Iterative functional equations: Marek Kuczma ; Bogdan Choczewski ; Roman Ger
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge u.a.
Cambridge Univ. Pr.
1990
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| Ausgabe: | 1. publ. |
| Schriftenreihe: | Encyclopedia of mathematics and its applications
32 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIX, 552 S. |
| ISBN: | 0521355613 |
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Datensatz im Suchindex
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| adam_text | Titel: Iterative functional equations
Autor: Kuczma, Marek
Jahr: 1990
CONTENTS Preface xv Symbols and conventions xviii 0 Introduction 1 0.0 Preliminaries 1 0.0A Types of equations considered 1 0.0B Problems of uniqueness 2 0.0C Fixed points 3 0.0D General solution 4 0.0E Solution depending on an arbitrary function 4 0.1 Special equations 5 0.1 A Change of variables 5 0.1B Schroder s, Abel s and Bottcher s equations 5 0.2 Applications 6 0.2A Synthesizing judgements 6 0.2B Clock-graduation and the concept of chronon 9 0.2C Sensation scale and Fechner s law 10 0.3 Iterative functional equations 11 1 Iteration 13 1.0 Introduction 13 1.1 Basic notions and some substantial facts 13 1.1 A Iterates, orbits and fixed points 13 1.1B Limit points of the sequence of iterates 15 1.1C Theorem of Sarkovskii 16 1.1D Attractive fixed points 17 1.2 Maximal domains of attraction 20
vi Contents 1.2 A Convergence of splinters 20 1,2B Analytic mappings 22 1.3 The speed of convergence of iteration sequences 23 1.3 A Some lemmas 24 1,3B Splinters behaving like geometric sequences 26 1.3C Slower convergence of splinters 27 1.3D Special cases 30 1.4 Iteration sequences of random-valued functions 32 1.4A Preliminaries 33 1AB Convergence of random splinters 35 1.5 Some fixed-point theorems 36 1.5A Generalizations of the Banach contraction principle 37 1,5B Case of product spaces 39 1.5C Equivalence statement 42 1.6 Continuous dependence 43 1.7 Notes 46 2 Linear equations and branching processes 51 2.0 Introduction 51 2.1 Galton- Watson processes 52 2.1 A Probability generating functions 52 2.1 B Limit distributions 54 2.1 C Stationary measures for processes with immigration 56 2.1 D Restricted stationary measures for simple processes 58 2.2 Nonnegative solutions 59 2.2A Negative g 60 2.2B Positive g 60 2.3 Monotonie solutions 61 2.3,4 Homogeneous equation 61 2.3B Special inhomogeneous equation 64 2.3C General inhomogeneous equation 65 2.3D An example 66 2.3E Homogeneous difference equation 68 2.3F Schroder s equation 68 2.4 Convex solutions 69 2.4A Lemmas 69 2.4B Existence-and-uniqueness result 71 2.4C A difference equation 12 2.4D Abel’s and Schroder’s equations 73 2.5 Regularly varying solutions 74 2.5A Regularly varying functions 75 2.5B Homogeneous equation 75
Contents vii 2.5C Special inhomogeneous equation 78 2.6 Application to branching processes 81 2.6A Conditional limit probabilities 81 2.6B Stationary measures 83 2.6C Restricted stationary measures 86 2.7 Convex solutions of higher order 89 2.7A Definitions and results 89 2.7B A characterization of polynomials 91 2.8 Notes 91 3 Regularity of solutions of linear equations 96 3.0 Introduction 96 3.1 Continuous solutions 96 3.1 A Homogeneous equation 97 3.IB General continuous solution of the homogeneous equation 99 3.1C Inhomogeneous equation 101 3.2 Continuous dependence of continuous solutions on given functions 106 3.3 Asymptotic properties of solutions 108 3.3A Solutions continuous at the origin 108 3.3B Sample proofs 111 3.3C Asymptotic series expansions 115 3.3D Solutions discontinuous at the origin 117 3.4 Differentiable solutions 119 3.5 Special equations 122 3.5A Schroder s equation 122 3.5B Julia s equation 124 3.5C Abel’s equation 127 3.5D A characterization of the cross ratio 129 3.6 Solutions of bounded variation 130 3.6A Preliminaries 130 3.6B Homogeneous equation 132 3.6C Inhomogeneous equation 134 3.6D Solutions of almost bounded variation 135 3.7 Applications 137 3.7A An Anosov diffeomorphism without invariant measure 137 3.7B Doubly stochastic measures supported on a hairpin 138 3.7C Phase and dispersion for second-order differential equations 141 3.8 Notes 143 4 Analytic and integrable solutions of linear equations 148 4.0 Introduction 148
