Approximate solutions of operator equations

APPROXIMATE SOLUTIONS OF OPERATOR EQUATIONS

APPROXIMATIONS AND DECOMPOSITIONS Editor-in-Chief: CHARLES K. CHUI

Vol. 1: Wavelets: An Elementary Treatment of Theory and Applications Tom H. Koornwinder, ed. Vol. 2: Approximate Kalman Filtering Guanrong Chen, ed. Vol. 3: Multivariate Approximation: From CAGD to Wavelets Kurt Jetter and Florencio I. Utreras, eds. Vol. 4: Advances in Computational Mathematics: New Delhi, India H. P. Dikshit and C. A. Micchelli, eds. Vol. 5: Computational Methods and Function Theory Proceedings of CMFT '94 Conference, Penang, Malaysia R. M. AH, St. Ruscheweyh and E. B. Saff, eds. Vol. 6: Approximation Theory VIII Approximation and Interpolation - Vol. 1 Wavelets and Multi-level Approximation - Vol. 2 C. K. Chui and L. L. Schumaker, eds. Vol. 7: Introduction to the Theory of Weighted Polynomial Approximation H. N. Mhaskar Vol. 8: Advanced Topics in Multivariate Approximation F. Fontanella, K. Jetter and P. J. Laurent, eds. Vol. 9: Approximate Solutions of Operator Equations M. J. Chen, Z. Y. Chen and G. R. Chen

Series in Approximations and Decompositions - Vol. 9

APPROXIMATE SOLUTIONS OF OPERATOR EQUATIONS Mingjun Chen Department of Computer Science Zhongshan University, P P China

Zhongying Chen Department of Computer Science Zhongshan University, P P China

Guanrong Chen Department of Electrical Engineering University of Houston, ,SA

World Scientific Singapore •New Jersey

London• Hong Kong

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Library of Congress Cataloging-in-Publication Data Chen, Mingjun, 1934Approximate solutions of operator equations / by Mingjun Chen, Zhongying Chen, Guanrong Chen. p. cm. — (Series in approximations and decompositions; vol. 9) Includes bibliographical references and index. ISBN 9810230648 (alk. paper) 1. Operator equations ~ Numerical solutions. 2. Approximation theory. I. Chen, Zhongying, 1946II. Chen, G. (Guanrong) III. Title. IV. Series. QA329.C475 1997 515'.724--dc21 97-6029 CIP

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Approximations and Decompositions During the past decade, Approximation Theory has reached out to encompass the approximation-theoretic and computational aspects of several exciting areas in applied mathematics such as wavelets, fractals, neural networks, and computer-aided-geometric design, as well as the modern mathematical development in science and technology. The objective of this book series is to capture this exciting development in the form of monographs, lecture notes, reprint volumes, text books, edited review volumes, and conference proceedings. Approximate Solutions of Operator Equations, the 8th volume of this series, represents one of the computational aspects of Approximation Theory. It emphasizes on efficient algorithms for numerical solutions of differential and integral equations, including various linear and nonlinear as well as evolution equations. The series editor would like to thank the authors for their very fine contributions to this series.

World Scientific Series in

APPROXIMATIONS A N D DECOMPOSITIONS Editor-in-Chief:

