Iterative Functional Equations

CONTENTS
PREFACE
SYMBOLS AND CONVENTIONS
0 Introduction
0.0 Preliminaries
0.0A. Types of equations considered
0.0B. Problems of uniqueness
0.0C. Fixed points
0.0D. General solution
0.1 Special equations
0.1 A. Change of variables
0.1B. Schroder's, Abel's and Boucher's equations
0.2 Applications
0.2A. Synthesizing judgements
0.2B. Clock-graduation and the concept of chronon
0.2C. Sensation scale and Fechner's law
0.3 Iterative functional equations
1 Iteration
1.0 Introduction
1.1 Basic notions and some substantial facts
1.1 A. Iterates, orbits and fixed points
1.1B. Limit points of the sequence of iterates
1.1C. Theorem of Sarkovskii
1.1D. Attractive fixed points
1.2 Maximal domains of attraction
1.2A. Convergence of splinters
1.2B. Analytic mappings
1.3 The speed of convergence of iteration sequences
1.3A. Some lemmas
1.3B. Splinters behaving like geometric sequences
1.3C. Slower convergence of splinters
1.4 Iteration sequences of random-valued functions
1.4A. Preliminaries
1.4B. Convergence of random splinters
1.5 Some fixed-point theorems
1.5A. Generalizations of the Banach contraction principle
1.5B. Case of product spaces
1.5C. Equivalence statement
1.6 Continuous dependence
1.7 Notes
2 Linear equations and branching processes
2.0 Introduction
2.1 Galton-Watson processes
2.1 A. Probability generating functions
2.IB. Limit distributions
2.1C. Stationary measures for processes with immigration
2.1D. Restricted stationary measures for simple processes
2.2 Nonnegative solutions
2.2A. Negative g
2.2B. Positive g
2.3 Monotonic solutions
2.3A. Homogeneous equation
2.3B. Special inhomogeneous equation
2.3C. General inhomogeneous equation
2.3D. An example
2.3E. Homogeneous difference equation
2.3F. Schroder's equation
2.4 Convex solutions
2.4A. Lemmas
2.4B. Existence-and-uniqueness result
2.4C. A difference equation
2.4D. Abel's and Schroder's equations
2.5 Regularly varying solutions
2.5A. Regularly varying functions
2.5B. Homogeneous equation
2.5C. Special inhomogeneous equation
2.6 Application to branching processes
2.6A. Conditional limit probabilities
2.6B. Stationary measures
2.6C. Restricted stationary measures
2.7 Convex solutions of higher order
2.7A. Definitions and results
2.7B. A characterization of polynomials
2.8 Notes
3 Regularity of solutions of linear equations
3.0 Introduction
3.1 Continuous solutions
3.1A. Homogeneous equation
3.1B. General continuous solution of the homogeneous equation
3.1 C. Inhomogeneous equation
3.2 Continuous dependence of continuous solutions on given functions
3.3 Asymptotic properties of solutions
3.3A. Solutions continuous at the origin
3.3B. Sample proofs
3.3C. Asymptotic series expansions
3.3D. Solutions discontinuous at the origin
3.4 Differentiable solutions
3.5 Special equations
3.5A. Schroder's equation
3.5B. Julia's equation
3.5C. Abel's equation
3.5D. A characterization of the cross ratio
3.6 Solutions of bounded variation
3.6A. Preliminaries
3.6B. Homogeneous equation
3.6C. Inhomogeneous equation
3.7 Applications
3.7A. An Anosov diffeomorphism without invariant measure
3.7B. Doubly stochastic measures supported on a hairpin
3.7C. Phase and dispersion for second-order differential equations
3.8 Notes
4 Analytic and integrable solutions of linear equations
4.0 Introduction
4.1 Linear equation in a topological space
4.2 Analytic solutions; the case |/(0)| < 1
4.2A. Extension theorems
4.2B. Existence and uniqueness results
4.2C. Continuous dependence on the data
4.3 Analytic solutions; the case |/'(0)| = 1
4.3A. The Siegel set
4.3B. Special inhomogeneous equation
4.3C. General homogeneous and inhomogeneous equations
4.4 Analytic solutions; the case /'(0) = 0
4.5 Meromorphic solutions
4.6 Special equations
4.6A. The Schroder equation
4.6B. The Abel equation
4.7 Integrable solutions
4.7A. Preliminaries
4.7B. A functional inequality
4.7C. Homogeneous equation
4.7D. Inhomogeneous equation
4.7E. Lebesgue measure
4.8 Absolutely continuous solutions
4.8A. Existence-and-uniqueness result
4.8B. A Goursat problem
4.9 Notes
5 Theory of nonlinear equations
5.0 Introduction
5.1 An extension theorem
5.2 Existence and uniqueness of continuous solutions
5.2A. Lipschitzian h
5.2B. Continuous dependence on the data
5.2C. Non-Lipschitzian h
5.2D. Existence via solutions of inequalities
5.3 Continuous solution depending on an arbitrary function