Contents viii 4.1 Linear equation in a topological space 148 4.2 Analytic solutions: the case | ƒ (0)| 1 150 4.2 A Extension theorems 150 4.2B Existence and uniqueness results 152 4.2C Continuous dependence on the data 154 4.3 Analytic solutions: the case |/ (0)| = 1 155 4.3A The Siegel set 156 4.3B Special inhomogeneous equation 156 4.3C General homogeneous and inhomogeneous equations 159 4.4 Analytic solutions: the case ƒ (0) = 0 160 4.5 Meromorphic solutions 165 4.6 Special equations — 167 4.6A The Schröder equation 167 4.6B The Abel equation 168 4.7 Integrable solutions 169 4.7A Preliminaries 170 4.7B A functional inequality 172 4.7C Homogeneous equation 174 4.7D Inhomogeneous equation 174 4.7E Lebesgue measure 175 4.8 Absolutely continuous solutions 177 4.8A Existence-and-uniqueness result 177 4.8B A Goursat problem 178 4.9 Notes 180 5 Theory of nonlinear equations 185 5.0 Introduction 185 5.1 An extension theorem 185 5.2 Existence and uniqueness of continuous solutions 187 5.2A Lipschitzian h 187 5.2B Continuous dependence on the data 190 5.2C Non-Lipschitzian h 190 5.2D Existence via solutions of inequalities 192 5.3 Continuous solution depending on an arbitrary function 194 5.3A Extension of solutions 195 5.3B Nonuniqueness theorems 195 5.3C Existence theorems 197 5.3D Comparison with the linear case 199 5.4 Asymptotic properties of solutions 200 5.4A Coincidence and existence theorems 200 5.4B Solutions differentiable at the origin 203 5.5 Lipschitzian solutions 204
Contents ix 5.5 A Existence and uniqueness 205 5.5B Lipschitzian Nemytskii operators 206 5.6 Smooth solutions 208 5.6A Preliminaries 208 5.6B Existence and uniqueness of C r solutions 210 5.6C Lack of uniqueness of C solutions 214 5.7 Local analytic solutions 216 5.7A Unique solution 217 5.7B Continuous dependence on the data 219 5.8 Equations in measure space 221 5.8A Existence and uniqueness of II solutions 222 5.8B 1} solutions 225 5.8C Extension theorems 226 5.8D II solution depending on an arbitrary function 228 5.9 Notes 229 6 Equations of higher orders and systems of linear equations 235 6.0 Introduction 235 6.1 Particular solutions of some special equations 236 6.1 A Cauchy functional equations on a curve 236 6.1 B The Gaussian normal distribution 237 6.1 C Equation of Nth order 237 6.1 D Proofs 239 6.2 Further applications 240 6.2A Invariant measures under piecewise linear transformations 240 6.2B Decomposition of two-place functions 243 6.3 Cyclic equations 244 6.3A Homogeneous equation with a finite group of substitutions 245 6.3B Compatibility 246 6.3C General solution 247 6.4 Matrix equation with constant g 248 6.4A Reduction to systems of scalar equations 248 6.4B Real solutions when some characteristic roots of g are complex numbers 249 6.4C Special system of two linear equations 250 6.4D Discussion of more general cases 252 6.5 Local C 00 solutions of a matrix equation 253 6.5A Preliminaries 253 6.5B Existence of a unique solution 254 6.5C Solution depending on an arbitrary function 255 6.5D Two existence theorems 257 6.6 Smooth solutions of the equation of Nth order 258
X Contents 6.6A Continuous and differentiable solutions 259 6.6B Integrable solutions 260 6.7 Equation of Nth order with iterates of one function 260 6.7A Reduction of order 261 6.7B Constant coefficients 262 6.7C Reduction to a matrix linear equation 262 6.8 Linear recurrence inequalities 263 6.8A System of inequalities 263 6.8B Consequences for single inequalities 264 6.9 Notes 265 7 Equations of infinite order and systems of nonlinear equations 270 7.0 Introduction 270 7.1 Extending solutions 271 7.1 A General extension theorem 271 7.IB Two sufficient conditions 273 7.1 C Extending of continuous solutions 275 7.2 Existence and uniqueness 276 7.2A Basic result 276 7.2B Important special case 279 7.2C An application 280 7.2D Lipschitzian solutions 282 7.2E Denumerable order 285 7.3 Stability 286 7.3A Main result 286 7.3B Special results 287 7.3C Comments 290 7.4 Approximate solutions 291 7.4A Two special equations 291 7.4B Approximation in Buck s sense 293 7 AC Uniform approximation of a continuous mapping by a Lipschitzian one 296 7.4D Polynomial approximate solutions 298 7.5 Continuous dependence 299 7.5A A general result 299 7.5B Important special case 301 7.6 A survey of results on systems of nonlinear equations of finite orders 302 7.6A Continuous solutions of h-systems 304 7.6B Solutions of h-systems with a prescribed asymptotic behaviour 305 7.6C Differentiable solutions of h-systems 307 7.6D Analytic solutions of h-systems 308
