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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Series in Pure and Applied Mathematics
- Sprache: Englisch
Features new results and up-to-date advances in modeling and solving differential equations Introducing the various classes of functional differential equations, Functional Differential Equations: Advances and Applications presents the needed tools and topics to study the various classes of functional differential equations and is primarily concerned with the existence, uniqueness, and estimates of solutions to specific problems. The book focuses on the general theory of functional differential equations, provides the requisite mathematical background, and details the qualitative behavior of solutions to functional differential equations. The book addresses problems of stability, particularly for ordinary differential equations in which the theory can provide models for other classes of functional differential equations, and the stability of solutions is useful for the application of results within various fields of science, engineering, and economics. Functional Differential Equations: Advances and Applications also features: * Discussions on the classes of equations that cannot be solved to the highest order derivative, and in turn, addresses existence results and behavior types * Oscillatory motion and solutions that occur in many real-world phenomena as well as in man-made machines * Numerous examples and applications with a specific focus on ordinary differential equations and functional differential equations with finite delay * An appendix that introduces generalized Fourier series and Fourier analysis after periodicity and almost periodicity * An extensive Bibliography with over 550 references that connects the presented concepts to further topical exploration Functional Differential Equations: Advances and Applications is an ideal reference for academics and practitioners in applied mathematics, engineering, economics, and physics. The book is also an appropriate textbook for graduate- and PhD-level courses in applied mathematics, differential and difference equations, differential analysis, and dynamics processes. CONSTANTIN CORDUNEANU, PhD, is Emeritus Professor in the Department of Mathematics at The University of Texas at Arlington, USA. The author of six books and over 200 journal articles, he is currently Associate Editor for seven journals; a member of the American Mathematical Society, Society for Industrial and Applied Mathematics, and the Romanian Academy; and past president of the American Romanian Academy of Arts and Sciences. YIZENG LI, PhD, is Professor in the Department of Mathematics at Tarrant County College, USA. He is a member of the Society for Industrial and Applied Mathematics. MEHRAN MAHDAVI, PhD, is Professor in the Department of Mathematics at Bowie State University, USA. The author of numerous journal articles, he is a member of the American Mathematical Society, Society for Industrial and Applied Mathematics, and the Mathematical Association of America.
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Table of Contents
COVER
TITLE PAGE
PREFACE
ACKNOWLEDGMENTS
1 INTRODUCTION, CLASSIFICATION, SHORT HISTORY, AUXILIARY RESULTS, AND METHODS
1.1 CLASSICAL AND NEW TYPES OF FEs
1.2 MAIN DIRECTIONS IN THE STUDY OF FDE
1.3 METRIC SPACES AND RELATED CONCEPTS
1.4 FUNCTIONS SPACES
1.5 SOME NONLINEAR AUXILIARY TOOLS
1.6 FURTHER TYPES OF FES
2 EXISTENCE THEORY FOR FUNCTIONAL EQUATIONS
2.1 LOCAL EXISTENCE FOR CONTINUOUS OR MEASURABLE SOLUTIONS
2.2 GLOBAL EXISTENCE FOR SOME CLASSES OF FUNCTIONAL DIFFERENTIAL EQUATIONS
2.3 EXISTENCE FOR A SECOND-ORDER FUNCTIONAL DIFFERENTIAL EQUATION
2.4 THE COMPARISON METHOD IN OBTAINING GLOBAL EXISTENCE RESULTS
2.5 A FUNCTIONAL DIFFERENTIAL EQUATION WITH BOUNDED SOLUTIONS ON THE POSITIVE SEMIAXIS
2.6 AN EXISTENCE RESULT FOR FUNCTIONAL DIFFERENTIAL EQUATIONS WITH RETARDED ARGUMENT
2.7 A SECOND-ORDER FUNCTIONAL DIFFERENTIAL EQUATION WITH BOUNDED SOLUTIONS ON THE POSITIVE SEMIAXIS
2.8 A GLOBAL EXISTENCE RESULT FOR A CLASS OF FIRST-ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS
2.9 A GLOBAL EXISTENCE RESULT IN A SPECIAL FUNCTION SPACE AND A POSITIVITY RESULT
2.10 SOLUTION SETS FOR CAUSAL FUNCTIONAL DIFFERENTIAL EQUATIONS
2.11 AN APPLICATION TO OPTIMAL CONTROL THEORY
2.12 FLOW INVARIANCE
2.13 FURTHER EXAMPLES/APPLICATIONS/COMMENTS
2.14 BIBLIOGRAPHICAL NOTES
3 STABILITY THEORY OF FUNCTIONAL DIFFERENTIAL EQUATIONS
3.1 SOME PRELIMINARY CONSIDERATIONS AND DEFINITIONS
3.2 COMPARISON METHOD IN STABILITY THEORY OF ORDINARY DIFFERENTIAL EQUATIONS
3.3 STABILITY UNDER PERMANENT PERTURBATIONS
3.4 STABILITY FOR SOME FUNCTIONAL DIFFERENTIAL EQUATIONS
3.5 PARTIAL STABILITY
3.6 STABILITY AND PARTIAL STABILITY OF FINITE DELAY SYSTEMS
3.7 STABILITY OF INVARIANT SETS
3.8 ANOTHER TYPE OF STABILITY
3.9 VECTOR AND MATRIX LIAPUNOV FUNCTIONS
3.10 A FUNCTIONAL DIFFERENTIAL EQUATION
3.11 BRIEF COMMENTS ON THE START AND EVOLUTION OF THE COMPARISON METHOD IN STABILITY
3.12 BIBLIOGRAPHICAL NOTES
4 OSCILLATORY MOTION, WITH SPECIAL REGARD TO THE ALMOST PERIODIC CASE
4.1 TRIGONOMETRIC POLYNOMIALS AND
AP
r
-SPACES
4.2 SOME PROPERTIES OF THE SPACES
4.3
AP
r
-SOLUTIONS TO ORDINARY DIFFERENTIAL EQUATIONS
4.4
AP
r
-SOLUTIONS TO CONVOLUTION EQUATIONS
4.5 OSCILLATORY SOLUTIONS INVOLVING THE SPACE
B
4.6 OSCILLATORY MOTIONS DESCRIBED BY CLASSICAL ALMOST PERIODIC FUNCTIONS
4.7 DYNAMICAL SYSTEMS AND ALMOST PERIODICITY
4.8 BRIEF COMMENTS ON THE DEFINITION OF SPACES AND RELATED TOPICS
4.9 BIBLIOGRAPHICAL NOTES
5 NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS
5.1 SOME GENERALITIES AND EXAMPLES RELATED TO NEUTRAL FUNCTIONAL EQUATIONS
5.2 FURTHER EXISTENCE RESULTS CONCERNING NEUTRAL FIRST-ORDER EQUATIONS
5.3 SOME AUXILIARY RESULTS
5.4 A CASE STUDY, I
5.5 ANOTHER CASE STUDY, II
5.6 SECOND-ORDER CAUSAL NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS, I
5.7 SECOND-ORDER CAUSAL NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS, II
5.8 A NEUTRAL FUNCTIONAL EQUATION WITH CONVOLUTION
5.9 BIBLIOGRAPHICAL NOTES
APPENDIX A: ON THE THIRD STAGE OF FOURIER ANALYSIS
A.1 INTRODUCTION