CHARLES K. CHUI

Texas A&M University, College Station,

Texas

This page is intentionally left blank

Preface The rapid development in mathematical analysis of complex real-world phys­ ical systems has led to many kinds of linear and nonlinear, ordinary and partial differential, integral or integro-differential, and functional evolution equations, formulated in elementary or abstract function spaces, with initialboundary value conditions and possible additional constraints. The classical approach to solving such equations for analytic solutions has proved impossi­ ble and, in reality, unnecessary with the great advances of modern computer facilities and technology. Approximate solutions to these problems by means of numerical computation have thus become the main stream of research and applications developed in recent years, in both academia and industry. In the endeavor of advancing efficient computational theories and methods, one successful and unified approach is to formulate various apparently different but intrinsically related equations as a certain type of operator equations in Banach spaces of real or complex functions. Approximate solutions to these operator equations can provide deep insights and feasible resolutions to many specific initial-boundary value problems of differential and integral equations. This book is designed as an elementary and self-contained introduc­ tion to some important notions such as the solvability issue, computational schemes, convergence analysis, stability conditions, and error estimates of approximate solutions for several types of operator equations in abstract Ba­ nach spaces. The operator equations studied in this treatise include various linear and nonlinear, ordinary and partial differential, integral and evolu­ tion equations that are frequently encountered in applied mathematics and engineering applications. The book serves also as a textbook for graduate students and as a refer­ ence for researchers and professionals in the fields of approximation theory, numerical analysis, scientific computation, applied mathematics, and engi­ neering. Among the many important and elegant results in the literature of approximate solutions of operator equations, we only include certain elemen­ tary and fundamental topics in this introductory text of modest size. Other results and more advanced topics can be found from the references provided vn

Vlll

Preface

at the end of the book. The presentation is organized as follows. Chapter 1 is an overview of the projection approximation technique, which is of fundamental importance for operator equations throughout the entire text. Projection operators, projective approximation schemes, and their properties are discussed in this chapter. In Chapter 2, compact linear operator equations and their approximate solutions in a Banach space setting are investigated. Projection approxima­ tion of eigenvalues of self-adjoint compact linear operators in Hilbert spaces is also studied. The Fredholm integral equation is used as a case study of the theory and computational methods. General linear operator equations and their perturbation problems are the main topics in Chapter 3. Some sufficient and necessary conditions for unique approximate solvability and stability are derived for bounded linear operator equations in reflexive Banach spaces. Several general computational frameworks for finding generalized solutions of densely defined linear opera­ tor equations are established, in Banach space and/or Hilbert space settings. Applications of these schemes to numerical solutions of boundary value prob­ lems of linear ordinary and partial differential equations are discussed. The basic theory of topological degrees is presented in Chapter 4. It includes the topological degree of continuous operators in Euclidean spaces (the Brouwer degree), topological degree of compact fields in Banach spaces (the Leray-Schauder degree), and the generalized topological degree of the so-called A-proper operators. Some important fixed-point theorems are given in this chapter, with applications to projective approximate solutions of non­ linear integral equations. Chapter 5 provides a review of some fundamental concepts of Calculus in Banach space, and studies the projective approximate solvability problem for monotone and K-monotone nonlinear operator equations, including their perturbation problems. Numerical solutions of boundary value problems of some typical nonlinear elliptic differential equations are discussed along with several examples. Finally, in Chapter 6, both continuous-time and discrete-time (semidiscrete as well as fully discrete) projection approximation methods are de­ veloped for approximate solutions of first and second order abstract evolution equations. Their applications to initial-boundary values problems of differ­ ential evolution equations are illustrated via concrete examples. The present book is a significant expansion and revision of the text Operator Equations and Their Projection Approximate Solutions by the first two authors, published in Chinese by the Guangdong Scientific Publishing Company in 1992. The revision involves complete reorganization, filling in many details, updating with certain new techniques, and inclusion of many illustrative examples and exercises.

Preface

ix

The authors wish to acknowledge several individuals, Professors Ronghua Li, Yuesheng Li, Guoshen Feng, and Jingan Lei, for their interest and en­ couragement during the preparation of this monograph. They would also like to express their gratitude to Professor Charles Chui, the series editor, for his continued support, and to Mrs. Margaret Chui for her assistance in the editorial work.

Guangzhou Houston Summer, 1996

Mingjun Chen Zhongying Chen Guanrong Chen

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Contents Preface

vii

Chapter 1 Introduction

1

1.1 Overview of Different Approximation Methods 1.2 Projection Operators and Their Properties 1.3 Projective Approximation Algorithms (I) 1.4 Projective Approximation Algorithms (II) 1.5 Examples of Projective Approximation Methods Exercises

