Preface ....................................................................... v List of Figures ............................................................ xvii 1 Metric Spaces and Banach Fixed Point Theorem ............................... 1 1.1 Introduction ........................................................... 1 1.2 Banach Contraction Fixed Point Theorem ................................. 1 1.3 Application of Banach Contraction Mapping Theorem ...................... 7 1.3.1 Application to Real-valued Equation .............................. 7 1.3.2 Application to Matrix Equation ................................... 8 1.3.3 Application to Integral Equation ................................ 10 1.3.4 Application to Differential Equation ............................ 12 1.4 Problems .............................................................. 14 References ................................................................... 17 2 Banach Spaces ............................................................. 19 2.1 Introduction .......................................................... 19 2.2 Definitions and Examples of Normed and Banach Spaces .................. 19 2.2.1 Examples of Normed and Banach Spaces ............................ 21 2.3 Basic Properties-Closure, Denseness and Separability .................. 25 2.3.1 Closed, Dense and Separable Sets ................................ 25 2.3.2 Riesz Theorem and Construction of a New Banach Space ............ 27 2.3.3 Dimension of Normed Spaces ...................................... 27 2.3.4 Open and Closed Spheres ......................................... 28 2.4 Bounded and Unbounded Operators ....................................... 32 2.4.1 Definitions and Examples ........................................ 32 2.4.2 Properties of Linear Operators .................................. 40 2.4.3 Unbounded Operators ............................................. 48 2.5 Representation of Bounded and Linear Functionals ...................... 50 2.6 Algebra of Operators .................................................. 51 2.7 Convex Functional ..................................................... 57 2.7.1 Convex Sets ..................................................... 58 2.7.2 Affine Operator ................................................. 60 2.7.3 Lower Semicontinuous and Upper Semicontinuous Functionals ....... 64 2.8 Problems .............................................................. 65 2.8.1 Solved Problems ................................................. 65 2.8.2 Unsolved Problems ............................................... 76 References ................................................................... 83 3 Hilbert Space ............................................................. 85 3.1 Introduction .......................................................... 85 3.2 Basic Definition and Properties ....................................... 86 3.2.1 Definitions, Examples and Properties of Inner Product Space ..... 86 3.2.2 Parallelogram Law and Characterization of Hilbert Space ......... 93 3.3 Orthogonal Complements and Projection Theorem ......................... 95 3.3.1 Orthogonal Complements and Projections .......................... 95 3.4 Orthogonal Projections and Projection Theorem ......................... 99 3.5 Projection on Convex Sets ............................................ 107 3.6 Orthonormal Systems and Fourier Expansion ............................ 110 3.7 Duality and Reflexivity .............................................. 118 3.7.1 Riesz Representation Theorem ................................... 118 3.7.2 Reflexivity of Hilbert Spaces .................................. 122 3.8 Operators in Hilbert Space ........................................... 124 3.8.1 Adjoint of Bounded Linear Operators on a Hilbert Space ......... 124 3.8.2 Self-Adjoint, Positive, Normal and Unitary Operators ........... 130 3.8.3 Adjoint of an Unbounded Linear Operator ........................ 139 3.9 Bilinear Forms and Lax-Milgram Lemma ................................. 142 3.9.1 Basic Properties ............................................... 142 3.10 Problems ............................................................ 151 3.10.1 Solved Problems ............................................... 151 3.10.2 Unsolved Problems ............................................. 160 References .................................................................. 165 4 Fundamental Theorems ..................................................... 167 4.1 Introduction ......................................................... 167 4.2 Hahn-Banach Theorem .................................................. 168 4.2.1 Extension Form of Hahn-Banach Theorem .......................... 168 4.2.2 Extension Form of the Hahn-Banach Theorem ...................... 173 4.3 Topologies on Normed Spaces .......................................... 178 4.3.1 Strong and Weak Topologies ..................................... 178 4.4 Weak Convergence ..................................................... 179 4.4.1 Weak Convergence in Banach Spaces .............................. 179 4.4.2 Weak Convergence in Hilbert Spaces ............................. 183 4.5 Banach-Alaoglu Theorem ............................................... 185 4.6 Principle of Uniform Boundedness and Its Applications ................ 187 4.6.1 Principle of Uniform Boundedness ............................... 