A.2 RECONSTRUCTION OF SOME CLASSICAL SPACES
A.3 CONSTRUCTION OF ANOTHER CLASSICAL SPACE
A.4 CONSTRUCTING SPACES OF OSCILLATORY FUNCTIONS: EXAMPLES AND METHODS
A.5 CONSTRUCTION OF ANOTHER SPACE OF OSCILLATORY FUNCTIONS
A.6 SEARCHING FUNCTIONAL EXPONENTS FOR GENERALIZED FOURIER SERIES
A.7 SOME COMPACTNESS PROBLEMS
BIBLIOGRAPHY
INDEX
PURE AND APPLIED MATHEMATICS
END USER LICENSE AGREEMENT
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PURE AND APPLIED MATHEMATICS
A Wiley Series of Texts, Monographs, and Tracts
Founded by RICHARD COURANTEditors Emeriti: MYRON B. ALLEN III, DAVID A. COX, PETER HILTON, HARRY HOCHSTADT, PETER LAX, JOHN TOLAND
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FUNCTIONAL DIFFERENTIAL EQUATIONS
Advances and Applications
CONSTANTIN CORDUNEANU
YIZENG LI
MEHRAN MAHDAVI
Copyright © 2016 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada
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Library of Congress Cataloging-in-Publication Data:
Names: Corduneanu, C., author. | Li, Yizeng, 1949– author. | Mahdavi, Mehran, 1959– author.Title: Functional differential equations : advances and applications/Constantin Corduneanu, Yizeng Li, Mehran Mahdavi.Description: Hoboken, New Jersey : John Wiley & Sons, Inc., [2016] | Series: Pure and applied mathematics | Includes bibliographical references and index.Identifiers: LCCN 2015040683 | ISBN 9781119189473 (cloth)Subjects: LCSH: Functional differential equations.Classification: LCC QA372 .C667 2016 | DDC 515/.35–dc23 LC record available at http://lccn.loc.gov/2015040683
This book is dedicated to the memory of three pioneers of the Theory of Functional Differential Equations:
A.D. MYSHKIS, the author of the first book on non-classical differential equations
N.N. KRASOVSKII, for substantial contributions, particularly for extending Liapunov’s method in Stability Theory, to the case of delay equations with finite delay, thus opening the way to the methods of Functional Analysis
J.K. HALE, for creating a vast theory in the study of functional differential equations, by using the modern tools of Functional Analysis
Each created a research school, with many distinguished contributors
PREFACE
The origin of this book is in the research seminar organized by the first coauthor (alphabetically), during the period September 1990 to May 1994, in the Department of Mathematics at The University of Texas at Arlington. The second and the third co-authors were, at that time, Ph.D. students working under the guidance of the first co-author. The seminar was also attended by another Ph.D. student, Zephyrinus Okonkwo, who had been interested in stochastic problems, and sporadically by other members of the Department of Mathematics (among them, Dr. Richard Newcomb II). Visitors also occasionally attended and presented their research results. We mention V. Barbu, Y. Hamaya, M. Kwapisz, I. Gyori, and Cz. Olech. Cooperation between the co-authors continued after the graduation of the co-authors Li and Mahdavi, who attained their academic positions at schools in Texas and Maryland, respectively. All three co-authors continued to pursue the topics that were presented at the seminar and that are included in this book through their Ph.D. theses as well as a number of published papers (single author or jointly). The list of references in this book contains almost all of the contributions that were made during the past two decades (1994–2014). Separation among the coauthors certainly contributed to the extended period that was needed to carry out the required work.
The book is a monograph, presenting only part of the results available in the literature, mainly mathematical ones, without any claim related to the coverage of the whole field of functional differential equations (FDEs). Paraphrasing the ancient dictum “mundum regunt numeri,” one can instead say, “mundum regunt aequationes.” That is why, likely, one finds a large number of reviews dedicated to this field of research, in all the publications concerned with the review of current literature (Mathematical Reviews, Zentralblatt fur Mathematik, Referativnyi Zhurnal, a. o.).
Over 550 items are listed in the Bibliography, all of which have been selected from a much larger number of publications available to the coauthors (i.e., university libraries, preprints, or reprints obtained from authors, papers received for review in view of insertion into various journals, or on the Internet).
The principles of selection were the connections with the topics considered in the book, but also the inclusion of information for themes similar to those treated by the co-authors, but not covered in the book because of the limitations imposed by multiple factors occurring during the preparation of the final text.
Mehran Mahdavi was in charge of the technical realization of the manuscript, a task he carried out with patience and skill. The other co-authors thank him for voluntarily assuming this demanding task, which he accomplished with devotion and competence.
Some developing fields of research related to the study of FDEs, but not included in this book’s presentations, are the stochastic equations, the fuzzy equations, and the fractional-order functional equations. Some sporadic references are made to discrete argument functional equations, also known as difference equations. While their presence is spotted in many publications, Springer recently dedicated an entire journal to the subject, Advances in Difference Equations (R. P. Agarwal, editor). In addition, Clarendon Press is publishing papers on functional differential equations of the time scale type, which is encountered in some applications.
The plan that was followed in presenting the discussed topics is imposed by the different types of applications the theory of FDEs is dealing with, including the following:
Existence, uniqueness, estimates of solutions, and some behavior, when they are globally defined (e.g., domain invariance).
Stability of solutions is also of great interest for those applying the results in various fields of science, engineering, economics, and others.
Oscillatory motion/solutions, a feature occurring in many real phenomena and in man-made machines.