Contents xi 7.6E Integrable solutions of h-systems 308 7.6F Continuous solutions of g-systems 310 7.6G Differentiable solutions of g-systems 311 7.7 Three sample proofs 313 7.8 Continuous solutions - deeper uniqueness conditions 319 7.8A A crucial inequality 320 7.8B Result for the testing equation 322 7.8C Proof of Theorem 7.8.1 324 7.8D Uniqueness implies existence 325 7.9 Notes 327 8 On conjugacy 332 8.0 Introduction 332 8.1 Conjugacy 333 8.1 A Change of variables 333 8.1 B Properties of the conjugacy relation 333 8.2 Linearization 334 8.2A The Schröder equation 335 8.2B Unique local C r solutions in 336 8.2C Further results on smooth solutions 339 8.3 The Böttcher equation 339 8.3A Complex case 339 8.3B Asymptotic behaviour and regularity of real solutions 341 8.4 Conjugate functions 342 8.4A N-dimensional case 342 8.4B One-dimensional case 343 8.5 Conjugate formal series and analytic functions 345 8.5 A Julia s equation and the iterative logarithm 346 8.5B Formally conjugate power series 347 8.5C Conjugate analytic functions 350 8.5D Abel’s equation 351 8.6 Permutable functions 353 8.7 Commuting formal series and analytic functions 355 8.7A Formal power series that commute with a given one 355 8.7B Convergence of formal power series having iterative logarithm 357 8.7C Permutable analytic functions 358 8.7D Conjugacy again 359 8.8 Notes 360 9 More on the Schröder and Abel equations 365 9.0 Introduction 365 9.1 Principal solutions 365
Contents xii 9.2 The pre-Schröder system 368 9.2 A An equivalent system and automorphic functions 368 9.2B Equivalence of the Schroder equation and the pre-Schröder system 369 9.3 Abel-Schröder systems and the associativity equation 371 9.3A Strict Archimedean associative functions 371 9.3B Abel and Schroder equations jointly 373 9.3C Existence of generators 374 9 .3D A solution of the Abel-Schröder system 375 9.4 Abel systems and differential equations with deviations 378 9.4A A group of transformations 379 9.4B Systems of simultaneous Abel equations 380 9.5 A Schröder system and a characterization of norms 382 9.5A Characterization of norms 382 9.5B A system of simultaneous Schröder equations 383 9.6 Notes 385 10 Characterization of functions 389 10.0 Introduction 389 10.1 Power functions 389 10.1 A Identity 389 10.IB Reciprocal 390 10.1C Roots 391 10.1 D Comments 392 10.2 Logarithmic and exponential functions 393 10.2A Logarithm 394 10.2B Exponential functions 395 10.2C An improper integral 396 10.2D Comments 398 10.3 Trigonometric and hyperbolic functions 399 10.3A Cosine and hyperbolic cosine 399 10.3B Periodic solutions of the cosine equation 399 10.3C Sine 403 10.3D An improper integral 404 10.3E Comments 406 10.4 Euler’s gamma function 407 10.4A The fundamental functional equation 407 10.4B Riemann-integrable solutions of an auxiliary equation 409 10.4C Gauss s multiplication theorem 410 10.4D Complex gamma function 412 10.4E Legendre’s duplication formula and Euler s functional equation 412 10.4F Logarithmic derivative of the gamma function 414
Contents xiii 10.5 Continuous nowhere differentiable functions 414 10.5A The Weierstrass c.n.d. functions 414 10.5B A characterization of S p by homogeneous equations 415 10.5C An inhomogeneous equation for S p 415 10.5D Comments 418 10.6 Notes 419 11 Iterative roots and invariant curves 421 11.0 Introduction 421 11.1 Purely set-theoretical case 422 11.2 Continuous and monotonic solutions 423 11.2A Strictly increasing continuous iterative roots 424 11.2B Strictly decreasing roots of strictly decreasing functions 425 11.2C Strictly decreasing roots of strictly increasing functions 427 11.3 Monotonic C r solutions 428 11.4 C 1 iterative roots with nonzero multiplier 430 11.4A A function with no smooth convex square roots 430 11.4B Necessary conditions 431 11.4C Main existence theorem 432 11.4D Convex and concave iterative roots 436 11.5 C 1 iterative roots with zero multiplier 438 11.5A Abundance of solutions 438 11.5B Convergence problem 440 11.5C Uniqueness conditions 442 11.6 Complex domain and local analytic solutions 443 11.6A Existence and uniqueness 444 11.6B Multiplier 1 445 11.6C Multiplier being a primitive root of unity 447 11,6D Fractional iterates of the roots of identity function 449 11.7 Babbage equation and involutions 450 11.7A Decreasing involutions 451 11.7B Primitive iterative roots, homographies 453 11.8 Another characterization of reciprocals 455 11.8A Dubikajtis theorem 455 11.8B Volkmann’s theorem 457 11.9 Invariant curves 459 11.9A Simplifications and assumptions 460 11.9B Equation of invariant curves and its Lipschitzian solutions 460 11.9C Lack of uniqueness of Lipschitzian solutions 463 11.9D Comments 465 11.9E Euler’s and other special equations 466 11.10 Notes 458
