Analytic solutions of an iterative functional equation

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

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

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]).

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.

A special solution of Abel's ordinary differential equation of the first kind is considered, which describes the behaviour of a broad spectrum of solutions of partial differential equations with a small parameter in the neighbourhood of cusp points of their slowly varying equilibrium positions. The existence of this special solution is demonstrated; an asymptotic formula for it as , is constructed and substantiated.

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

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.

It ist shown that the ordinary non‐linear second order differential equation of the phenomenological theory of the one‐dimensional steady shock wave in heat conducting viscous gases can be transformed into the Abelian differential equation of the second class. Among the known special solutions of Abel's equation is one applicable to the shock wave and leads to a hitherto unknown closed form solution.

Following a 70-year old suggestion of Paul Lévy, a condition is formulated for the regularity of growth of real functions. The condition, which is quite explicit, makes use of the iterates of the function and solutions of Abel's functional equation, and is well adapted to a computer testing. Numerous computer experiments reveal interesting properties of the proposed regularity criterion.

Schroder developed several notions which are central to the study of complex dynamics. Despite his failure to rigorously establish his fixed point theorem, it is a fundamental result, and his belief that iteration of an arbitrary function ∅(z) could be reduced to the solution of the so-called Abel and Schroder functional equations was prophetic. Not only was the next phase in the development of complex dynamics ushered in by an interest in the solution of the Schroder and Abel equations, but the study and solution of functional equations is fundamental in many contemporary studies of iteration.

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>

Hilbert [13] suggested in connection with his fifth problem that, while the theory of differential equations provides elegant and powerful techniques for solving functional equations, the inherent differentiability assumptions are typically unnatural (see [4]). Such shortcomings can sometimes be overcome by appealing to regularity theorems of the type "continuity implies differentiability" (see e.g. [4], and w of [1]) and of the type "measurability implies continuity" beginning with Fr~chet [9] and continued by scorses of others (see e.g. [4], [5], [16] and [19] for references to further such results). The theory of distribution of Laurent Schwartz has also been used, perhaps first by Feny5 [8], (see also [2], [6], [11], [12], [20] and [25]) to partially overcome this deficiency. The main aim of the present paper is to introduce another distributional method, based on approximating distributional derivatives by differences, which allows one to find the distributional solutions to distributional analogues of many functional equations by reduction to (distributional) differential equations. This method, when combined with regularity theorems, allows one to find all the Continuous (or even locally Lebesgue integrable) solutions of certain functional equations. References to some of the vast literature concerning many of the functional equations considered herein can be found in [1] and [3]. While many of our results concerning functional equations may be considered as bu"t slight generalizations of known results, our main aim is to illustrate a method of solution. A novelty of our approach is that it permits "local" consideration. Unless otherwise indicated, our notation and terminology is that of Rudin [24]. The set of all Schwartz test function on R n will be denoted by 79~. We let D~ denote the set of Schwartz distributions on R n. All measure-theoretic language refers to Lebesgue measure.

This paper discussed some existence theorems for nonlinear functional integral equations in the space L^1 of Lebesgue integrable functions,by using the Darbo fixed point theorem associated with the Hausdorff measure of noncompactness. Also, as an application, we discuss the existence of solutions for some nonlinear integral equations with fractional order.

In this paper the functional equation¶¶\( F\Bigl(x + G(t(H(y - x))\Bigr) = S\Bigl(A(t)K(x),B(t)K(y)\Bigr), \qquad x,y \in I, x \le y, t \in [0,1] \),¶is solved under monotonicity and injectivity assumptions for the unkown functions involved. This equation turned out to be useful in the axiomatic treatment of various joint-receipt utility structures. After several reduction steps, the solution of the above equation is reduced to that of¶¶\( \Gamma^{-1}\bigl(\Gamma\bigl(\frac{H(u)}{H(u + v)}\bigr) \cdot \frac{H(x)}{H(x + u + v)}\bigr) = \frac{H(u)H(x + u + v)}{H(u + v)H(x + u)},\qquad x,u,v > 0,\,x + u + v\in I \).¶Assuming that H and \( \Gamma \) are strictly monotonic, it is shown that H is a log-concave function, and as a consequence, using Lebesgue's theorem, the everywhere-differentiability of \( \Gamma \) is deduced. Then, differentiating this equation, a functional differential equation for H is derived. The solutions of this new equation are then determined and the form of \( \Gamma \) is described. Thus, the general solution of the original equation is also obtained.