Chapter 1 is introductory and is aimed at providing the readers with the tools necessary to conduct the study of various classes of FDEs. Chapter 2 is primarily concerned with the existence of solutions, the uniqueness (not always present), and estimate for solutions, in view of their application to specific problems. Chapter 3, the most extended, deals with problems of stability, particularly for ordinary DEs (in which case the theory is the most advanced and can provide models for other classes of FDEs). In this chapter, the interest is of concern not only to mathematicians, but also to other scientists, engineers, economists, and others deeply engaged in related applied fields. Chapter 4 deals with oscillatory properties, especially of almost periodic type (which includes periodic). The choice of spaces of almost periodic functions are not, as usual, the classical Bohr type, but a class of spaces forming a scale, starting with the simplest space (Poincaré) of those almost periodic functions whose Fourier series are absolutely convergent and finishes with the space of Besicovitch B2, the richest one known, for which we have enough meaningful tools. Chapter 5 contains results of any nature, available for the so-called neutral equations. There are several types of FDEs belonging to this class, which can be roughly defined as the class of those equations that are not solved with respect to the highest-order derivative involved.
For those types of equations, this book proves existence results, some kinds of behavior (e.g., boundedness), and stability of the solutions (especially asymptotic stability).
Appendix A, written by C. Corduneanu, introduces the reader to what is known about generalized Fourier series of the form with and λk(t) some real-valued function on R. Such series intervene in studying various applied problems and appears naturally to classify them as belonging to the third stage of development of the Fourier analysis (after periodicity and almost periodicity). The presentation is descriptive, less formal, and somewhat a survey of problems occurring in the construction of new spaces of oscillatory functions.
Since the topics discussed in this book are rather specialized with respect to the general theory of FDEs, the book can serve as source material for graduate students in mathematics, science, and engineering. In many applications, one encounters FDEs that are not of a classical type and, therefore, are only rarely taken into consideration for teaching. The list of references in this book contains many examples of this situation. For instance, the case of equations in population dynamics is a good illustration (see Gopalsamy [225]). Also, the book by Kolmanovskii and Myshkis [292] provides a large number of applications for FDEs, which may interest many categories of readers.
ACKNOWLEDGMENTS
We would like to express our appreciation to John Wiley & Sons, Inc., for publishing our book and to Ms. Susanne Steitz-Filler, senior editor, and the editorial and production staff at Wiley.
Constantin [email protected]
Yizeng [email protected]
Mehran [email protected]
1INTRODUCTION, CLASSIFICATION, SHORT HISTORY, AUXILIARY RESULTS, AND METHODS
Generally speaking, a functional equation is a relationship containing an unknown element, usually a function, which has to be determined, or at least partially identifiable by some of its properties. Solving a functional equation (FE) means finding a solution, that is, the unknown element in the relationship. Sometimes one finds several solutions (solutions set), while in other cases the equation may be deprived of a solution, particularly when one provides the class/space to which it should belong.
Since a relationship could mean the equality, or an inequality, or even the familiar “belongs to,” designated by , , or , the description given earlier could also include the functional inequalities or the functional inclusions, rather often encountered in the literature. Actually, in many cases, their theory is based on the theory of corresponding equations with which they interact. For instance, the selection of a single solution from a solution set, especially in case of inclusions.
In this book we are mainly interested in FEs, in the proper/usual sense. We send the readers to adequate sources for cases of related categories, like inequalities or inclusions.Functional Differential Equations: Advances and Applications, First Edition.Constantin Corduneanu, Yizeng Li and Mehran Mahdavi© 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.
1.1 CLASSICAL AND NEW TYPES OF FEs
The classical types of FEs include the ordinary differential equations (ODEs), the integral equations (IEs) of Volterra or Fredholm and the integro-differential equations (IDEs). These types, which have been thoroughly investigated since Newton’s time, constitute the classical part of the vast field of FEs, or functional differential equations (FDEs).
The names Bernoulli, Newton, Riccati, Euler, Lagrange, Cauchy (analytic solutions), Dini, and Poincaré as well as many more well-known mathematicians, are usually related to the classical theory of ODE. This theory leads to a large number of applications in the fields of science, engineering, economics, in cases of the modeling of specific problems leading to ODE.
A large number of books/monographs are available in the classical field of ODE: our list of references containing at least those authored by Halanay [237], Hale [240], Hartman [248], Lefschetz [323], Petrovskii [449], Sansone and Conti [489], Rouche and Mawhin [475], Nemytskii and Stepanov [416], and Coddington and Levinson [106].
Another classical type of FEs, closely related to the ODEs, is the class of IEs, whose birth is related to Abel in the early nineteenth century. They reached an independent status by the end of nineteenth century and the early twentieth century, with Volterra and Fredholm. Hilbert is constituting his theory of linear IEs of Fredholm’s type, with symmetric kernel, providing a successful start to the spectral theory of completely continuous operators and orthogonal function series.
Classical sources in regard to the basic theory of integral equations include books/monographs by Volterra [528], Lalesco [319], Hilbert [261], Lovitt [340], Tricomi [520], Vath [527]. More recent sources are Corduneanu [135], Gripenberg et al. [228], Burton [80, 84], and O’Regan and Precup [430].
A third category of FEs, somewhat encompassing the differential and the IEs, is the class of IDEs, for which Volterra [528] appears to be the originator. It is also true that E. Picard used the integral equivalent of the ODE , under initial condition , Cauchy’s problem, namely
obtaining classical existence and uniqueness results by the method of successive approximations.
A recent reference, mostly based on classical analysis and theories of DEs and IEs, is Lakshmikantham and R. M. Rao [316], representing a rather comprehensive picture of this field, including some significant applications and indicating further sources.
The extended class of FDEs contains all preceding classes, as well equations involving operators instead of functions (usually from R into R). The classical categories are related to the use of the so-called Niemytskii operator, defined by the formula , with or in an interval of R, while in the case of FDE, the right-hand side of the equation
implies a more general type of operator F. For instance, using Hale’s notation, one can take , where , represents a restriction of the function x(t), to the interval . This is the finite delay case. Another choice is
where V represents an abstract Volterra operator (see definition in Chapter 2), also known as causal operator.
Many other choices are possible for the operator F, leading to various classes of FDE. Bibliography is very rich in this case, and exact references will be given in the forthcoming chapters, where we investigate various properties of equations with operators.
The first book entirely dedicated to FDE, in the category of delay type (finite or infinite) is the book by A. Myshkis [411], based on his thesis at Moscow State University (under I. G. Petrovskii). This book was preceded by a survey article in the Uspekhi Mat. Nauk, and one could also mention the joint paper by Myshkis and Eĺsgoĺtz [412], reviewing the progress achieved in this field, due to both authors and their followers. The book Myshkis [411] is the first dedicated entirely to the DEs with delay, marking the beginnings of the literature dealing with non-traditional FEs.
