Maple Package itsol for Formal Solutions of Iterative Functional Equations

Some sufficient conditions which allow to extend every local solution of a simultaneous system of equations in a single variable of the form \[ \varphi(x) = h (x, \varphi[f_1(x)],\ldots,\varphi[f_m(x)]),\] \[\varphi(x) = H (x, \varphi[F_1(x)],\ldots,\varphi[F_m(x)]),\] to a global one are presented. Extensions of solutions of functional equations, both in single and in several variables, play important role (cf. for instance [M. Kuczma, Functional equations in a single variable, Monografie Mat. 46, Polish Scientific Publishers, Warsaw, 1968, M. Kuczma, B. Choczewski, R. Ger, Iterative functional equations, Encyclopedia of Mathematics and Its Applications v. 32, Cambridge, 1990, J. Matkowski, Iteration groups, commuting functions and simultaneous systems of linear functional equations, Opuscula Math. 28 (2008) 4, 531-541]).

This paper is concerned with an iterative functional equation from the problem of invariant curves. By reducing the equation with the Schröder transformation to another functional equation without iteration of the unknown function f. For technical reasons, in previous work the constant λ given in the Schröder transformation is required to fulfil that λ is off the unit circle or lies on the circle with the Diophantine condition. In this paper, we obtain analytic solutions in the case of λ at resonance, i.e., at a root of the unity and the case of λ near resonance under the Brjuno condition.KeywordsIterative functional equationConvergenceBrjuno conditionAnalytic solutionMajorant series

Following M. Laczkovich [5] we consider the iterative functional equation $$\phi (x) = \sum\limits_{i = 1}^k {A_{i\phi (x + a_i ),} } $$ and its non-negative measurable solutions ϕ. In the mentioned paper the author studies non-negative Lebesgue measurable solutions defined on the real line. It is our goal to consider this problem both in more dimensions and from a more abstract point of view, in particular to get also topological version of his result.

Fractal and Hausdorff dimensions for systems of iterative functional equations

Functional equations are equations in which the unknown (or unknowns) are functions. We consider equations of generalized associativity, mediality (bisymmetry, entropy), paramediality, transitivity as well as the generalized Kolmogoroff equation. Their usefulness was proved in applications both in mathematics and in other disciplines, particularly in economics and social sciences (see J. Aczél, On mean values, Bull. Amer. Math. Soc. 54 (1948) 392–400; J. Aczél, Remarques algebriques sur la solution donner par M. Frechet a l’equation de Kolmogoroff, Pupl. Math. Debrecen 4 (1955) 33–42; J. Aczél, A Short Course on Functional Equations Based Upon Recent Applications to the Social and Behavioral Sciences, Theory and decision library, Series B: Mathematical and statistical methods (D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, Tokyo, 1987); J. Aczél, Lectures on Functional Equations and Their Applications (Supplemented by the author, ed. H. Oser) (Dover Publications, Mineola, New York, 2006); J. Aczél, V. D. Belousov and M. Hosszu, Generalized associativity and bisymmetry on quasigroups, Acta Math. Acad. Sci. Hungar. 11 (1960) 127–136; J. Aczél and J. Dhombres, Functional Equations in Several Variables (Cambridge University Press, New York, 1991); J. Aczél and T. L. Saaty, Procedures for synthesizing ratio judgements, J. Math. Psych. 27(1) (1983) 93–102). We use unifying approach to solve these equations for division and regular operations generalizing the classical quasigroup case.

We present a survey of outcomes on Ulam stability of functional equations in non-Archimedean normed spaces. We focus mainly on functional equations in several variables (including the Cauchy equation, the Jordan–von Neumann equation, the Jensen equation, and their generalizations), but we also report a result on a general equation in a single variable, which can be applied to the very important linear functional equation. Let us note that one can observe the symmetry between the presented results and the analogous ones obtained for both classical and two-normed spaces.

In this paper, we establish Hyers–Ulam–Rassias stability results belonging to two different set valued functional equations in several variables, namely additive and cubic. The results are obtained in the contexts of Banach spaces. The work is in the domain of set valued analysis.

