Methods of Nonlinear Analysis

Methods of Nonlinear Analysis ......Page 2
Copyright Page ......Page 3
Contents ......Page 14
Preface ......Page 6
1.1. Introduction ......Page 22
1.2. The First-order Linear Differential Equation ......Page 23
1.3. Fundamental Inequality ......Page 24
1.4. Second-order Linear Differential Equations ......Page 26
1.5. Inhomogeneous Equation ......Page 28
1.6. Lagrange Variation of Parameters ......Page 29
1.7. Two-point Boundary Value Problem ......Page 31
1.8. Connection with Calculus of Variations ......Page 32
1.9. Greens Functions ......Page 33
1.10. Riccati Equation ......Page 35
1.11. The Cauchy…Schwarz Inequality ......Page 37
1.12. Perturbation and Stability Theory ......Page 39
1.13. A Counter-example ......Page 41
1.14. ∫^∞ | f'(t)| dt < ∞ ......Page 42
1.15. ∫^∞|f'(t)| dt < ∞ ......Page 43
1.1 6. Asymptotic Behavior ......Page 44
1.17. The Equation u"… (1 + f (t))u = 0 ......Page 45
1.18. More Refined Asymptotic Behavior ......Page 47
1.19. ∫^∞ f^2 dt < ∞ ......Page 48
1.20. The Second Solution ......Page 50
1.21. The Liouville Transformation ......Page 51
1.22. Elimination of Middle Term ......Page 52
1.25. uŽ + (1 + f (t))u = 0; Asymptotic Behavior ......Page 54
1.26. Asymptotic Series ......Page 56
1.27. The Equation u = p(u, t)/q(u, t ) ......Page 58
1.28. Monotonicity of Rational Functions of u and t ......Page 59
1.29. Asymptotic Behavior of Solutions of u = p(u, t)/q( u, t ) ......Page 60
Miscellaneous Exercises ......Page 63
Bibliography and Comments ......Page 72
2.1. Introduction ......Page 75
2.2. Determinantal Solution ......Page 76
2.3. Elimination ......Page 79
2.4. Ill-conditioned Systems ......Page 80
2.6. Vector Notation ......Page 81
2.8. Vector Inner Product ......Page 82
2.9. Matrix Notation ......Page 84
2.10. Noncommutativity ......Page 85
2.12. The Inverse Matrix ......Page 86
2.13. Matrix Norm ......Page 88
2.14. Relative Invariants ......Page 89
2.15. Constrained Minimization ......Page 92
2.16. Symmetric Matrices ......Page 93
2.17. Quadratic Forms ......Page 95
2.18. Multiple Characteristic Roots ......Page 96
2.19. Maximization and Minimization of Quadratic Forms ......Page 97
2.20. Min-Max Characterization of the λ_k ......Page 98
2.21. Positive Definite Matrices ......Page 100
2.22. Determinantal Criteria ......Page 102
2.24. Canonical Representation for Arbitrary A ......Page 103
2.25. Perturbation of Characteristic Frequencies ......Page 105
2.26. Separation and Reduction of Dimensionality ......Page 106
2.27. Ill-conditioned Matrices and Tychonov Regularization ......Page 107
2.29. Positive Matrices ......Page 109
2.30. Variational Characterization of λ(A) ......Page 110
2.31. Proof of Minimum Property ......Page 112
2.32. Equivalent Definition of λ(A) ......Page 113
Miscellaneous Exercises ......Page 115
Bibliography and Comments ......Page 122
3.2. Vector-Matrix Calculus ......Page 125
3.3. Existence and Uniqueness of Solution ......Page 126
3.4. The Matrix Exponential ......Page 128
3.5. Commutators ......Page 129
3.6. Inhomogeneous Equation ......Page 131
3.7. The Euler Solution ......Page 132
3.8. Stability of Solution ......Page 134
3.9. Linear Differential Equation with Variable Coefficients ......Page 135
3.10. Linear Inhomogeneous Equation ......Page 137
3.12. The Equation X' = AX + X B ......Page 139
3.13. Periodic Matrices: the Floquet Representation ......Page 141
3.14. Calculus of Variations ......Page 142
3.15. Two-point Boundary Condition ......Page 143
3.17. The Matrix Riccati Equation ......Page 144
3.18. Kronecker Products and Sums ......Page 145
3.19. AX + XB = C ......Page 146
Miscellaneous Exercises ......Page 148
Bibliography and Comments ......Page 152
4.1. Introduction ......Page 155
4.2. Dini-Hukuhara Theorem„I ......Page 156
4.3. Dini-Hukuhara Theorem„II ......Page 159
4.5. Existence and Uniqueness of Solution ......Page 161
4.6. Poincaré-Lyapunov Stability Theory ......Page 163
4.7. Proof of Theorem ......Page 164
4.8. Asymptotic Behavior ......Page 167
4.9. The Function φ(c) ......Page 169
4.10. More Refined Asymptotic Behavior ......Page 170
4.11. Analysis of Method of Successive Approximations ......Page 171
4.13. Time-dependent Equations over Finite Intervals ......Page 173
4.14. Alternative Norm ......Page 176
4.1 5. Perturbation Techniques ......Page 177
4.17. Solution of Linear Systems ......Page 178
4.18. Origins of Two-point Boundary Value Problems ......Page 179
4.19. Stability Theorem for Two-point Boundary Value Problem ......Page 180
