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The book is devoted to the foundations of the theory of
boundary-value problems for various classes of systems of
differential-operator equations whose linear part is represented by
Fredholm operators of the general form. A common point of view on
numerous classes of problems that were traditionally studied
independently of each other enables us to study, in a natural way,
the theory of these problems, to supplement and improve the
existing results, and in certain cases, study some of these
problems for the first time. With the help of the technique of
generalized inverse operators, the Vishik- Lyusternik method, and
iterative methods, we perform a detailed investigation of the
problems of existence, bifurcations, and branching of the solutions
of linear and nonlinear boundary-value problems for various classes
of differential-operator systems and propose new procedures for
their construction. For more than 11 years that have passed since
the appearance of the first edition of the monograph, numerous new
publications of the authors in this direction have appeared. In
this connection, it became necessary to make some additions and
corrections to the previous extensively cited edition, which is
still of signifi cant interest for the researchers. For
researchers, teachers, post-graduate students, and students of
physical and mathematical departments of universities. Contents:
Preliminary Information Generalized Inverse Operators in Banach
Spaces Pseudoinverse Operators in Hilbert Spaces Boundary-Value
Problems for Operator Equations Boundary-Value Problems for Systems
of Ordinary Differential Equations Impulsive Boundary-Value
Problems for Systems of Ordinary Differential Equations Solutions
of Differential and Difference Systems Bounded on the Entire Real
Axis
The monograph gives a detailed exposition of the theory of general
elliptic operators (scalar and matrix) and elliptic boundary value
problems in Hilbert scales of Hormander function spaces. This
theory was constructed by the authors in a number of papers
published in 2005 2009. It is distinguished by a systematic use of
the method of interpolation with a functional parameter of abstract
Hilbert spaces and Sobolev inner product spaces. This method, the
theory and their applications are expounded for the first time in
the monographic literature. The monograph is written in detail and
in a reader-friendly style. The complete proofs of theorems are
given. This monograph is intended for a wide range of
mathematicians whose research interests concern with mathematical
analysis and differential equations."
This monograph is devoted to the systematic presentation of
foundations of the quantum field theory. Unlike numerous monographs
devoted to this topic, a wide range of problems covered in this
book are accompanied by their sufficiently clear interpretations
and applications. An important significant feature of this
monograph is the desire of the author to present mathematical
problems of the quantum field theory with regard to new methods of
the constructive and Euclidean field theory that appeared in the
last thirty years of the 20th century and are based on the rigorous
mathematical apparatus of functional analysis, the theory of
operators, and the theory of generalized functions. The monograph
is useful for students, post-graduate students, and young
scientists who desire to understand not only the formality of
construction of the quantum field theory but also its essence and
connection with the classical mechanics, relativistic classical
field theory, quantum mechanics, group theory, and the theory of
path integral formalism.
Significant interest in the investigation of systems with
discontinuous trajectories is explained by the development of
equipment in which significant role is played by impulsive control
systems and impulsive computing systems. Impulsive systems are also
encountered in numerous problems of natural sciences described by
mathematical models with conditions reflecting the impulsive action
of external forces with pulses whose duration can be neglected.
Differential equations with set-valued right-hand side arise in the
investigation of evolution processes in the case of measurement
errors, inaccuracy or incompleteness of information, action of
bounded perturbations, violation of unique solvability conditions,
etc. Differential inclusions also allow one to describe the
dynamics of controlled processes and are widely used in the theory
of optimal control. This monograph is devoted to the investigation
of impulsive differential equations with set-valued and
discontinuous right-hand sides. It is intended for researchers,
lecturers, postgraduate students, and students of higher schools
specialized in the field of the theory of differential equations,
the theory of optimal control, and their applications.
A thorough, self-contained and easily accessible treatment of the
theory on the polynomial best approximation of functions with
respect to maximum norms. The topics include Chebychev theory,
Weierstrass theorems, smoothness of functions, and continuation of
functions.
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