The next important step in this direction has been made by N. N. Krasovskii [299], English translation of 1959 Russian edition. In his doctoral thesis (under N. G. Chetayev), Krasovskii introduced the method of Liapunov functionals (not just functions!), which permitted a true advancement in the theory of FDEs, especially in the nonlinear case and stability problems. The research school in Ekaterinburg has substantially contributed to the progress of the theory of FDEs (including Control Theory), and names like Malkin, Barbashin, and Krasovskii are closely related to this progress.
The third remarkable step in the development of the theory of FDE has been made by Jack Hale, whose contribution should be emphasized, in respect to the constant use of the arsenal of Functional Analysis, both linear and nonlinear. A first contribution was published in 1963 (see Hale [239]), utilizing the theory of semigroups of linear operators on a Banach function space. This approach allowed Hale to develop a theory of linear systems with finite delay, in the time-invariant framework, dealing with adequate concepts that naturally generalize those of ODE with constant coefficients (e.g., characteristic values of the system/equation). Furthermore, many problems of the theory of nonlinear ODE have been formulated and investigated for FDE (stability, bifurcation, and others (a.o.)). The classical book of Hale [240] appears to be the first in this field, with strong support of basic results, some of them of recent date, from functional analysis.
In the field of applications of FDE, the book by Kolmanovskii and Myshkis [292] illustrates a great number of applications to science (including biology), engineering, business/economics, environmental sciences, and medicine, including the stochastic factors. Also, the book displays a list of references with over 500 entries.
In concluding this introductory section, we shall mention the fact that the study of FDE, having in mind the nontraditional types, is the focus for a large number of researchers around the world: Japan, China, India, Russia, Ukraine, Finland, Poland, Romania, Greece, Bulgaria, Hungary, Austria, Germany, Great Britain, Italy, France, Morocco, Algeria, Israel, Australia and the Americas, and elsewhere.
The Journal of Functional Differential Equations is published at the College of Judea and Samaria, but its origin was at Perm Technical University (Russia), where N. V. Azbelev created a school in the field of FDE, whose former members are currently active in Russia, Ukraine, Israel, Norway, and Mozambique.
Many other journals are dedicated to the papers on FDE and their applications. We can enumerate titles like Nonlinear Analysis (Theory, Methods & Applications), published by Elsevier; Journal of Differential Equations; Journal of Mathematical Analysis and Applications, published by Academic Press; Differentsialuye Uravnenja (Russian: English translation available); and Funkcialaj Ekvacioj (Japan). Also, there are some electronic journals publishing papers on FDE: Electronic Journal of Qualitative Theory of Differential Equations, published by Szeged University; EJQTDE, published by Texas State University, San Marcos.
1.2 MAIN DIRECTIONS IN THE STUDY OF FDE
This section is dedicated to the description of various types of problems arising in the investigation of FDE, at the mathematical side of the problem as well as the application of FDE in various fields, particularly in science and engineering.
A first problem occurring in relationship with an FDE is the existence or absence of a solution. The solution is usually sought in a certain class of functions (scalar, vector, or even Banach space valued) and “a priori” limitations/restrictions may be imposed on it.
In most cases, besides the “pure” existence, we need estimates for the solutions. Also, it may be necessary to use the numerical approach, usually approximating the real values of the solution. Such approximations may have a “local” character (i.e., valid in a neighborhood of the initial/starting value of the solution, assumed also unique), or they may be of “global” type, keeping their validity on the whole domain of definition of the solution.
Let us examine an example of a linear FDE, of the form
with a linear, casual continuous map, while . As shown in Corduneanu [149; p. 85], the unique solution of equation (1.1), such that , is representable by the formula
In (1.2), the Cauchy matrix is given, on , by the formula
where stands for the conjugate kernel associated to the kernel k(t, s), the latter being determined by the relationship
For details, see the reference indicated earlier in the text.
Formula (1.2) is helpful in finding various estimates for the solution x(t) of the initial value problem considered previously.
Assume, for instance, that the Cauchy matrix X(t, s) is bounded on by M, that is, ; hence , then (1.2) yields the following estimate for the solution x(t):
with and f continuous on [0, T]. We derive from (1.5) the estimate
which means an upper bound of the norm of the solutions, in terms of data.
We shall also notice that (1.6) keeps its validity in case , that is, we consider the problem on the semiaxis . This example shows how, assuming also , all solutions of (1.1) remain bounded on the positive semiaxis.
Boundedness of all solutions of (1.1), on the positive semiaxis, is also assured by the conditions , , and
The readers are invited to check the validity of the following estimate:
Estimates like (1.6) or (1.7), related to the concept of boundedness of solutions, are often encountered in the literature. Their significance stems from the fact that the motion/evolution of a man-made system takes place in a bounded region of the space. Without having estimates for the solutions of FDE, it is practically impossible to establish properties of these solutions.
One of the best examples in this regard is constituted by the property of stability of an equilibrium state of a system, described by the FDE under investigation. At least, theoretically, the problem of stability of a given motion of a system can be reduced to that of an equilibrium state. Historically, Lagrange has stated a result of stability for the equilibrium for a mechanical system, in terms of a variational property of its energy. This idea has been developed by A. M. Liapunov [332] (1857–1918), who introduced the method of an auxiliary function, later called Liapunov function method. Liapunov’s approach to stability theory is known as one of the most spectacular developments in the theory of DE and then for larger classes of FDE, starting with N. N. Krasovskii [299].
The comparison method, on which we shall rely (in Chapter 3), has brought new impetus to the investigation of stability problems. The schools created by V. V. Rumiantsev in Moscow (including L. Hatvani and V. I. Vorotnikov), V. M. Matrosov in Kazan, then moved to Siberia and finally to Moscow, have developed a great deal of this method, concentrating mainly on the ODE case. Also, V. Lakshmikantham and S. Leela have included many contributions in their treaty [309]. They had many followers in the United States and India, publishing a conspicuous number of results and developments of this method. One of the last contributions to this topic [311], authored by Lakshmikantham, Leela, Drici, and McRae, contains the general theory of equations with causal operators, including stability problems.
The comparison method consists of the simultaneous use of Liapunov functions (functionals), and differential inequalities. Started in its general setting by R. Conti [110], it has been used to prove global existence criteria for ODE. In short time, the use has extended to deal with uniqueness problems for ODE by F. Brauer [75] and Corduneanu [114, 115] for stability problems. The method is still present in the literature, with contributions continuing those already included in classical references due to Sansone and Conti [489], Hahn [235], Rouche and Mawhin [475], Matrosov [376–378], Matrosov and Voronov [387], Lakshimikantham and Leela [309], and Vorotnikov [531].
