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