Continuous solutions to two iterative functional equations

Equations including unknown functions are called function al equations([1]). Among them, functionalequations containing only one independent variable are cal led iterative functional equations([9]). On theother hand, functional equations, like Cauchy's equation( [10]), containing two or more independent vari-ables for unknown functions of a single variable are called f unctional equations of several variables([2]).Among iterative functional equations, the problem of itera tive roots is important. If the n -th iterateof a function f is equal to g, i.e.f

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]).

Fractal and Hausdorff dimensions for systems of iterative functional equations

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.

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 .

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.

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.

Schauder's fixed point theorem and the Banach contraction principle are used to study the polynomial-like iterative functional equation with variable coefficients. We give sufficient conditions for the existence, uniqueness, and stability of the strongly convex and strongly concave solutions. Finally, some examples were considered by our results.

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.

Iterative differential equations provide a new idea to study functional differential equations. The study of iterative equations can provide new methods for the study of differential equations with state-dependent delays. In this paper, we are concerned with proving the existence of infinitely many positive solutions to nonlinear first-order iterative systems of singular BVPs on time scales by using Krasnoselskii’s cone fixed point theorem in a Banach space. It is worth pointing out that in this paper, we can use the symmetry of the iterative process and Green’s function to transform the considered differential equation into an equivalent integral equation, which plays a key role in the proof of the theorem in this paper.

The paper contains sufficient conditions for the oscillation of all solutions of linear functional iterative equations.

Solutions of mixed-type functional differential equations with state-dependence

<p style='text-indent:20px;'>In this paper we consider the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions for the general iterative functional differential equation <inline-formula><tex-math id="M1">\begin{document}$ \dot{x}(t) = f(t,x(t),x^{[2]}(t),...,x^{[n]}(t)). $\end{document}</tex-math></inline-formula> As <inline-formula><tex-math id="M2">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula>, this equation can be regarded as a mixed-type functional differential equation with state-dependence <inline-formula><tex-math id="M3">\begin{document}$ \dot{x}(t) = f(t,x(t),x(T(t,x(t)))) $\end{document}</tex-math></inline-formula> of a special form but, being a nonlinear operator, <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula>-th order iteration makes more difficulties in estimation than usual state-dependence. Then we apply our results to the existence, uniqueness, boundedness, asymptotics and continuous dependence of solutions for the mixed-type functional differential equation. Finally, we present two concrete examples to show the boundedness and asymptotics of solutions to these two types of equations respectively.</p>

By Faà di Bruno’s formula, using the fixed‐point theorems of Schauder and Banach, we study the existence and uniqueness of smooth solutions of an iterative functional differential equation x′(t) = 1/(c0x[0](t) + c1x[1](t) + ⋯+cmx[m](t)).