5.3A. Extension of solutions
5.3B. Nonuniqueness theorem
5.3C. Existence theorem
5.3D. Comparison with the linear case
5.4 Asymptotic properties of solutions
5.4A. Coincidence and existence theorems
5.4B. Solutions differentiable at the origin
5.5 Lipschitzian solutions
5.5A. Existence and uniqueness
5.5B. Lipschitzian Nemitskii operators
5.6 Smooth solutions
5.6A. Preliminaries
5.6B. Existence and uniqueness of Cr solutions
5.6C. Lack of uniqueness of C solutions
5.7 Local analytic solutions
5.7A. Unique solution
5.7B. Continuous dependence on the data
5.8 Equations in measure spaces
5.8A. Existence and uniqueness of IF solutions
5.8B. L1 solutions
5.8C. Extension theorems
5.8D. IP solution depending on an arbitrary function
5.9 Notes
6 Equations of higher orders and systems of linear equations
6.0 Introduction
6.1 Particular solutions of some special equations
6.1A. Cauchy's functional equations on a curve
6.IB. The Gaussian normal distribution
6.1C. Equation of Nth order
6.1 D. Proofs
6.2 Further applications
6.2A. Invariant measures under piecewise linear transformations
6.2B. Decomposition of two-place functions
6.3 Cyclic equations
6.3A. Homogeneous equation with a finite group of substitutions
6.3B. Compatibility
6.3C. General solution
6.4 Matrix equation with constant g
6.4A. Reduction to systems of scalar equations
6.4B. Real solutions when some characteristic roots of g are complex numbers
6.4C. Special system of two linear equations
6.4D. Discussion of more general cases
6.5 Local C00 solutions of a matrix equation
6.5A. Preliminaries
6.5B. Existence of a unique solution
6.5C. Solution depending on an arbitrary function
6.5D. Two existence theorems
6.6 Smooth solutions of the equation of Nth order
6.6A. Continuous and differentiable solutions
6.6B. Integrable solutions
6.7 Equation of Nth order with iterates of one function
6.7A. Reduction of order
6.7B. Constant coefficients
6.7C. Reduction to a matrix linear equation
6.8 Linear recurrence inequalities
6.8A. System of inequalities
6.8B. Consequences for single inequalities
6.9 Notes
7 Equations of infinite order and systems of nonlinear equations
7.0 Introduction
7.1 Extending solutions
7.1 A. General extension theorem
7.IB. Two sufficient conditions
7.1 C. Extending of continuous solutions
7.2 Existence and uniqueness
7.2A. Basic result
7.2B. Important special case
7.2C. An application
7.2D. Lipschitzian solutions
7.2E. Denumerable order
7.3 Stability
7.3A. Main result
7.3B. Special results
7.3C. Comments
7.4 Approximate solutions
7.4A. Two special equations
7.4B. Approximation in Buck's sense
7.4C. Uniform approximation of a continuous mapping by a Lipschitzian one
7.4D. Polynomial approximate solutions
7.5 Continuous dependence
7.5A. A general result
7.5B. Important special case
7.6 A survey of results on systems of nonlinear equations of finite orders
7.6A. Continuous solutions of /t-systems
7.6B. Solutions of /i-systems with a prescribed asymptotic behaviour
7.6C. Differentiable solutions of /t-systems
7.6D. Analytic solutions of /t-systems
7.6E. Integrable solutions of /t-systems
7.6F. Continuous solutions of g -systems
7.6G. Differentiable solutions of g-systems
7.7 Three sample proofs
7.8 Continuous solutions - deeper uniqueness conditions
7.8A. A crucial inequality
7.8B. Result for the testing equation
7.8C. Proof of Theorem 7.8.1
7.8D. Uniqueness implies existence
7.9 Notes
8 On conjugacy
8.0 Introduction
8.1 Conjugacy
8.1 A. Change of variables
8.1B. Properties of the conjugacy relation
8.2 Linearization
8.2A. The Schroder equation
8.2B. Unique local Cr solution in UN
8.2C. Further results on smooth solutions
8.3 The Bottcher equation
8.3A. Complex case
8.3B. Asymptotic behaviour and regularity of real solutions
8.4 Conjugate functions
8.4A. /V-dimensional case
8.4B. One-dimensional case
8.5 Conjugate formal series and analytic functions
8.5A. Julia's equation and the iterative logarithm
8.5B. Formally conjugate power series
8.5C. Conjugate analytic functions
8.5D. Abel's equation
8.6 Permutable functions
8.7 Commuting formal series and analytic functions
8.7A. Formal power series that commute with a given one
8.7B. Convergence of formal power series having iterative logarithm
8.7C. Permutable analytic functions
8.7D. Conjugacy again
8.8 Notes
9 More on the Schroder and Abel equations
9.0 Introduction