2 6 7 12 16 21

Chapter 2 Operator Equations and Their Approximate Solutions (I): Compact Linear Operators 27 2.1 Compact Operators and Their Equations 28 2.2 Projection Algorithms: The Banach Space Setting 32 2.3 Approximate Solutions of Fredholm Integral Equation and Boundary Value Problems of Higher-Order Ordinary Differential Equations .. 36 2.3.1 Fredholm integral equation and its approximate solutions 36 2.3.2 Approximate solutions for boundary value problems of higher-order ordinary differential equations 42 2.4 Projection Algorithms: The Hilbert Space Setting 46 Exercises 57 Chapter 3 Operator Equations and Their Approximate Solutions (II): Other Linear Operators 3.1 Bounded Linear Operator Equations and Their Approximate Solvability 3.1.1 The approximate solvability problem 3.1.2 The perturbed operator equation 3.1.3 Operator equations on reflexive Banach space and Hilbert space

xi

63 64 64 68 69

Xll

3.2 Densely Defined Linear Operators and Their Equations 3.2.1 Closable linear operator equation in a Banach space setting 3.2.2 Closed linear operator equation in a Hilbert space setting 3.2.3 Definite linear operator equation in a Hilbert space setting 3.2.4 K-positive definite operator equation in a Hilbert space setting 3.3 Stability of Approximation Schemes 3.4 Numerical Solutions of Boundary Value Problems 3.4.1 Ordinary differential equations 3.4.2 Partial differential equations Exercises

Contents 73 74 76 80 87 94 96 96 100 105

Chapter 4 Topological Degrees and Fixed Point Equations .. .113 4.1 Topological Degrees of Continuous Operators in Euclidean Spaces. 114 4.1.1 Topological degrees of regular operators and their integral representations 114 4.1.2 Basic properties of topological degrees 122 4.1.3 Topological degrees of continuous operators 124 4.2 Topological Degrees of Compact Fields 132 4.3 Generalized Topological Degrees of A-Proper Operators 139 4.4 Fixed Point Theorems 144 4.4.1 Brouwer fixed point and open-set invariant theorems 145 4.4.2 Schauder and Krasnosel'skii fixed point theorems 147 4.4.3 Leray-Schauder fixed point theorem 148 4.4.4 Boundary conditions and fixed point theorems 149 4.5 Approximate Solutions of Nonlinear Fixed Point Equations 151 4.5.1 Projective approximate solvability of fixed point equations 151 4.5.2 Projective solutions of nonlinear integral equations 154 Exercises 158 Chapter 5 Nonlinear Monotone Operator Equations and Their Approximate Solutions 5.1 Continuity, Derivative, and Differential of Operators 5.1.1 Continuity of operators 5.1.2 Derivative and differential of operators 5.2 Monotone Operators from a Banach Space to Its Dual Space 5.2.1 Monotone operators 5.2.2 Monotonicity and semicontinuities 5.2.3 Strongly monotone operators 5.3 Approximate Solvability of Monotone Operator Equations 5.3.1 Monotone operator equations 5.3.2 The perturbation problem 5.3.3 Some remarks on the complex Banach space setting

163 164 164 165 174 174 180 183 185 185 192 194

Contents

xiii

5.4

Solvability and Approximate Solutions of K-Monotone Operator Equations 5.4.1 K-monotone operator equations 5.4.2 The perturbation problem 5.5 Application Examples: Numerical Solutions of Boundary Value Problems Exercises Operator Evolution Equations and Their Projective Approximate Solutions 6.1 Preliminaries 6.1.1 Strongly and weakly measurable functions 6.1.2 Bochner integrals 6.1.3 Abstract functions on the Lp space 6.1.4 Smoothing operator and smooth approximation 6.1.5 Generalized derivatives of abstract functions and the Hm space 6.2 Projective Solutions of First Order Evolution Equations 6.2.1 Initial-boundary value problem of a linear parabolic equation 6.2.2 Continuous-time projection methods 6.2.3 Discrete-time projection methods 6.2.4 Initial-boundary value problems of nonlinear parabolic equations 6.3 Projective Solutions of Second Order Evolution Equations Exercises

195 195 200 202 221

Chapter 6

References

227 228 228 229 229 231 233 235 235 237 252 259 264 280 285

Appendix A:

Fundamental Functional Analysis

289

Appendix B:

Introduction to Sobolev Spaces

317

Subject Index

339

CHAPTER 1 Introduction

The study of operator equations is an important branch of mathematics. The fundamental theory of operator equations, linear or nonlinear, formu­ lated in a Hilbert or a Banach space setting, is originated from the classical theory of differential and integral equations. This modern theory of differen­ tial and integral equations has been well developed in the last few decades, in which many profound concepts, results, methods, and algorithms were established with considerable generality. On the one hand, various types of mathematical equations, such as linear and nonlinear differential, integral, integro-differential, and functional equations, can all be unified under the same framework of abstract operator equations. On the other hand, many well-known theories and methods in Functional Analysis and Operator The­ ory have proven very effective and useful in the study of basic solvability problems in operator equations, including not only the existence and unique­ ness of a solution but also efficient numerical algorithms for approximating the solution. The theory and methodology of operator equations have now played a very important role in computational mathematics, applied sciences and engineering. In applications, one is particularly concerned with finding a solution of an operator equation, either exactly or approximately. Searching for exact ana­ lytic solutions of an operator equation, especially nonlinear, high-dimensional and complicated equations, is always impractical. Instead, numerical or ap­ proximate solutions are very common in modern industries and scientific research, with the aid of the high-speed and high-accuracy computers. There are some mature techniques for approximate solutions of operator equations,

1

2

Chapter 1

including several finite difference methods, the finite element method, spline function methods, and some of their combinations. It has been observed, however, that these methods are not completely independent. As a matter of fact, they are connected via some efficient techniques, where the projection approximation scheme is a very useful one, among others, which is the main theme of study in this treatise. This chapter introduces the projection approximation technique and some of its special schemes for solving operator equations. Meanwhile, it will familiarize the reader the general setting of operator equations. Section 1.1 offers an overview of different approximation methods, such as the Galerkin method, finite element method, collocation method, Galerkin- Petrov method, least-squares method, and Bubnov-Galerkin method. Section 1.2 discusses projection operators and their fundamental properties, which will be essential throughout the entire text. Sections 1.3 and 1.4 introduce projective approx­ imation algorithms and discuss their properties. Some concrete examples are then given in Section 1.5 to illustrate the theory and methods. A certain amount of insightful exercises are provided in the last section, some of which serve also as additional examples for the text. 1.1 Overview of Different Approximation Methods Let X and Y be normed linear spaces and T : X —► Y a nonlinear (not necessarily linear) operator. For a given / e Y, suppose that we want to find a u € X such that Tu = f. (1.1) This is the most general setting of an operator equation formulated in the normed spaces. Under such a general framework, what one can say about the operator Equation (1.1) is that it has a unique solution if and only if the operator T is invertible and, in this case, the solution is given by u = T - 1 / . Otherwise, Equation (1.1) may not even be solvable. Thus, it raises a few questions like under what conditions this equation is solvable (the solvability problem) and, if so, is the solution unique (the uniqueness problem), and moreover, no matter if it is unique or not, how to find a solution (the computation problem). Obviously, in order to be able to answer such questions, more precise description of the equation, such as some properties of the two spaces and the operator as well as certain additional conditions on them, is needed. In the case that finding an analytic solution for Equation (1.1) is very difficult, or even impossible, one tends to find its approximate solutions. Approximation is always possible, however, the questions about approximate solutions are what kind of approximation is good, where the "goodness" has to be first defined, of course, and how to find such a good approximation.