A historical account on the development of the stability concept has been accurately given by Leine [325], covering the period from Lagrange to Liapunov. The mechanical/physical aspects are emphasized, showing the significance of the stability concept in modern science. The original work of Liapunov [332] marks a crossroad in the development of this concept, with so many connections in the theory of evolutionary systems occurring in the mathematical description in contemporary science.
In Chapter 3, we shall present stability theory for ODE and FDE, particularly for the equations with finite delay. The existing literature contains results related to the infinite delay equations, a theory that has been originated by Hale and Kato [241]. An account on the status of the theory, including stability, is to be found in Corduneanu and Lakshmikantham [167]. We notice the fact that a theory of stability, for general classes of FDE, has not yet been elaborated. As far as special classes of FDE are concerned, the book [84] by T. Burton presents the method of Liapunov functionals for integral equations, by using modern functional analytic methods. The book [43] by Barbashin, one of the first in this field, contains several examples of constructing Liapunov functions/functionals.
The converse theorems in stability theory, in the case of ODE, have been obtained, in a rather general framework, by Massera [373], Kurzweil [303], and Vrkoč [532]. Early contributions to stability theory of ODE were brought by followers of Liapunov, (see Chetayev [103] and Malkin [356]). In Chapter 3, the readers will find, besides some basic results on stability, more bibliographical indications pertaining to this rich category of problems.
As an example, often encountered in some books containing stability theory, we shall mention here the classical result (Poincaré and Liapunov) concerning the differential system , , , and a continuous map. If we admit the commutativity condition
then the solution, under initial condition , can be represented by
From this representation formula one derives, without difficulty, the following results:
Stability of the solution the zero vector in Rn is equivalent to boundedness, on , of the matrix function .
Asymptotic stability of the solution is equivalent to the condition
Both statements are elementary consequences of formula (1.8). The definitions of various types of stability will be done in Chapter 3. We notice here that the already used terms, stability and asymptotic stability, suggest that the first stands for the property of the motion to remain in the neighborhood of the equilibrium point when small perturbations of the initial data are occurring, while the second term tells us that besides the property of stability (as intuitively described earlier), the motion is actually “tending” or approaching indefinitely the equilibrium state, when .
Remark 1.1
The aforementioned considerations help us derive the celebrated stability result, known as Poincaré–Liapunov stability theorem for linear differential systems with constant coefficients.
Indeed, if constant is an matrix, with real or complex coefficients, with characteristic equation , the unit matrix of type , then we denote by λ1, λ2, …, λk its distinct roots (). From the elementary theory of DEs with constant coefficients, we know that the entries of the matrix eA t are quasi-polynomials of the form
with pj(t), , some algebraic polynomials.
Since the commutativity condition (1.8) is valid when constant, there results that (1.10) can hold if and only if the condition
is satisfied. Condition (1.12) is frequently used in stability theory, particularly in the case of linear systems encountered in applications, but also in the case of nonlinear systems of the form
when f—using an established odd term—is of “higher order” with respect to x (say, for instance, ).
We will conclude this section with the discussion of another important property of motion, encountered in nature and man-made systems. This property is known as oscillation or oscillatory motion. Historically, the periodic oscillations (of a pendulum, for instance) have been investigated by mathematicians and physicists.
Gradually, more complicated oscillatory motions have been observed, leading to the apparition of almost periodic oscillations/vibrations. In the third decade of the twentieth century, Harald Bohr (1887–1951), from Copenhagen, constructed a wider class than the periodic one, called almost periodic.
In the last decade of the twentieth century, motivated by the needs of researchers in applied fields, even more complex oscillatory motions have emerged. In the books by Osipov [432] and Zhang [553, 554], new spaces of oscillatory functions/motions have been constructed and their applications illustrated.
In case of the Bohr–Fresnel almost periodic functions, a new space has been constructed, its functions being representable by generalized Fourier series of the form
with and , .
In the construction of Zhang, the attached generalized Fourier series has the form
with , , and , Q(R) denoting the algebra of polynomial functions of the form
and for ; , while denote arbitrary reals.
The functions (on R) obtained by uniform approximation with generalized trigonometric polynomials of the form
are called strong limit power functions and their space is denoted by .
A discussion of these generalizations of the classical trigonometric series and attached “sum” are presented in Appendix. The research work is getting more and more adepts, contributing to the development of this third stage in the history of oscillatory motions/functions.
In order to illustrate, including some applications to FDEs, the role of almost periodic oscillations/motions, we have chosen to present in Chapter 4 only the case of APr-almost periodic functions, , constituting a relatively new class of almost periodic functions, related to the theory of oscillatory motions. Their construction is given, in detail, in Chapter 4, as well as several examples from the theory of FDEs.
Concerning the first two stages in the development of the theory of oscillatory functions, the existing literature includes the treatises of Bary [47] and Zygmund [562]. These present the main achievements of the first stage of development (from Euler and Fourier, to contemporary researchers). With regard to the second stage in the theory of almost periodic motions/functions, there are many books/monographs dedicated to the development, following the fundamental contributions brought by Harald Bohr. We shall mention here the first books presenting the basic facts, Bohr [72] and Besicovitch [61], Favard [208], Fink [213], Corduneanu [129, 156], Amerio and Prouse [21], and Levitan [326], Levitan and Zhikov [327]. These references contain many more indications to the work of authors dealing with the theory of almost periodic motions/functions. They will be mentioned in Section 4.9.
As an example of an almost periodic function, likely the first in the literature but without naming it by its name, seems to be due to Poincaré [454], who dealt with the representations of the form
Supposing that the series converges uniformly to f on R (which situation can occur, for instance, when ), Poincaré found the formula for the coefficients ak, introducing simultaneously the concept of mean value of a function on R:
This concept was used 30 years later by H. Bohr, to build up the theory of almost periodic functions (complex-valued). The coefficients were given by the formula
1.3 METRIC SPACES AND RELATED CONCEPTS
One of the most frequent tools encountered in modern mathematical analysis is a metric space, introduced at the beginning of the twentieth century by Maurice Fréchet (in his Ph.D. thesis at Sorbonne). This concept came into being after G. Cantor laid the bases of the set theory, opening a new era in mathematics. The simple idea, exploited by Fréchet, was to consider a “distance” between the elements of an abstract set.
Definition 1.1
A set S, associated with a map, is called a metric space, if the following axioms are adopted:
,
with
only when
;
, ;
, .
Several consequences can be drawn from Definition 1.1. Perhaps, the most important is contained in the following definition:
Definition 1.2
Consider a sequence of elements/points. If
then one says that the sequenceconverges to x in S.
