非线性泛函分析及其应用 第2B卷 非线性单调算子 英文版PDF电子书下载

GENERALIZATION TO NONLINEAR STATIONARY PROBLEMS 469

Basic Ideas of the Theory of Monotone Operators 471

CHAPTER 25 Lipschitz Continuous,Strongly Monotone Operators,the Projection-Iteration Method,and Monotone Potential Operators 495

25.1．Sequences of k-Contractive Operators 497

25.2．The Projection-Iteration Method for k-Contractive Operators 499

25.3．Monotone Operators 500

25.4．The Main Theorem on Strongly Monotone Operators,and the Projection-Iteration Method 503

25.5．Monotone and Pseudomonotone Operators,and the Calculus of Variations 506

25.6．The Main Theorem on Monotone Potential Operators 516

25.7．The Main Theorem on Pseudomonotone Potential Operators 518

25.8．Application to the Main Theorem on Quadratic Variational Inequalities 519

25.9．Application to Nonlinear Stationary Conservation Laws 521

25.10．Projection-Iteration Method for Conservation Laws 527

25.11．The Main Theorem on Nonlinear Stationary Conservation Laws 535

25.12．Duality Theory for Conservation Laws and Two-sided aposteriori Error Estimates for the Ritz Method 537

25.13．The Ka？arov Method for Stationary Conservation Laws 542

25.14．The Abstract Ka？anov Method for Variational Inequalities 545

CHAPTER 26 Monotone Operators and Quasi-Linear Elliptic Differential Equations 553

26.1．Hemicontinuity and Demicontinuity 554

26.2．The Main Theorem on Monotone Operators 556

26.3．The Nemyckii Operator 561

26.4．Generalized Gradient Method for the Solution of the Galerkin Equations 564

26.5．Application to Quasi-Linear Elliptic Differential Equations of Order 2m 567

26.6．Proper Monotone Operators and Proper Quasi-Linear Elliptic Differential Operators 576

CHAPTER 27 Pseudomonotone Operators and Quasi-Linear Elliptic Differential Equations 580

27.1．The Conditions(M)and(S),and the Convergence of the Galerkin Method 583

27.2．Pseudomonotone Operators 585

27.3．The Main Theorem on Pseudomonotone Operators 589

27.4．Application to Quasi-Linear Elliptic Differential Equations 590

27.5．Relations Between Important Properties of Nonlinear Operators 595

27.6．Dual Pairs of B-Spaces 598

27.7．The Main Theorem on Locally Coercive Operators 598

27.8．Application to Strongly Nonlinear Differential Equations 604

CHAPTER 28 Monotone Operators and Hammerstein Integral Equations 615

28.1．A Factorization Theorem for Angle-Bounded Operators 619

28.2．Abstract Hammerstein Equations with Angle-Bounded Kernel Operators 620

28.3．Abstract Hammerstein Equations with Compact Kernel Operators 625

28.4．Application to Hammerstein Integral Equations 627

28.5．Application to Semilinear Elliptic Differential Equations 632

CHAPTER 29 Noncoercive Equations,Nonlinear Fredholm Alternatives,Locally Monotone Operators,Stability,and Bifurcation 639

29.1．Pseudoresolvent,Equivalent Coincidence Problems,and the Coincidence Degree 643

29.2．Fredholm Alternatives for Asymptotically Linear,Compact Perturbations of the Identity 650

29.3．Application to Nonlinear Systems of Real Equations 652

29.4．Application to Integral Equations 653

29.5．Application to Differential Equations 653

29.6．The Generalized Antipodal Theorem 654

29.7．Fredholm Alternatives for Asymptotically Linear(S)-Operators 657

29.8．Weak Asymptotes and Fredholm Alternatives 657

29.9．Application to Semilinear Elliptic Differential Equations of the Landesman-Lazer Type 661

29.10．The Main Theorem on Nonlinear Proper Fredholm Operators 665

29.11．Locally Strictly Monotone Operators 677

29.12．Locally Regularly Monotone Operators,Minima,and Stability 679

29.13．Application to the Buckling of Beams 697

29.14．Stationary Points of Functionals 706

29.15．Application to the Principle of Stationary Action 708

29.16．Abstract Statical Stability Theory 709

29.17．The Continuation Method 712

29.18．The Main Theorem of Bifurcation Theory for Fredholm Operators of Variational Type 712

29.19．Application to the Calculus of Variations 722

29.20．A General Bifurcation Theorem for the Euler Equations and Stability 730

29.21．A Local M ultiplicity Theorem 733

29.22．A Global Multiplicity Theorem 735

GENERALIZATION TO NONLINEAR NONSTATIONARY PROBLEMS 765

CHAPTER 30 First-Order Evolution Equations and the Galerkin Method 767

30.1．Equivalent Formulations of First-Order Evolution Equations 767

30.2．The Main Theorem on Monotone First-Order Evolution Equations 770

30.3．Proof of the Main Theorem 771

30.4．Application to Quasi-Linear Parabolic Differential Equations of Order 2m 779

