Iterative functional equations

Title page......Page 1
Preface......Page 13
Symbols and conventions......Page 16
0.0A Types of equations considered......Page 18
0.0B Problems of uniqueness......Page 19
0.0C Fixed points......Page 20
0.0E Solution depending on an arbitrary function......Page 21
0.1 B Schröder's, Abel's and Boucher's equations......Page 22
0.2A Synthesizing judgements......Page 23
0.2B Clock-graduation and the concept of chronon......Page 26
0.2C Sensation scale and Fechner's law......Page 27
0.3 Iterative functional equations......Page 28
1.1A Iterates, orbits and fixed points......Page 30
1.1B Limit points of the sequence of iterates......Page 32
1.1C Theorem of Sarkovskii......Page 33
1.1D Attractive fixed points......Page 34
1.2A Convergence of splinters......Page 37
1.2B Analytic mappings......Page 39
1.3 The speed of convergence of iteration sequences......Page 40
1.3A Some lemmas......Page 41
1.3B Splinters behaving like geometric sequences......Page 43
1.3C Slower convergence of splinters......Page 44
1.3D Special cases......Page 47
1.4 Iteration sequences of random-valued functions......Page 49
1.4A Preliminaries......Page 50
1.4B Convergence of random splinters......Page 52
1.5 Some fixed-point theorems......Page 53
1.5A Generalizations of the Banach contraction principle......Page 54
1.5B Case of product spaces......Page 56
1.5C Equivalence statement......Page 59
1.6 Continuous dependence......Page 60
1.7 Notes......Page 63
2.0 Introduction......Page 68
2.1A Probability generating functions......Page 69
2.1B Limit distributions......Page 71
2.1C Stationary measures for processes with immigration......Page 73
2.1D Restricted stationary measures for simple processes......Page 75
2.2 Nonnegative solutions......Page 76
2.2B Positive g......Page 77
2.3A Homogeneous equation......Page 78
2.3B Special inhomogeneous equation......Page 81
2.3C General inhomogeneous equation......Page 82
2.3D An example......Page 83
2.3F Schröder's equation......Page 85
2.4A Lemmas......Page 86
2.4B Existence-and-uniqueness result......Page 88
2.4C A difference equation......Page 89
2.4D Abel's and Schröder's equations......Page 90
2.5 Regularly varying solutions......Page 91
2.5B Homogeneous equation......Page 92
2.5C Special inhomogeneous equation......Page 95
2.6A Conditional limit probabilities......Page 98
2.6B Stationary measures......Page 100
2.6C Restricted stationary measures......Page 103
2.7A Definitions and results......Page 106
2.8 Notes......Page 108
3.1 Continuous solutions......Page 113
3.1A Homogeneous equation......Page 114
3.1B General continuous solution of the homogeneous equation......Page 116
3.1C Inhomogeneous equation......Page 118
3.2 Continuous dependence of continuous solutions on given functions......Page 123
3.3A Solutions continuous at the origin......Page 125
3.3B Sample proofs......Page 128
3.3C Asymptotic series expansions......Page 132
3.3D Solutions discontinuous at the origin......Page 134
3.4 Differentiate solutions......Page 136
3.5A Schröder's equation......Page 139
3.5B Julia's equation......Page 141
3.5C Abel's equation......Page 144
3.5D A characterization of the cross ratio......Page 146
3.6A Preliminaries......Page 147
3.6B Homogeneous equation......Page 149
3.6C Inhomogeneous equation......Page 151
3.6D Solutions of almost bounded variation......Page 152
3.7A An Anosov diffeomorphism without invariant measure......Page 154
3.7B Doubly stochastic measures supported on a hairpin......Page 155
3.7C Phase and dispersion for second-order differential equations......Page 158
3.8 Notes......Page 160
4.1 Linear equation in a topological space......Page 165
4.2A Extension theorems......Page 167
4.2B Existence and uniqueness results......Page 169
4.2C Continuous dependence on the data......Page 171
4.3 Analytic solutions: the case |f'(0)| = 1......Page 172
4.3B Special inhomogeneous equation......Page 173
4.3C General homogeneous and inhomogeneous equations......Page 176
4.4 Analytic solutions: the case f'(0) = 0......Page 177
4.5 Meromorphic solutions......Page 182
4.6A The Schröder equation......Page 184
4.6B The Abel equation......Page 185
4.7 Integrable solutions......Page 186
4.7A Preliminaries......Page 187
4.7B A functional inequality......Page 189
4.7D Inhomogeneous equation......Page 191
4.7E Lebesgue measure......Page 192
4.8A Existence-and-uniqueness result......Page 194
4.8B A Goursat problem......Page 195
4.9 Notes......Page 197
5.1 An extension theorem......Page 202
5.2A Lipschitzian h......Page 204
5.2C Non-Lipschitzian h......Page 207
5.2D Existence via solutions of inequalities......Page 209