The paper consists of two parts. At first, assuming that (Ω, A, P) is a probability space and (X, ϱ) is a complete and separable metric space with the σ-algebra 𝒝 of all its Borel subsets we consider the set 𝒭 c of all 𝒝 ⊗ 𝒜-measurable and contractive in mean functions f : X × Ω → X with finite integral ∫ Ω ϱ (f(x, ω), x) P (dω) for x ∈ X, the weak limit π f of the sequence of iterates of f ∈ 𝒭 c , and investigate continuity-like property of the function f ↦ π f, f ∈ 𝒭 c , and Lipschitz solutions φ that take values in a separable Banach space of the equation φ ( x ) = ∫ Ω φ ( f ( x , ω ) ) P ( d ω ) + F ( x ) . \varphi \left( x \right) = \int_\Omega {\varphi \left( {f\left( {x,\omega } \right)} \right)P\left( {d\omega } \right)} + F\left( x \right). Next, assuming that X is a real separable Hilbert space, Λ: X → X is linear and continuous with ||Λ || < 1, and µ is a probability Borel measure on X with finite first moment we examine continuous at zero solutions φ : X → 𝔺 of the equation φ ( x ) = μ ⌢ ( x ) φ ( Λ x ) \varphi \left( x \right) = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \mu } \left( x \right)\varphi \left( {\Lambda x} \right) which characterizes the limit distribution π f for some special f ∈ 𝒭 c .

Based on iteration of random-valued functions we study the problem of solvability in the class of continuous and Hölder continuous functions varphi of the equations φ(x)=F(x)-∫Ωφ(f(x,ω))P(dω),φ(x)=F(x)+∫Ωφ(f(x,ω))P(dω),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\varphi (x)=F(x)-\\int _{\\Omega }\\varphi \\big (f(x,\\omega )\\big )P(d\\omega ),\\\\ \\varphi (x)=F(x)+\\int _{\\Omega }\\varphi \\big (f(x,\\omega )\\big )P(d\\omega ), \\end{aligned}$$\\end{document}where P is a probability measure on a sigma -algebra of subsets of Omega .

Preface Symbols and conventions 1. Introduction 2. Iteration 3. Linear equations and branching processes 4. Regularity of solutions of linear equations 5. Analytic and integrable solutions of linear equations 6. Theory of nonlinear equations 7. Equations of higher orders and systems of linear equations 8. Equations of infinite order and systems of nonlinear equations 9. On conjugacy 10. More on the Schroder and Abel equations 11. Characterization of functions 12. Iterative roots and invariant curves 13. Linear iterative functional inequalities References Author index Subject index.

Spectral synthesis and spectral analysis.- Spectral analysis and spectral synthesis on discrete Abelian groups.- Spectral synthesis and functional equations.- Mean periodic functions.- Difference equations in several variables.- Spectral analysis and synthesis on polynomial hypergroups in a single variable.- Spectral analysis and synthesis on multivariate polynomial hypergroups.

This article concerns primarily the study of functional equations of the form f(x) + f(y) − f(x + y) = g(H(x, y)), where H is a given function of two (real or complex) variables, and where f and g are unknown functions. The problem of finding general methods to solve equations of this type was posed at the 37th International Symposium on Functional Equations in 1999 by R. Ger and L. Reich. We present results including the following. Let I be a nonvoid real interval or complex region with I + I ⊂ I. Suppose there exists a particular solution (f0, g0) on I such that f0 is analytic and not affine (or, in the real case, continuously differentiable and strictly convex). If g0 and H are analytic (or, in the real case, continuously differentiable), then all solutions of the equation are of the form \(g(u) = A_{1} \circ g_{0} (u) + c,\;f(x) = A_{1} \circ f_{0} (x) + A_{2} (x) + c,\) where A1, A2 are arbitrary additive functions and c is an arbitrary constant. Analogous results hold for functional equations of the form \(f(x) + f(y) - f \circ \theta ^{{ - 1}} (\theta (x) + \theta (y)) = g(H(x,y))\) with invertible smooth θ.

We are dealing with those continuous solutions $ \varphi $ of the functional equation¶¶ $ \varphi\circ f=g\cdot \varphi+h $ ¶that are asymptotically comparable at the origin (the fixed point of f) with the function h. Connections with a linear iterative functional inequality of second order are also mentioned.

This paper is concerned with the functional equation f (p (z) + bf (z)) = h (z) for f. By constructing a convergent power series solution of an auxiliary equation of the form \( g\left(\beta ^{2}z\right) -p\left(g(\beta z)\right) =bh\left( g(z)\right), \) analytic solutions of the form \( f(z) = \left\{ g\left( \beta g^{-1}(z)\right) -p(z)\right\} /b \) for the original equation are obtained.

Making use of some transparent convolutional approach we consider the sine functional equation in several variables.