30.5．The Main Theorem on Semibounded Nonlinear Evolution Equations 783

30.6．Application to the Generalized Korteweg-de Vries Equation 790

CHAPTER 31 Maximal Accretive Operators,Nonlinear Nonexpansive Semigroups,and First-Order Evolution Equations 817

31.1．The Main Theorem 819

31.2．Maximal Accretive Operators 820

31.3．Proof of the Main Theorem 822

31.4．Application to Monotone Coercive Operators on B-Spaces 827

31.5．Application to Quasi-Linear Parabolic Differential Equations 829

31.6．A Look at Quasi-Linear Evolution Equations 830

31.7．A Look at Quasi-Linear Parabolic Systems Regarded as Dynamical Systems 832

CHAPTER 32 Maximal Monotone Mappings 840

32.1 Basic Ideas 843

32.2．Definition of Maximal Monotone Mappings 850

32.3．Typical Examples for Maximal Monotone Mappings 854

32.4．The Main Theorem on Pseudomonotone Perturbations of Maximal Monotone Mappings 866

32.5．Application to Abstract Hammerstein Equations 873

32.6．Application to Hammerstein Integral Equations 874

32.7．Application to Elliptic Variational Inequalities 874

32.8．Application to First-Order Evolution Equations 876

32.9．Application to Time-Periodic Solutions for Quasi-Linear Parabolic Differential Equations 877

32.10．Application to Second-Order Evolution Equations 879

32.11．Regularization of Maximal Monotone Operators 881

32.12．Regularization of Pseudomonotone Operators 883

32.13．Local Boundedness of Monotone Mappings 884

32.14．Characterization of the Surjectivity of Maximal Monotone Mappings 886

32.15．The Sum Theorem 888

32.16．Application to Elliptic Variational Inequalities 892

32.17．Application to Evolution Variational Inequalities 893

32.18．The Regularization Method for Nonuniquely Solvable Operator Equations 894

32.19．Characterization of Linear Maximal Monotone Operators 897

32.20．Extension of Monotone Mappings 899

32.21．3-Monotone Mappings and Their Generalizations 901

32.22．The Range of Sum Operators 906

32.23．Application to Hammerstein Equations 908

32.24．The Characterization of Nonexpansive Semigroups in H-Spaces 909

CHAPTER 33 Second-Order Evolution Equations and the Galerkin Method 919

33.1．The Original Problem 921

33.2．Equivalent Formulations of the Original Problem 921

33.3．The Existence Theorem 923

33.4．Proof of the Existence Theorem 924

33.5．Application to Quasi-Linear Hyperbolic Differential Equations 928

33.6．Strong Monotonicity,Systems of Conservation Laws,and Quasi-Linear Symmetric Hyperbolic Systems 930

33.7．Three Important General Phenomena 934

33.8．The Formation of Shocks 935

33.9．Blowing-Up Effects 937

33.10．Blow-Up of Solutions for Semilinear Wave Equations 944

33.11．A Look at Generalized Viscosity Solutions of Hamilton-Jacobi Equations 947

GENERAL THEORY OF DISCRETIZATION METHODS 959

CHAPTER 34 Inner Approximation Schemes,A-Proper Operators,and the Galerkin Method 963

34.1．Inner Approximation Schemes 963

34.2．The Main Theorem on Stable Discretization Methods with Inner Approximation Schemes 965

34.3．Proof of the Main Theorem 968

34.4．Inner Approximation Schemes in H-Spaces and the Main Theorem on Strongly Stable Operators 969

34.5．Inner Approximation Schemes in B-Spaces 972

34.6．Application to the Numerical Range of Nonlinear Operators 974

CHAPTER 35 External Approximation Schemes,A-Proper Operators,and the Difference Method 978

35.1．External Approximation Schemes 980

35.2．Main Theorem on Stable Discretization Methods with External Approximation Schemes 982

35.3．Proof of the Main Theorem 984

35.4．Discrete Sobolev Spaces 985

35.5．Application to Difference Methods 988

35.6．Proof of Convergence 990

CHAPTER 36 Mapping Degree for A-Proper Operators 997

36.1．Definition of the Mapping Degree 998

36.2．Properties of the Mapping Degree 1000

36.3．The Antipodal Theorem for A-Proper Operators 1000

36.4．A General Existence Principle 1001

Appendix 1009

References 1119

List of Symbols 1163

List of Theorems 1174

List of the Most Important Definitions 1179

List of Schematic Overviews 1182

List of Important Principles 1183

Index 1189