Then x is called the limit of the sequence.
It is common knowledge that the limit of a convergent sequence in S is unique.
Since the concept of a metric space has gained wide acceptance in Mathematics, Science and Engineering, we will send the readers to the book of Friedman [214] for further elementary properties of metric spaces and the concept of convergence.
It is important to mention the fact that the concept of convergence/limit helps to define other concepts, such as compactness of a subset . Particularly, the concept of a complete metric space plays a significant role.
Definition 1.3
The metric space (S, d) is called complete, if any sequencesatisfying the Cauchy condition, “for each, there exists an integer, such thatfor, is convergent in (S, d).”
Definition 1.4
The metric space (S, d) is called compact, according to Fréchet, iff any sequencecontains a convergent subsequence, that is, such that, for some.
Definition 1.4 leads easily to other properties of a compact metric space. For instance, the diameter of a compact metric space S is finite: . Also, every compact metric space is complete.
We rely on other properties of the metric spaces, sending the readers to the aforementioned book of Friedman [214], which contains, in a concise form, many useful results we shall use in subsequent sections of this book. Other references are available in the literature: see, for instance, Corduneanu [135], Zeidler [551], Kolmogorov and Fomin [295], Lusternik and Sobolev [343], and Deimling [190].
Almost all books mentioned already contain applications to the theory of FEs, particularly to differential equations and to integral equations. Other sources can be found in the titles referenced earlier in the text.
The metric spaces are a particular case of topological spaces. The latter represent a category of mathematical objects, allowing the use of the concept of limit, as well as many other concepts derived from that of limit (of a sequence of a function, limit point of a set, closure of a set, closed set, open set, a.o.)
If we take the definition of a topological space by means of the axioms for the family of open sets, then in case of metric spaces the open sets are those subsets A of the space S, defined by the property that any point x of A belongs to A, together with the “ball” of arbitrary small radius r, .
It is easy to check that the family of all open sets, of a metric space S, verifies the following axioms (for a topological space):
The
union
of a family of open sets is also an open set.
The
intersection
of a finite family of open sets is also an open set.
The space
S
and the empty set belong to the family of open sets.
Such a family, satisfying axioms 1, 2, and 3, induces a topology τ on S. Returning to the class of metric spaces, we shall notice that the couple (S, d) is inducing a topology on S and, therefore, any property of topological nature of this space is the product of the metric structure (S, d). The converse problem, to find conditions on a topological space to be the product of a metric structure, known as metrizability, has kept the attention of mathematicians for several decades of the past century, being finally solved. The result is known as the theorem of Nagata–Smirnov.
Substantial progress has been made, with regard to the enrichment of a metric structure, when Banach [39] introduced the new concept of linear metric space, known currently as Banach space.
Besides the metric structure/space (S, d), one assumes that S is a linear space (algebraically) over the field of reals R, or the field of complex numbers C. Moreover, there must be some compatibility between the metric structure and the algebraic one. Accordingly, the following system of axioms is defining a Banach space, denoted , with a map from S into , , called a norm.
S
is a linear space over
R
, in additive notation.
S
is a
normed
space, that is, there is a map, from
S
into , , satisfying the following conditions:
for , iff ;
, for , ;
for .
It is obvious that , , is a distance/metric on S.
(
S
,
d
), with
d
defined earlier, is a complete metric space.
Also, traditional notations for a Banach space, frequently encountered in literature, are or , in the latter case, the generic element of X being denoted by x.
The most commonly encountered Banach space is the vector space Rn (or Cn), the norm being usually defined by
and called the Euclidean norm. Another norm is defined by .
Both norms mentioned previously lead to the same kind of convergence in Rn, because , . This is the usual convergence on coordinates, that is, .
A special type of Banach space is the Hilbert space. The prototype has been constructed by Hilbert, and it is known as ℓ2(R), or ℓ2(), space. This fact occurred long before Banach introduced his concept of space in the 1920s. The ℓ2-space appeared in connection with the theory of orthogonal function series, generated by the Fredholm–Hilbert theory of integral equations with symmetric kernel (in the complex case, the condition is . It is also worth mentioning that the first book on Hilbert spaces, authored by M. Stone [508], shortly preceded the first book on Banach space theory, Banach [39], 1932. The Banach spaces reduce to Hilbert spaces, in the real case, if and only if the rule of the parallelogram is valid:
Of course, the parallelogram involved is the one constructed on the vectors x and y as sides.
What is really specific for Hilbert spaces is the fact that the concept of inner product is defined for , as follows: it is a map from into R (or ), such that
;
, the value 0 leading to ;
, with , .
In the complex case, one should change to , remains the same, and must be changed accordingly.
If one starts with a Banach space satisfying condition (1.22), then the inner product is given by
Conditions , , and , stated in the text, can be easily verified by the product given by formula (1.23).
A condition verified by the inner product is known as Cauchy inequality, and it looks
It is easily obtained starting from the obvious inequality , which is equivalent to , which, regarded as a quadratic polynomial in a, must take only nonnegative values. This would be possible only in case the discriminant is nonpositive, that is, , which implies (1.24). Using (1.24), prove that , for any .
In concluding this section, we will define another special case of linear metric spaces, whose metric is invariant to translations. These spaces are known as linear Fréchet spaces. We will use them in Chapter 2.
If instead of a norm, satisfying conditions II and III in the definition of a Banach space, we shall limit the imposed properties to , accepting the possibility that there may be elements , we obtain what is called a semi-norm. One can operate with a semi-norm in the same way we do with a norm, the difference appearing in the part that the limit of a convergent sequence is not necessarily unique.
Here precisely, the semi-norm is defined by the means of the following axioms related to a linear space E:
for ;
, , ;
, .
In order to define a metric/distance on E, we need this concept: a family of semi-norms on E is called sufficient, if and only if from , , there results .
By means of a countable family/sequence of semi-norms, one can define on E the metric by
Indeed, for and imply , . Hence, , which means that the distance between two elements is zero, if and only if the elements coincide. The symmetry is obvious while the triangle inequality for d(x, y) follows from the elementary inequality for nonnegative reals, .
It is interesting to mention the fact that d(x, y) is bounded on by 1, regardless of the (possible) situation when each , is unbounded on E. This is related to the fact that a metric d(x, y), on E, generates another bounded metric , with the same kind of convergence in E.
1.4 FUNCTIONS SPACES
Since our main preoccupation in this book is the study of solutions of various classes of FEs (existence, uniqueness, and local or global behavior), it is useful to give an account on the type of functions spaces we will encounter in subsequent chapters. As proceeded in the preceding sections, we will not provide all the details, but we will indicate adequate sources available in the existing literature.
