## Iterative Functional EquationsA cohesive and comprehensive account of the modern theory of iterative functional equations. Many of the results included have appeared before only in research literature, making this an essential volume for all those working in functional equations and in such areas as dynamical systems and chaos, to which the theory is closely related. The authors introduce the reader to the theory and then explore the most recent developments and general results. Fundamental notions such as the existence and uniqueness of solutions to the equations are stressed throughout, as are applications of the theory to such areas as branching processes, differential equations, ergodic theory, functional analysis and geometry. Other topics covered include systems of linear and nonlinear equations of finite and infinite ORD various function classes, conjugate and commutable functions, linearization, iterative roots of functions, and special functional equations. |

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### Contents

Introduction | 1 |

2B Clockgraduation and the concept of chronon | 9 |

LIB Limit points of the sequence of iterates | 15 |

2B Analytic mappings | 22 |

3D Special cases | 30 |

5A Generalizations of the Banach contraction principle | 37 |

Linear equations and branching processes | 51 |

2A Negative g | 60 |

6A Continuous and differentiable solutions | 259 |

Equations of infinite order and systems of nonlinear equations | 270 |

2B Important special case | 279 |

2E Denumerabie order | 285 |

4B Approximation in Bucks sense | 293 |

5B Important special case | 301 |

6C Differentiable solutions of hsystems | 307 |

6E Integrable solutions of hsystems | 308 |

3D An example | 66 |

4C A difference equation | 72 |

5C Special inhomogeneous equation | 78 |

6C Restricted stationary measures | 86 |

Regularity of solutions of linear equations | 96 |

functions | 106 |

3C Asymptotic series expansions | 115 |

5B Julias equation | 124 |

6B Homogeneous equation | 132 |

IB Doubly stochastic measures supported on a hairpin | 138 |

Analytic and integrable solutions of linear equations | 148 |

2B Existence and uniqueness results | 152 |

3C General homogeneous and inhomogeneous equations | 159 |

6B The Abel equation | 168 |

7C Homogeneous equation | 174 |

Theory of nonlinear equations | 185 |

2D Existence via solutions of inequalities | 192 |

3D Comparison with the linear case | 199 |

5JB Lipschitzian Nemytskii operators | 206 |

6C Lack of uniqueness of C solutions | 214 |

8A Existence and uniqueness of Lp solutions | 222 |

8D U solution depending on an arbitrary function | 228 |

Equations of higher orders and systems of linear equations | 235 |

2B Decomposition of twoplace functions | 243 |

4B Real solutions when some characteristic roots of g | 249 |

5C Solution depending on an arbitrary function | 255 |

8A A crucial inequality | 320 |

On conjugacy | 332 |

2C Further results on smooth solutions | 339 |

5A Julias equation and the iterative logarithm | 346 |

7B Convergence of formal power series having iterative logarithm | 357 |

More on the Schroder and Abel equations | 365 |

Characterization of functions | 389 |

2B Exponential functions | 395 |

3C Sine | 403 |

4B Riemannintegrable solutions of an auxiliary equation | 409 |

5A The Weierstrass c n d functions | 414 |

Iterative roots and invariant curves | 421 |

1I 2C Strictly decreasing roots of strictly increasing functions | 427 |

AD Convex and concave iterative roots | 436 |

5C Uniqueness conditions | 442 |

6D Fractional iterates of the roots of identity function | 449 |

8B Volkmann s theorem | 457 |

9C Lack of uniqueness of Lipschitzian solutions | 463 |

2A Comparison theorems | 475 |

12AB Regular solution | 483 |

5B Properties of solutions of the inequality | 489 |

7B A property of the particular solution | 497 |

504 | |

546 | |

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### Common terms and phrases

Abel equation analytic function analytic solutions arbitrary function assume assumptions asymptotic Banach space characteristic roots class C1 compact complete metric space concave conjugate consider constant continuous and strictly continuous function continuous solution converges decreasing defined denote equation p(f(x equation p(x extension finite fixed point formal power series formula fulfilling the condition function h Galton-Watson process Hence holds ie{l implies induction inequality integrable interval iterative functional equations Kuczma 26 Lebesgue measure Lemma Let f Let hypotheses i)-(iii mapping Math Matkowski matrix meN0 metric space monotonic solutions Moreover neN0 nonnegative obtain origin polynomial positive proof of Theorem prove radius of convergence Remark resp root of unity satisfies equation Schroder equation Section self-mapping sequence Smajdor solution of equation solution q strictly increasing Subsection Suppose topological space uniform convergence unique one-parameter family unique solution whence x0eX yields Zdun