5.3 Continuous solution depending on an arbitrary function......Page 211
5.3B Nonuniqueness theorems......Page 212
5.3C Existence theorems......Page 214
5.3D Comparison with the linear case......Page 216
5.4A Coincidence and existence theorems......Page 217
5.4B Solutions differentiable at the origin......Page 220
5.5 Lipschitzian solutions......Page 221
5.5A Existence and uniqueness......Page 222
5.5B Lipschitzian Nemytskii operators......Page 223
5.6A Preliminaries......Page 225
5.6B Existence and uniqueness of C^r solutions......Page 227
5.6C Lack of uniqueness of C^r solutions......Page 231
5.7 Local analytic solutions......Page 233
5.7A Unique solution......Page 228
5.7B Continuous dependence on the data......Page 236
5.8 Equations in measure space......Page 238
5.8A Existence and uniqueness of L^p solutions......Page 239
5.8B L¹ solutions......Page 242
5.8C Extension theorems......Page 243
5.8D L^p solution depending on an arbitrary function......Page 245
5.9 Notes......Page 246
6.0 Introduction......Page 252
6.1A Cauchy functional equations on a curve......Page 253
6.1C Equation of Nth order......Page 254
6.1D Proofs......Page 256
6.2A Invariant measures under piecewise linear transformations......Page 257
6.2B Decomposition of two-place functions......Page 260
6.3 Cyclic equations......Page 261
6.3A Homogeneous equation with a finite group of substitutions......Page 262
6.3B Compatibility......Page 263
6.3C General solution......Page 264
6.4A Reduction to systems of scalar equations......Page 265
6.4B Real solutions when some characteristic roots of g are complex numbers......Page 266
6.4C Special system of two linear equations......Page 267
6.4D Discussion of more general cases......Page 269
6.5A Preliminaries......Page 270
6.5B Existence of a unique solution......Page 271
6.5C Solution depending on an arbitrary function......Page 272
6.5D Two existence theorems......Page 274
6.6 Smooth solutions of the equation of Nth order......Page 275
6.6A Continuous and differentiable solutions......Page 276
6.7 Equation of Nth order with iterates of one function......Page 277
6.7A Reduction of order......Page 278
6.7C Reduction to a matrix linear equation......Page 279
6.8A System of inequalities......Page 280
6.8B Consequences for single inequalities......Page 281
6.9 Notes......Page 282
7.0 Introduction......Page 287
7.1A General extension theorem......Page 288
7.1B Two sufficient conditions......Page 290
7.1C Extending of continuous solutions......Page 292
7.2A Basic result......Page 293
7.2B Important special case......Page 296
7.2C An application......Page 297
7.2D Lipschitzian solutions......Page 299
7.2E Denumerable order......Page 302
7.3A Main result......Page 303
7.3B Special results......Page 304
7.3C Comments......Page 307
7.4A Two special equations......Page 308
7.4B Approximation in Buck's sense......Page 310
7.4C Uniform approximation of a continuous mapping by a Lipschitzian one......Page 313
7.4D Polynomial approximate solutions......Page 315
7.5A A general result......Page 316
7.5B Important special case......Page 318
7.6 A survey of results on systems of nonlinear equations of finite orders......Page 319
7.6A Continuous solutions of h-systems......Page 321
7.6B Solutions of h-systems with a prescribed asymptotic behaviour......Page 322
7.6C Differentiate solutions of h-systems......Page 324
7.6E Integrable solutions of h-systems......Page 325
7.6F Continuous solutions of g-systems......Page 327
7.6G Differentiable solutions of g-systems......Page 328
7.7 Three sample proofs......Page 330
7.8 Continuous solutions - deeper uniqueness conditions......Page 336
7.8A A crucial inequality......Page 337
7.8B Result for the testing equation......Page 339
7.8C Proof of Theorem 7.8.1......Page 341
7.8D Uniqueness implies existence......Page 342
7.9 Notes......Page 344
8.0 Introduction......Page 349
8.1B Properties of the conjugacy relation......Page 350
8.2 Linearization......Page 351
8.2A The Schröder equation......Page 352
8.2B Unique local C^r solutions in R^N......Page 353
8.3A Complex case......Page 356
8.3B Asymptotic behaviour and regularity of real solutions......Page 358
8.4A N-dimensional case......Page 359
8.4B One-dimensional case......Page 360
8.5 Conjugate formal series and analytic functions......Page 362
8.5A Julia's equation and the iterative logarithm......Page 363
8.5B Formally conjugate power series......Page 364
8.5C Conjugate analytic functions......Page 367
8.5D Abel's equation......Page 368
8.6 Permutable functions......Page 370
8.7A Formal power series that commute with a given one......Page 372