Contents xiv 12 Linear iterative functional inequalities 472 12.0 Introduction 472 12.1 {Æ’ }-monotonic functions 473 12.2 Inequalities in the uniqueness case for associated equations 474 12.2A Comparison theorems 475 12.2B Representation theorems 476 12.3 Asymptotic behaviour of nonnegative CSs y of the inhomogeneous inequality 477 12 3A Some properties of y All 12.3B Unique solution of the associated equation 479 12.3C Asymptotic behaviour of y 479 12.4 Regular solutions of the homogeneous inequality 481 12.4A Estimates 481 12.4B Regular solution 483 12.4C Comparison theorems 484 12.4D Representation theorems 485 12.5 The inhomogeneous inequality in the nonuniqueness case 488 12.5A The best lower bound and the regular solution 488 12.5B Properties of solutions of the inequality 489 12.5C Representation theorem 490 12.6 Regular solutions of the homogeneous inequality determined by asymptotic properties 490 12.6A One-parameter family of solutions of the associated equation 491 12.6B Regular solutions of the inequality 492 12.6C Special behaviour of given functions 492 12.6D Conditions equivalent to regularity 494 12.7 A homogeneous inequality of second order 495 12.7A An equivalent system 495 12.7B A property of the particular solution 497 12.7C Reduction of order 498 12.8 An inequality of infinite order 499 12.9 Notes 501 References 504 Author index 546 Subject index 549
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| spelling | Kuczma, Marek 1935-1991 Verfasser (DE-588)12711159X aut Iterative functional equations Marek Kuczma ; Bogdan Choczewski ; Roman Ger 1. publ. Cambridge u.a. Cambridge Univ. Pr. 1990 XIX, 552 S. txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 32 Equations de fonctions ram Fonctions d'une variable réelle ram Fonctions de variables réelles Équations fonctionnelles Functional equations Functions of real variables Funktionalgleichung (DE-588)4018923-5 gnd rswk-swf Funktionalgleichung (DE-588)4018923-5 s DE-604 Choczewski, Bogdan Verfasser aut Ger, Roman Verfasser aut Encyclopedia of mathematics and its applications 32 (DE-604)BV000903719 32 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005917288&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kuczma, Marek 1935-1991 Choczewski, Bogdan Ger, Roman Iterative functional equations Marek Kuczma ; Bogdan Choczewski ; Roman Ger Encyclopedia of mathematics and its applications Equations de fonctions ram Fonctions d'une variable réelle ram Fonctions de variables réelles Équations fonctionnelles Functional equations Functions of real variables Funktionalgleichung (DE-588)4018923-5 gnd |
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| title | Iterative functional equations Marek Kuczma ; Bogdan Choczewski ; Roman Ger |
| title_auth | Iterative functional equations Marek Kuczma ; Bogdan Choczewski ; Roman Ger |
| title_exact_search | Iterative functional equations Marek Kuczma ; Bogdan Choczewski ; Roman Ger |
| title_full | Iterative functional equations Marek Kuczma ; Bogdan Choczewski ; Roman Ger |
| title_fullStr | Iterative functional equations Marek Kuczma ; Bogdan Choczewski ; Roman Ger |
| title_full_unstemmed | Iterative functional equations Marek Kuczma ; Bogdan Choczewski ; Roman Ger |
| title_short | Iterative functional equations |
| title_sort | iterative functional equations marek kuczma bogdan choczewski roman ger |
| title_sub | Marek Kuczma ; Bogdan Choczewski ; Roman Ger |
| topic | Equations de fonctions ram Fonctions d'une variable réelle ram Fonctions de variables réelles Équations fonctionnelles Functional equations Functions of real variables Funktionalgleichung (DE-588)4018923-5 gnd |
| topic_facet | Equations de fonctions Fonctions d'une variable réelle Fonctions de variables réelles Équations fonctionnelles Functional equations Functions of real variables Funktionalgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005917288&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000903719 |
| work_keys_str_mv | AT kuczmamarek iterativefunctionalequationsmarekkuczmabogdanchoczewskiromanger AT choczewskibogdan iterativefunctionalequationsmarekkuczmabogdanchoczewskiromanger AT gerroman iterativefunctionalequationsmarekkuczmabogdanchoczewskiromanger |