4.20. Asymptotic Behavior ......Page 181
4.21. Numerical Aspects of Linear Two-point Boundary Value Problems ......Page 182
4.22. Difference Methods ......Page 184
4.24. Proof of Stability ......Page 186
4.25. Analysis of Stability Proof ......Page 187
4.27. Irregular Stability Problems ......Page 189
4.28. The Emden…Fowler…Fermi…Thomas Equation ......Page 191
Miscellaneous Exercises ......Page 192
Bibliography and Comments ......Page 203
5.1. Introduction ......Page 208
5.2. Example of the Bubnov…Galerkin Method ......Page 209
5.3. Validity of Method ......Page 210
5.5. The General Approach ......Page 211
5.6. Two Nonlinear Differential Equations ......Page 213
5.7. The Nonlinear Spring ......Page 214
5.9. Straightforward Perturbation ......Page 217
5.11. The Van der Pol Equation ......Page 219
5.13. The Linear Equation L(u) = g ......Page 221
5.15. Nonlinear Case ......Page 223
5.16. Newton…Raphson Method ......Page 225
5.17. Multidimensional Newton…Raphson ......Page 228
5.18. Choice of Initial Approximation ......Page 229
5.19. Nonlinear Extrapolation and Acceleration of Convergence ......Page 231
5.20. Alternatives to Newton…Raphson ......Page 232
5.21. Lagrange Expansion ......Page 233
5.22. Method of Moments Applied to Partial Differential Equations ......Page 235
Miscellaneous Exercises ......Page 236
Bibliography and Comments ......Page 243
6.2. Differential Approximation ......Page 246
6.4. Computational Aspects„I ......Page 247
6.5. Computational Aspects„II ......Page 248
6.6. Degree of Approximation ......Page 249
6.7. Orthogonal Polynomials ......Page 250
6.9. Extension of Classical Approximation Theory ......Page 252
6.10. Riccati Approximation ......Page 253
6.12. Application to Renewal Equation ......Page 254
6.13. An Example ......Page 257
6.14. Differential-Difference Equations ......Page 259
6.15. An Example ......Page 260
6.16. Functional-Differential Equations ......Page 261
6.19. Mean-square Approximation ......Page 263
6.20. Validity of the Method ......Page 264
6.22. The Nonlinear Spring ......Page 265
6.23. The Van der Pol Equation ......Page 267
6.25. The Riccati Equation ......Page 269
6.26. Higher-order Approximation ......Page 271
6.27. Mean-square Approximation„Periodic Solutions ......Page 272
Miscellaneous Exercises ......Page 274
Bibliography and Comments ......Page 276
7.2. The Euler Equation ......Page 280
7.3. The Euler Equation and the Variational Problem ......Page 281
7.4. Quadratic Functionals: Scalar Case ......Page 282
7.5. Positive Definiteness for Small T ......Page 284
7.6. Discussion ......Page 285
7.8. Validity of the Method ......Page 286
7.9. Monotone Behavior and Convergence ......Page 288
7.10. Estimation of | u … v | in Terms of J(v) - J(u) ......Page 289
7.11. Convergence of Coefficients ......Page 290
7.12. Alternate Estimate ......Page 291
7.13. Successive Approximations ......Page 292
7.14. Determination of the Cofficients ......Page 293
7.15. Multidimensional Case ......Page 294
7.16. Reduction of Dimension ......Page 295
7.17. Minimization of Inequalities ......Page 296
7.18. Extension to Quadratic Functionals ......Page 298
7.19. Linear Integral Equations ......Page 300
7.20. Nonlinear Euler Equation ......Page 301
7.21. Existence and Uniqueness ......Page 302
7.23. Convexity and Uniqueness ......Page 303
7.24. Implied Boundedness ......Page 304
7.26. Functional Analysis ......Page 305
7.27. The Euler Equation and Haar's Device ......Page 307
7.28. Discussion ......Page 308
7.30. Lagrange Multiplier ......Page 309
7.32. Raising the Price Diminishes the Demand ......Page 310
7.33. The Courant Parameter ......Page 311
Miscellaneous Exercises ......Page 312
Bibliography and Comments ......Page 322
8.1. Equations Involving Parameters ......Page 325
8.2. Stationary Values ......Page 326
8.3. Characteristic Values and Functions ......Page 327
8.4. Properties of Characteristic Values and Functions ......Page 328
8.5. Generalized Fourier Expansion ......Page 333
8.6. Discussion ......Page 334
8.7. Rigorous Formulation of Variational Problem ......Page 335
8.8. Rayleigh…Ritz Method ......Page 336
8.10. Transplantation ......Page 337
8.11. Positive Definiteness of Quadratic Functionals ......Page 338
8.12. Finite Difference Approximations ......Page 339
8.13. Monotonicity ......Page 340
8.14. Positive Kernels ......Page 341
Miscellaneous Exercises ......Page 343
Bibliography and Comment ......Page 350
Author Index ......Page 352
Subject Index ......Page 358
Mathematics in Science and Engineering ......Page 362