9.1 Principal solutions
9.2 The pre-Schroder system
9.2A. An equivalent system and automorphic functions
9.2B. Equivalence of the Schroder equation and the pre-Schroder system
9.3 Abel-Schroder systems and the associativity equation
9.3A. Strict Archimedean associative functions
9.3B. Abel and Schroder equations jointly
9.3C. Existence of generators
9.3D. A solution of the Abel-Schroder system
9.4 Abel systems and differential equations with deviations
9.4A. A group of transformations
9.4B. Systems of simultaneous Abel equations
9.5 A Schroder system and a characterization of norms
9.5A. Characterization of norms
9.5B. A system of simultaneous Schroder equations
9.6 Notes
10 Characterization of functions
10.0 Introduction
10.1 Power functions
10.1A. Identity
10.IB. Reciprocal
10.1C. Roots
10.1D. Comments
10.2 Logarithmic and exponential functions
10.2A. Logarithm
10.2B. Exponential functions
10.2C. An improper integral
10.2D. Comments
10.3 Trigonometric and hyperbolic functions
10.3A. Cosine and hyperbolic cosine
10.3B. Periodic solutions of the cosine equation
10.3C. Sine
10.3D. An improper integral
10.3E. Comments
10.4 Euler's gamma function
10.4A. The fundamental functional equation
10.4B. Riemann integrable solutions of an auxiliary equation
10.4C. Gauss' multiplication theorem
10.4D. Complex gamma function
10.4E. Legendre's duplication formula and Euler's functional equation
10.4F. Logarithmic derivative of the gamma function
10.5 Continuous nowhere differentiable functions
10.5A. The Weierstrass c.n.d. functions
10.5B. A characterization of 5p by homogeneous equations
10.5C. An inhomogeneous equation for Sp
10.5D. Comments
10.6 Notes
11 Iterative roots and invariant curves
11.0 Introduction
11.1 Purely set-theoretical case
11.2 Continuous and monotonic solutions
11.2A. Strictly increasing continuous iterative roots
11.2B. Strictly decreasing roots of strictly decreasing functions
11.2C. Strictly decreasing roots of strictly increasing functions
11.3 Monotonic Cr solutions
11.4 C1 iterative roots with nonzero multiplier
11.4A. A function with no smooth convex square roots
11.4B. Necessary conditions
11.4C. Main existence theorem
11.4D. Convex and concave iterative roots
11.5 C1 iterative roots with zero multiplier
11.5A. Abundance of solutions
11.5B. Convergence problem
11.5C. Uniqueness conditions
11.6 Complex domain and local analytic solutions
11.6A. Existence and uniqueness
11.6B. Multiplier 1
11.6C. Multiplier being a primitive root of unity
11.6D. Fractional iterates of the roots of identity function
11.7 Babbage equation and involutions
11.7A. Decreasing involutions
11.7B. Primitive iterative roots, homographies
11.8 Another characterization of reciprocals
11.8A. Dubikajtis' theorem
11.8B. Volkmann's theorem
11.9 Invariant curves
11.9A. Simplifications and assumptions
11.9B. Equation of invariant curves and its Lipschitzian solutions
11.9C. Lack of uniqueness of Lipschitzian solutions
11.9D. Comments
11.9E. Euler's and other special equations
11.10 Notes
12 Linear iterative functional inequalities
12.0 Introduction
12.1 {f}-monotonic functions
12.2 Inequalities in the uniqueness case for associated equations
12.2A. Comparison theorems
12.2B. Representation theorems
12.3 Asymptotic behaviour of nonnegative CSs y of the inhomogeneous inequality
12.3A. Some properties of y
12.3B. Unique solution of the associated equation
12.3C. Asymptotic behaviour of 3;
12.4 Regular solutions of the homogeneous inequality
12.4 Regular solutions of the homogeneous inequality
12.4A. Estimates
12.4B. Regular solution
12.4C. Comparison theorems
12.4D. Representation theorems
12.5 The inhomogeneous inequality in the nonuniqueness case
12.5A. The best lower bound and the regular solutions
12.5B. Properties of solutions of the inequality
12.5C. Representation theorem
12.6 Regular solutions of the homogeneous inequality determined by asymptotic properties
12.6A. One-parameter family of solutions of the associated equation
12.6B. Regular solutions of the inequality
12.6C. Special behaviour of given functions
12.6D. Conditions equivalent to regularity
12.7 A homogeneous inequality of second order
12.7A. An equivalent system
12.7B. A property of the particular solution
12.7C. Reduction of order
12.8 An inequality of infinite order
12.9 Notes
REFERENCES
AUTHOR INDEX
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