Introduction

3

For Equation (1.1), if both X and Y are infinite-dimensional normed linear spaces, then a natural measure of the goodness for any kind of ap­ proximation would be their norms, and a general approach to finding a good approximation would be the following: First, we select a sequence of linear subspaces {Xn} of X and a sequence of linear subspaces {Yn} of Y, and select two associate mappings Pn : X —► Xn and Qn : Y —► Y^, respectively, accord­ ing to certain meaningful criteria (to be further discussed later). Then we use the mappings Qn : Y —» Y^ to define a sequence of operators Tn : X n —► Yn by Tn = Q n T | X n , (1.2) where, for any operator A defined on any normed linear space Z, A\z0 denotes the restriction of A on a subspace Zo of Z. Finally, we use Tnun = Qnf,

uneXn,

n = l , 2 , •••,

(1.3)

as a sequence of approximate operator equations for the exact operator Equation (1.1). If a computational scheme can be found to compute a se­ quence of solutions {un} of the operator Equations (1.3), then we call it an approximation scheme for the operator Equation (1.1), and denote it by 1 n

=

(^U)

Mii

* n , Cjn j .

In order for a well-defined approximation scheme to be useful and effec­ tive, several important questions have to be addressed: (1) Does the approximate Equation (1.3) have a (unique) solution for each ra? (2) If each equation in (1.3) has a solution un, then does the solution se­ quence {un} converge under certain measure? (3) If the solution sequence {un} converges under the chosen measure, is its limit the (unique) solution of the exact Equation (1.1)? A definite answer to these three questions is referred to the approximate solvability of the operator Equation (1.1). To be more precise, we give the following definitions. Definition 1.1. In the normed linear space X, let its norm be denoted by || • ||x- A sequence un € X is said to converge strongly to a u € X, denoted s un — ► u or simply un —► u, if lim

\\un-u\\x=0.

n—*-oo

A sequence un G X is said to converge weakly to a u € X, denoted un w-^ u if lim l{un - u) = 0, V I e X* ,

4

Chapter 1

where X* is the dual space of X. Definition 1.2. The operator Equation (1.1) is said to be uniquely and strongly (resp., weakly) approximate-solvable under the approximation scheme T n , if there exists an integer N > 0 such that the operator Equation (1.3) Tn, has a unique solution un e€ Xn for every n > N, and the sequence {«„} {un} converges strongly (resp., weakly) to a u € X, where u is the unique solution of Equation (1.1). Definition 1.3. The operator Equation (1.1) is said to be strongly (resp., weakly) approximate-solvable weaKiyj approximate-soivaoie under under the trie approximation approximation scheme scheme T T nn,, ifn there there exists an integer N > 0 such that the operator Equation (1.3) has exists an integer N > 0 such that the operator Equation (1.3) has aa solution solution uun €€ X Xnn for for every every nn > > N, N, and and the the sequence sequence {u {unn}} has has aa subsequence subsequence convergconvergn ing strongly (resp., weakly) to a u € X, where u is a ing strongly (resp., weakly) to a u € X, where u is a solution solution of of Equation Equation (i.i). (i.i). Obviously, vuviv-iuoij', in i n order u i u c i to I U gguarantee u c n a i i t c t ; the LIIC approximate <xppiUAiiiicii;c; solvability ot-uveiL/iiiLy and ciiua the tiic (strong or or weak) weak) convergence convergence of of aa sequence sequence of of approximate approximate solutions, solutions, the the (strong four elements elements in in the the approximation approximation scheme scheme T Tnn = = {X {Xnn,P ,Pnn;Y ;Ynn,Q ,Qnn}} have to to four have satisfy satisfy certain certain conditions. conditions. For For example, example, aa necessary necessary condition condition is is U™=1 X. . U~ tXX nn = X

(1.4) (1.4)

This condition says that the union of the sequence {Xn} is ultimately dense in X. Other conditions will be studied later in the following chapters for different specific problems. For most approximation schemes, the operators Pn and Qn introduced above are linear operators. If P Xn and Q„ Qn : Y -— >„ >n ara linear P.n : X -» X„ projection operators, then the corresponding approximation scheme gives a projection computational method, and the scheme is called a projective approximation algorithm. In particular, according to different conditions on the operator T and the four r„ = {X ;Y }, n}, we we have have the the following following four elements elements in in the the scheme scheme F_ Tn = = fX.. {Xnn,P ,PnnP-.:K..O„^ ; n,Q Ynn,Q classification [24]: (i) If X = Y and Xn =Yn for all n = = 1,2, • • •, ,hen this projecttve scheme gives the well-known Galerkin method. (ii) If T is a differential or integral operator, and Xn are spaces of spline functions, then this yields the finite element method. (iii) If X and Y are some function spaces defined on a region fi C Rm, and Qn :Y -> Yn are interpolation operators denned by n n