8.7B Convergence of formal power series having iterative logarithm......Page 374
8.7C Permutable analytic functions......Page 375
8.7D Conjugacy again......Page 376
8.8 Notes......Page 377
9.1 Principal solutions......Page 382
9.2A An equivalent system and automorphic functions......Page 385
9.2B Equivalence of the Schroder equation and the pre-Schröder system......Page 386
9.3A Strict Archimedean associative functions......Page 388
9.3B Abel and Schröder equations jointly......Page 390
9.3C Existence of generators......Page 391
9.3D A solution of the Abel-Schröder system......Page 392
9.4 Abel systems and differential equations with deviations......Page 395
9.4A A group of transformations......Page 396
9.4B Systems of simultaneous Abel equations......Page 397
9.5A Characterization of norms......Page 399
9.5B A system of simultaneous Schröder equations......Page 400
9.6 Notes......Page 402
10.1A Identity......Page 406
10.1B Reciprocal......Page 407
10.1C Roots......Page 408
10.1D Comments......Page 409
10.2 Logarithmic and exponential functions......Page 410
10.2A Logarithm......Page 411
10.2B Exponential functions......Page 412
10.2C An improper integral......Page 413
10.2D Comments......Page 415
10.3B Periodic solutions of the cosine equation......Page 416
10.3C Sine......Page 420
10.3D An improper integral......Page 421
10.3E Comments......Page 423
10.4A The fundamental functional equation......Page 424
10.4B Riemann-integrable solutions of an auxiliary equation......Page 426
10.4C Gauss's multiplication theorem......Page 427
10.4D Complex gamma function 4ii......Page -99
10.5A The Weierstrass c.n.d. functions......Page 431
10.5C An inhomogeneous equation for Sp......Page 432
10.5D Comments......Page 435
10.6 Notes......Page 436
11.0 Introduction......Page 438
11.1 Purely set-theoretical case......Page 439
11.2 Continuous and monotonic solutions......Page 440
11.2A Strictly increasing continuous iterative roots......Page 441
11.2B Strictly decreasing roots of strictly decreasing functions......Page 442
11.2C Strictly decreasing roots of strictly increasing functions......Page 444
11.3 Monotonic C^r solutions......Page 445
11.4A A function with no smooth convex square roots......Page 447
11.4B Necessary conditions......Page 448
11.4C Main existence theorem......Page 449
11.4D Convex and concave iterative roots......Page 453
11.5A Abundance of solutions......Page 455
11.5B Convergence problem......Page 457
11.5C Uniqueness conditions......Page 459
11.6 Complex domain and local analytic solutions......Page 460
11.6A Existence and uniqueness......Page 461
11.6B Multiplier 1......Page 462
11.6C Multiplier being a primitive root of unity......Page 464
11.6D Fractional iterates of the roots of identity function......Page 466
11.7 Babbage equation and involutions......Page 467
11.7A Decreasing involutions......Page 468
11.7B Primitive iterative roots, homographies......Page 470
11.8A Dubikajtis'theorem......Page 472
11.8B Volkmann's theorem......Page 474
11.9 Invariant curves......Page 476
11.9B Equation of invariant curves and its Lipschitzian solutions......Page 477
11.9C Lack of uniqueness of Lipschitzian solutions......Page 480
11.9D Comments......Page 482
11.9E Euler's and other special equations......Page 483
11.10 Notes......Page 475
12.0 Introduction......Page 489
12.1 {f}-monotonic functions......Page 490
12.2 Inequalities in the uniqueness case for associated equations......Page 491
12.2A Comparison theorems......Page 492
12.2B Representation theorems......Page 493
12.3 Asymptotic behaviour of nonnegative CSs y of the inhomogeneous inequality......Page 494
12.3A Some properties of y......Page 428
12.3C Asymptotic behaviour of y......Page 496
12.4A Estimates......Page 498
12.4B Regular solution......Page 500
12.4C Comparison theorems......Page 501
12.4D Representation theorems......Page 502
12.5A The best lower bound and the regular solution......Page 505
12.5B Properties of solutions of the inequality......Page 506
12.6 Regular solutions of the homogeneous inequality determined by asymptotic properties......Page 507
12.6A One-parameter family of solutions of the associated equation......Page 508
12.6C Special behaviour of given functions......Page 509
12.6D Conditions equivalent to regularity......Page 511
12.7A An equivalent system......Page 512
12.7B A property of the particular solution......Page 514
12.7C Reduction of order......Page 515
12.8 An inequality of infinite order......Page 516
12.9 Notes......Page 518
References......Page 521
Author index......Page 563
Subject index......Page 566