187 4.7 Open Mapping and Closed Graph Theorems ............................... 189 4.7.1 Graph of a Linear Operator and Closedness Property ............. 189 4.7.2 Open Mapping Theorem ........................................... 192 4.7.3 The Closed-Graph Theorem ....................................... 193 4.8 Problems ............................................................. 194 4.8.1 Solved Problems ................................................ 194 4.8.2 Unsolved Problems .............................................. 196 References .................................................................. 199 5 Differential and Integral Calculus in Banach Spaces ...................... 201 5.1 Introduction ......................................................... 201 5.2 The Gateaux and Frechet Derivatives .................................. 201 5.2.1 The Gateaux Derivative ......................................... 201 5.2.2 The Frechet Derivative ......................................... 206 5.3 Generalized Gradient (Subdifferential) ............................... 215 5.4 Some Basic Results from Distribution Theory and Sobolev Spaces ....... 217 5.4.1 Distributions .................................................. 218 5.4.2 Sobolev Space .................................................. 232 5.4.3 The Sobolev Embedding Theorems ................................. 238 5.5 Integration in Banach Spaces ......................................... 241 5.6 Problems ............................................................. 245 5.6.1 Solved Problems ................................................ 245 5.7 Unsolved Problems .................................................... 250 References .................................................................. 255 6 Optimization Problems .................................................... 257 6.1 Introduction ......................................................... 257 6.2 General Results on Optimization ...................................... 257 6.3 Special Classes of Optimization Problems ............................. 261 6.3.1 Convex, Quadratic and Linear Programming ....................... 261 6.3.2 Calculus of Variations and Euler-Lagrange Equation ............. 262 6.3.3 Minimization of Energy Functional (Quadratic Functional) ....... 263 6.4 Algorithmic Optimization ............................................. 265 6.4.1 Newton Algorithm and Its Generalization ........................ 265 6.4.2 Conjugate Gradient Method ...................................... 275 6.5 Problems ............................................................. 277 References .................................................................. 281 7 Operator Equations and Variational Methods ............................... 283 7.1 Introduction ......................................................... 283 7.2 Boundary Value Problems .............................................. 283 7.3 Operator Equations and Solvability Conditions ........................ 287 7.3.1 Equivalence of Operator Equation and Minimization Problem ...... 287 7.3.2 Solvability Conditions ......................................... 289 7.3.3 Existence Theorem for Nonlinear Operators ...................... 292 7.4 Existence of Solutions of Dirichlet and Neumann Boundary Value Problems ............................................................. 293 7.5 Approximation Method for Operator Equations .......................... 297 7.5.1 Galerkin Method ................................................ 297 7.5.2 Rayleigh-Ritz-Galerkin Method .................................. 300 7.6 Eigenvalue Problems .................................................. 301 7.6.1 Eigenvalue of Bilinear Form .................................... 301 7.6.2 Existence and Uniqueness ....................................... 302 7.7 Boundary Value Problems in Science and Technology .................... 303 7.8 Problems ............................................................. 309 References .................................................................. 313 8 Finite Element and Boundary Element Methods .............................. 315 8.1 Introduction ......................................................... 315 8.2 Finite Element Method ................................................ 318 8.2.1 Abstract Problem and Error Estimation .......................... 318 8.2.2 Internal Approximation of H1 (Ω) ............................... 325 8.2.3 Finite Elements ................................................ 327 8.3 Applications of the Finite Method in Solving Boundary Value Problems ............................................................. 332 8.4 Basic Ingredients of Boundary Element Method ......................... 338 8.4.1 Weighted Residuals Method ...................................... 338 8.4.2 Inverse Problem and Boundary Solutions ......................... 340 8.4.3 Boundary Element Method ........................................ 341 8.5 Problems ............................................................. 349 References .................................................................. 353 9 Variational Inequalities and Applications ................................ 359 9.1 Motivation and Historical Remarks .................................... 359 9.1.1 Contact Problem (Signorini Problem) ............................ 