We shall dwell on the spaces of continuous functions on R, or intervals in R, using the notations that are established in literature. Generally speaking, by C(A, B) we mean the space of continuous maps from A into B, when continuity has a meaning. An index may be used for C, in case we have an extra property to be imposed. This is a list of spaces, consisting of continuous maps, we shall encounter in the book.
C([a, b], Rn) will denote the Banach space of continuous maps from [a, b] into Rn, with the norm
where is the Euclidean norm in Rn. This space is frequently encountered in problems related to FEs, especially when we look for continuous solutions. But even in case of ODEs, the Cauchy problem , , the proof is conducted by showing the existence of a continuous solution to the integral equation . Of course, the differentiability of x(t) follows from the special form of the integral equation, equivalent to Cauchy problem, within the class of continuous functions. A basic property of the space C([a, b], Rn), necessary in the sequel, is the famous criterion of compactness, for subsets , known under the names of Ascoli-Arzelà criterion of compactness in C([a, b], Rn): necessary and sufficient conditions, for the compactness of a set are the boundedness of M and the equicontinuity of its elements on [a, b].
The first property means that for the set, M, there exists a positive number, μ, such that implies , .
The second property means the following: for any , there exists , such that for , , . It is also called equi-uniform continuity.
Proofs can be found in many textbooks, including Corduneanu [123]. The book by Kolmogorov and Fomin [295] contains the criterion but also an interesting proof of Peano’s existence for Cauchy’s problem, without transforming the problem into an integral equation, as a direct application of Ascoli-Arzela’s result.
Let us notice that the compactness result of Ascoli–Arzelà is actually concerned with the concept of relative compactness.
What happens when the interval [a, b] is replaced by , or even ? The supremum norm used in (1.26) cannot be considered on the semi-open interval or on the half-axis .
In this case, in order to obtain a distance between maps defined on , the case , , being totally similar to the half-axis, we shall make recourse to the semi-norms
which obviously form a sufficient family. On the linear space of continuous maps, from into Rn, we have the sufficient family of semi-norms defined by (1.27). Therefore, we can apply formula (1.25), which in this case becomes
with defined by (1.27).
Therefore, the linear space of continuous maps, from into Rn, is a metric space. The property of completeness is the result of the fact that each is a norm on the restricted space, , and this (plus continuity) implies the completeness of the space under discussion, which shall be denoted by and sometimes by , the index denoting that the convergence induced by the metric is uniform on compact sets in .
In summary, a metric structure as indicated earlier, is better—and natural—for , even though normed space (Banach structures are possible and useful for “parts” of .
We shall present now a class of Banach function spaces, denoted , where is a continuous function, whose role is to serve as a weight for the concept of boundedness. Namely, is defined by
The norm in is given by
Obviously, , with Ax in (1.29). It is shown (see, for instance, Corduneanu [120]) that is a Banach space, with the norm given by (1.30). The special case on leads to the space of bounded functions on , with values in Rn, the norm being the supremum norm. This space of bounded continuous functions on is denoted by . It contains as subspaces several important Banach spaces, from into Rn, such as the space of functions with limit at , , which is encountered when we deal with the so-called transient solutions to FEs. This space is usually denoted by , and it is isomorphic and isometric to the space C([0, 1], Rn). Prove this statement! The subspace of , for which is denoted , and it is the space of asymptotic stability (each motion, described by elements of , tends to the equilibrium point ). We will also deal with the subspace of space BC(R, Rn), known as the space of almost periodic functions on R, with values in Rn (Bohr almost periodicity). It is denoted by AP(R, Rn) and contains all continuous maps from R into Rn, such that they can be uniformly approximated on R by vector trigonometric polynomials: for each and , there exists vectors and reals λ1, λ2, …, λm, such that
Inequality (1.31) shows that is as close as we want from oscillatory functions. For the classical types of almost periodic functions, see the books authored by Bohr [72], Besicovitch [61], Favard [208], Levitan [326], Corduneanu [129, 156], Fink [213], Amerio and Prouse [21], Levitan and Zhikov [327], Zaidman [547], and Malkin [355]. Appendix to this book will be dedicated to some new developments (not necessarily continuous functions).
We shall continue to enumerate function spaces, this time, having in mind the measurable functions/elements. These spaces of great importance in the development of modern analysis have appeared at the beginning of the past century, primarily due to Lebesgue’s discovery of measure theory. Actually, the first function spaces amply investigated in the literature are known as Lebesgue’s spaces or Lp-spaces.
The space Lp(R, Rn), , is the linear space of all measurable maps from R into Rn, such that , the ds representing the Lebesgue measure on R. The norm of this space is
The case is characterized by
The theory of these Banach spaces, whose elements are, in fact, equivalence classes of functions (i.e., two functions are equivalent, if and only if they coincide, except on set of points of Lebesgue measure zero) is largely diffused in many books/textbooks available. Let us indicate only Yosida [541], Lang [320], and Amann and Escher [20].
Let us consider now the problem of compactness (or relative compactness) in the space Lp([a, b], Rn), providing a useful result (Riesz):
Let , , be a subset; necessary and sufficient conditions for the (relative) compactness of M are as follows:
M
is a bounded set in
L
p
, that is, , for each ;
as .
We notice the fact that must be extended to be zero, outside [a, b].
There are other criteria of compactness in Lp-spaces, due to Kolmogorov a.o. See, for instance, the items mentioned already, in this section, or Kantorovich and Akilov [274].
Some results concerning Lp-spaces, with a weight function, have been obtained by Milman [399], in connection with stability theory for integral equations.
Also, Kwapisz [305] introduced and applied to integral equations normed spaces of measurable functions, with a mixed norm, such as , on finite intervals or on the semiaxis . Such spaces are useful when fixed-point theorems are used for existence of solutions to FEs. (See Kwapisz [304, 305]).
From the Lp-spaces theory, many other classes of measurable functions have been constructed, with important applications to the theory of FEs. An example, frequently appearing in literature, are the spaces . These spaces occur naturally when dealing with global existence of solutions.
For instance, the space will consist of all measurable maps from R into Rn, such that is determined by
where g(t) is measurable from R to . The norm adequate for this space is, obviously,
By using various weight functions g, one can achieve more generality in regard to the behavior/global properties of the solution.
Finally, we will mention the definition of . This is a linear metric space (Fréchet), which belongs to the larger class of linear locally convex topological spaces.
