator acting in a Banach space. Chapter 7 is concerned
with the Liapunov type operator equation. Chapter 8 deals with
estimates for the spectral radiuses of concrete operators, in
particular, for infinite matrices. These bounds enable the
formulation of explicit stability conditions. In Chapters 9 and 10
we consider nonautonomous (time-variant) linear equations. An
essential role in this chapter is played by the evolution operator.
In addition, we use the "freezing" method and
multiplicativerepresentations of solutions to construct the
majorants for linear equations. Chapters 11 and 12 are devoted to
semilinear autonomous and nonautonomous equations. Chapters 13 and
14 are concerned with linear and nonlinear higher order difference
equations. Chapter 15 is devoted to the input-to-state stability.
In Chapter 16 we study periodic solutions of linear and nonlinear
difference equations in a Banach space, as well as the global
orbital stability of solutions of vector difference equations.
Chapters 17 and 18 deal with linear and nonlinear Volterra discrete
equations in a Banach space. An important role in these chapter is
played by operator pencils. Chapter 19 deals with a class of the
Stieltjes differential equations.
These equations generalize difference and differential equations.
We apply estimates for norms of operator valued functions and
properties of the multiplicative integral to certain classes of
linear and nonlinear Stieltjes differential equations to obtain
solution estimates that allow us to study the stability and
boundedness of solutions. We also show the existence and uniqueness
of solutions as well as the continuous dependence of the solutions
on the time integrator. Chapter 20 provides some results regarding
the Volterra--Stieltjes equations. The Volterra--Stieltjes
equations include Volterra difference and Volterra integral
equations. We obtain estimates for the norms of solutions of the
Volterra--Stieltjes equation. Chapter 21 is devoted to difference
equations with continuous time. In Chapter 22, we suggest some
conditions for the existence of nontrivial and positive steady
states of difference equations, as well as bounds for the
stationary solutions.
- Deals systematically with difference equations in normed spaces
- Considers new classes of equations that could not be studied in
the frameworks of ordinary and partial difference equations
- Develops the freezing method and presents recent results on
Volterra discrete equations
- Contains an approach based on the estimates for norms of operator
functions
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