x x {Qny) (x) = Y^y( i)M ) >

Introduction

5

where x, xu • • •, xn € ft, y(x) € Y and {^i, • • •, ipn} are a basis of Yn, then the approximate operator Equation (1.3) becomes n

n

53 Tti »L=*>( x ) = 5^/ L >i(x) =x

for all x € ft, which is equivalent to Tw

«L =JBi =/L =a ..>

*=l,---,n.

(1.5)

This is the collocation method. (iv) If X and Y are Hilbert spaces with the inner product (♦, •) and Xn = span {i, • •, <j)n}

and

Yn = span {i, • • •, tpn} ,

respectively, and P n and Qn are both orthogonal projection operators, then the approximate operator Equation (1.3) is equivalent to (Tun - f,i/>i) = 0 ,

i = l,-..,n.

(1.6)

This gives the Galerkin-Petrov method. Especially, if ipi = Mi>i = 1,2, • • •, for some appropriately chosen linear operator M : X —> Y, then this is the moment method. On the other hand, if T : X —> Y is a linear operator and ^ = T X is called a projection operator if it satisfies the identity P2 = P on X. Throughout this book, we use the notation P(-)» ^ ( ) and Af(-) for the domain, range and null set of an operator, where the null set of a linear operator is a linear space and so is also called a null space or kernel. The following are basic properties of a projection operator. Since they are well-known fundamental results in functional analysis [13, 39, 40], we only state the results without proofs. Projection operators on Hilbert spaces have some other elementary yet more precise properties which are very useful in applications. Proposition 1.1. Let P : X —> X be a projection operator. Then (i) X = n(P)®Af{P); (ii) K{P) = M(I - P), X(P) = 11(1 - P); and (hi) Px = x, Vxeft(P); where I is the identity operator and © is the direct sum of two subspaces. Proposition 1.2. Let M and N be closed subspaces of X, with X =

M®N.

If there exists a bounded linear operator P : X —► X such that Px = x,

Vx € M

and

Px = 0,

V x e N,

then P is a projection operator, with M = 7£(P) and N = Af(P). Prom an application point of view, the following question is important: for a finite-dimensional subspace M of a Banach space X, can we find a projection operator P : X —► X with K(P) = Ml To answer this question, we suppose that the dimension of M is n > 1 with a basis {e1, • • •, e n } , namely, M = span {ei, • • - , e n } . According to the Hahn-Banach theorem [39, 40] (see Appendix A.6.1), there exist elements £±, • • •, £n e X*, the dual space of X, such that £i(ej) = 6ij ,

i,j = 1, ••-,ri,

Introduction ntroduction Introduction

77

where /here % is the Kronecker delta. Now, define a mapping P : X -> M by n

P x = ^y "44( (xx))eeff cc ,

V x1 €6 XI .

fc=l fe=i

Then, it can be verified that P is a bounded linear operator, SE satisfying

pSx = X>(*)^ = £ £ ^ ( ^ ; n

= = J2e^i(x)e x Wi = =P Px, Px, x, = V£>(x)e i =

V V xx €e X .. VxeX.

i=l

This This means that the answer to the above question is ves. yes. We summarize this result in the following proposition. Proposition Proposition 1.3. any Mite-dimensional finite-dimensional Proposition 1.3. For tor any nnite-aimensionai X, X, there exists a projection such X, there exists a projection operator operator P P such

subspace M space suospace m of or aa Banach tsanacn space that 1l{P) = M. that U(P) = M.

1.3 Projective Approximation Algorithms (I) Now, operator Equation Equation (1.1) (1.1) and and its its approximation approximation scheme scheme iiow, we we return return to to the me operator P ;Y , = {X , P„;Y Qn } defined in (1.2)-(1.3). For a projective approximation n n n n Irr „„ = (A„,P„;y ,