360 9.1.2 Variational Inequalities in Social, Financial and Management Sciences ....................................................... 361 9.2 Variational Inequalities and Their Relationship with Other Problems ............................................................. 362 9.2.1 Classes of Variational Inequalities ............................ 362 9.2.2 Formulation of a Few Problems in Terms of Variational Inequalities ................................................... 364 9.3 Elliptic Variational Inequalities ..................................... 370 9.3.1 Lions-Stampacchia Theorem ...................................... 371 9.3.2 Variational Inequalities for Monotone Operators ................ 373 9.4 Finite Element Methods for Variational Inequalities .................. 380 9.4.1 Convergence and Error Estimation ............................... 380 9.4.2 Error Estimation in Concrete Cases ............................. 384 9.5 Evolution Variational Inequalities and Parallel Algorithms ........... 386 9.5.1 Solution of Evolution Variational Inequalities ................. 386 9.5.2 Decomposition Method and Parallel Algorithms ................... 389 9.6 Obstacle Problem ..................................................... 396 9.6.1 Obstacle Problem ............................................... 396 9.6.2 Membrane Problem ............................................... 398 9.7 Problems ............................................................. 400 References .................................................................. 403 10 Wavelet Theory ........................................................... 407 10.1 Introduction ........................................................ 407 10.2 Continuous and Discrete Wavelet Transforms .......................... 409 10.2.1 Continuous Wavelet Transforms ................................ 409 10.2.2 Discrete Wavelet Transform and Wavelet Series ................ 420 10.3 Multiresolution Analysis, Wavelets Decomposition and Reconstruction ...................................................... 426 10.3.1 Multiresolution Analysis (MRA) ............................... 426 10.3.2 Decomposition and Reconstruction Algorithms .................. 430 10.3.3 Connection with Signal Processing ............................ 434 10.3.4 The Fast Wavelet Transform Algorithm ......................... 437 10.4 Wavelets and Smoothness of Functions ................................ 438 10.4.1 Lipschitz Class and Wavelets ................................. 438 10.4.2 Approximation and Detail Operators ........................... 442 10.4.3 Scaling and Wavelet Filters .................................. 449 10.4.4 Approximation by MRA Associated Projections .................. 457 10.5 Compactly Supported Wavelets ........................................ 460 10.5.1 Daubechies Wavelets .......................................... 460 10.5.2 Approximation by Family of Daubechies Wavelets ............... 466 10.6 Wavelet Packets ..................................................... 476 10.7 Problems ............................................................ 477 References .................................................................. 481 11 Wavelet Method for Partial Differential Equations and Image Processing ... 485 11.1 Introduction ........................................................ 485 11.2 Wavelet Methods in Partial Differential and Integral Equations ...... 486 11.2.1 Introduction ................................................. 486 11.2.2 General Procedure ............................................ 487 11.2.3 Miscellaneous Examples ....................................... 491 11.2.4 Error Estimation Using Wavelet Basis ......................... 497 11.3 Introduction to Signal and Image Processing ......................... 499 11.4 Representation of Signals by Frames ................................. 501 11.4.1 Functional Analytic Formulation .............................. 501 11.4.2 Iterative Reconstruction ..................................... 502 11.5 Noise Removal from Signals .......................................... 505 11.5.1 Introduction ................................................. 505 11.5.2 Model and Algorithm .......................................... 506 11.6 Wavelet Methods for Image Processing ................................ 510 11.6.1 Besov Space .................................................. 511 11.6.2 Linear and Nonlinear Image Compression ....................... 512 11.7 Problems ............................................................ 515 References .................................................................. 519 Appendices .................................................................. 523 A Set Theoretic Concepts .................................................... 523 B Topological Concepts ...................................................... 529 C Elements of Metric Spaces ................................................. 533 D Notations and Definitions of Concrete Spaces .............................. 539 E Vector Spaces ............................................................. 551 F Fourier Analysis .......................................................... 555 References .................................................................. 563 Symbols and Abbreviations ................................................... 565 Index ....................................................................... 567