In order to obtain the linearly invariant distance function for , which consists of all locally integrable maps from R into Rn, that is, such that each integral
we will use a formula similar to (1.28).Since
is a semi-norm, the distance function on will be given (by definition)
The convergence in is, therefore, the L2-convergence on each finite interval of R (or, on each compact set in R).
This definition for can be extended from Rn to Hilbert spaces, or even to Banach spaces.
The variety of function spaces encountered in investigating the solutions of functional differential equations is considerable, and we do not attempt to give a full list. We will mention one more category of function spaces, containing the Lp-spaces, for the sake of their frequent use in the theory of FDE. Apparently, these spaces have been first used by N. Wiener. Their definition and systematic use is given in Massera–Schäffer [374].
The space consists of all locally measurable functions on , such that
The space with the same norm contains only those elements for which
Such functions are important in connection with the concept of almost periodicity and other applications in the theory of FDE.
1.5 SOME NONLINEAR AUXILIARY TOOLS
The development of Functional Analysis has brought to the investigation of various classes of FEs, such as integral equations, for instance, but later to the theory of partial differential equations, many tools and methods. Volterra, who started the systematic investigation of integral equations in the 1890s, considered the problem of existence for these equations as the inversion of an integral operator (generally nonlinear). At the beginning of the twentieth century, Fredholm created the theory of linear integral equations, with special impact on the spectral side. Hilbert went further in this regard, contributing substantially to the birth of the theory of orthogonal series, related to symmetric/hermitian kernels of the Fredholm type equations. Due to remarkable contributions from Fréchet, Riesz (F and M), and other mathematicians, the new field of functional analysis (i.e., dealing with spaces/classes whose elements are functions) made substantial progress, but it was centered around the linear problems.
Starting with the third decade of the twentieth century, essentially nonlinear results have appeared in the literature. One of the first items in nonlinear functional analysis is the theory (or method) of fixed points, generally speaking for nonlinear operators/maps. The history takes us back to Poincaré and Brouwer, when only finite dimensional (Euclidean) spaces were involved. The problem of fixed points was considered by mathematicians starting the third decade of the twentieth century, during which period both best known results have been obtained.
The contraction mapping principle was the first tool, due to Banach in the case of what we call now Banach spaces. The case of complete metric spaces, as it is usually encountered nowadays, is due to V. V. Nemytskii (Uspekhi Mat. Nauk, 1927). This principle, usually encompassing all the results that can be obtained by the iteration method (successive approximations), can be stated as follows:
Let (S, d) denote a complete metric space and a map, such that
for fixed a, . Then, there exists a unique , such that .
The proof of this statement is based on the iterative process
the initial term x0 being arbitrarily chosen in S. One easily shows that is a Cauchy sequence in (S, d), whose limit is x*. The uniqueness follows directly by application of (1.37).
Proofs are available in many books on Functional Analysis or FEs (see, for instance, Corduneanu [120, 135]).
Let us mention that an estimate for the “error” is easily obtained, namely
The fixed point theorem due to Schauder is formulated for Banach spaces and can be stated as follows:
Let B be a Banach space (over reals) and a continuous map/operator such that
with a closed convex set, while TM is relatively compact. Then, there exists at least a fixed point of T, that is, .
The definition of a convex set, say M, is implies , for each .
For a proof of the Schauder fixed-point result, see the proof of a more general result (the next statement) in Corduneanu [120]. Schauder–Tychonoff fixed point theorem. Let E be a locally convex Hausdorff space and a continuous map, such that
where K denotes a convex set, A being compact. Then, there exists at least one , such that .
Both fixed point results formulated already imply the relative compactness of the image , and we mention the fact that uniqueness does not, generally, hold.
The set of fixed points of a map, under appropriate conditions, satisfies certain interesting and useful properties, such as compactness, convexity, and others.
We invite the readers to check the property of compactness, under aforementioned conditions. Another fixed-point result, known as the Leray–Schuader Principle also involves the concept of compactness of the operator, but also the idea of an “a priori” estimate for searched solution. Namely, one considers the equation
with B a Banach space and a compact operator (i.e., taking bounded sets in B, into relatively compact sets). One associates to (1.42) the parameterized equation
assuming that (1.43) is solvable in B for each . Moreover, each solution x satisfies the “a priori” estimate
where K is a fixed number. Then, equation (1.42), which corresponds to in (1.43), possesses a solution in B.
The proof of the principle can be found in Brézis [77] and Zeidler [550, 551], including also some applications.
Several other methods/principles in functional analysis are known and widely applied in the study (particularly, in existence results) of various classes of FEs. We will mention here the method based on monotone operators (see Barbu [45], Deimling [190], and Zeidler [551]).
The definition of a strongly monotone operator , with H a Hilbert space, is
for some . A very useful result can be stated as follows:
Consider in H the equation
with strongly monotone. Further, assume A satisfies on H the Lipschitz condition
Then, for each , Equation (1.45) has a unique solution .
The proof of this result can be found in Zeidler [550], and it is done by means of Banach fixed point (contraction).
An alternate statement of the solvability property of (1.45), for any , is obviously the property of A to be onto H (or surjective).
A somewhat similar result, still working in a Hilbert space H, can be stated as follows: Equation (1.45) is solvable for each , when A is continuous, monotone, that is,
and coercive
See the proof, under slightly more general conditions, in Deimling [190] or Barbu and Precupanu [46].
All aforementioned references, in regard to monotone operators, contain applications to various types of FEs.
Other methods/procedures leading to the existence of solutions of various classes of FEs are based on diverse form of the implicit functions theorem (in Banach spaces and Hilbert Spaces); see Zeidler [550].
When we deal with FDE with finite delay, which we will consider in Chapters 2 and 3, we use another constructive method called the step method. We briefly discuss this method, which is frequently used to construct solutions to FDE of the form
with the delay, or time delay. In order to make the first step in constructing the solution, it is necessary, assuming the initial moment is , to know x(t) on the interval . In other words, one has to associate to (1.49) the initial condition , . Using the notation , , this condition can be written in the form
If we assign the initial function ϕ from a certain function space, say , then equation (1.49) becomes an ODE, on the interval [0, h]:
and the initial condition at will be
The second step, after finding, from (1.51) to (1.52), x(t) on [0, h] will require to solve the ODE (1.49) on [h, 2h], starting at x(h), as found from x(t), on [0, h].
The process continues and, at each step, one finds x(t), on an interval , , solving the ODE (1.49),
under initial condition at mh, as determined from the preceding step. In this way, the solution x(t) of (1.49), appears as a chain, say
each term in (1.54) being found from (1.53), for different values of m, taking as initial value for xm(t), the final value of , . In this way, one obtains a continuous solution on for (1.49)
