Solutions Of Certain Functional Equations Research Articles

Using Aichinger’s equation to characterize polynomial functions

Aichinger’s equation is used to give simple proofs of several well-known characterizations of polynomial functions as solutions of certain functional equations. Concretely, we use that Aichinger’s equation characterizes polynomial functions to solve, for arbitrary commutative groups, Ghurye–Olkin’s functional equation, Wilson’s functional equation, the Kakutani–Nagumo–Walsh functional equation, and a general version of Fréchet’s unmixed functional equation.

Non-existence of measurable solutions of certain functional equations via probabilistic approaches

This paper deals with functional equations in the form of $f(x) + g(y) = h(x,y)$ where $h$ is given and $f$ and $g$ are unknown. We will show that if $h$ is a Borel measurable function associated with characterizations of the uniform or Cauchy distributions, then there is no measurable solutions of the equation. Our proof uses a characterization of the Dirac measure and it is also applicable to the arctan equation.

Characterization of polynomials as solutions of certain functional equations

In this paper we present several characterizations of ordinary polynomials as the solution sets of certain functional equations related to the equation∑i=0mfi(bix+ciy)=∑i=1nai(y)vi(x), where x,y∈Rd and bi,ci∈GLd(C), whose solution set is, typically, formed by exponential polynomials. Some of these equations are important because of their connection with the Characterization Problem of distributions in Probability Theory.

Symmetric difference operators on fuzzy sets

Based on the properties of symmetric difference of sets, a symmetric difference operator for fuzzy sets is defined to be a continuous and associative binary operator on the closed unit interval with some boundary condition. Structures and properties of these operators are investigated in this paper. The main results are: (1) It is proved that a symmetric difference operator is determined by a continuous t-conorm and a strong negation operator on the unit interval. (2) Two models of these operators are discussed. These models are related to the solutions of certain functional equations on the unit interval. In particular, the results presented here provide a partial answer to a problem raised by Alsina, Frank and Schweizer in 2003 about functional equations on the unit interval.

Arithmetic Properties of Solutions of Certain Functional Equations with Transformations Represented by Matrices Including a Negative Entry

Mahler's method gives algebraic independence results for the values of functions of several variables satisfying certain functional equations under the transformations of the variables represented as a kind of the multiplicative action of matrices with integral entries. In the Mahler's method, the entries of those matrices must be nonnegative; however, in the special case stated in this paper, one can admit those matrices to have a negative entry. We show the algebraic independence of the values of certain functions satisfying functional equations under the transformation represented by such matrices, expressing those values as linear combinations of the values of ordinary Mahler functions.

Meromorphic Solutions of Certain Functional Equations

By utilizing Nevanlinna's value distribution theory, we study the existence or solvability of meromorphic solutions of functional equations of the type $P(f) f'P(g) g' = 1$, where $P(z)$ is a polynomial with two distinct zeros at least. We show that such type of equations have no meromorphic solutions $f$ and $g$ when $P(z)$ has at least three distinct zeros. Moreover, for some polynomials $P(z)$ with two distinct zeros only, such type of equations possess transcendental meromorphic solutions which can be expressed by Weierstrass $\wp$ function.

Meromorphic Solutions of Certain Functional Equations

By utilizing Nevanlinna's value distribution theory, we study the existence or solvability of meromorphic solutions of functional equations of the type $P(f)f'P(g)g'=1$, where $P(z)$ is a polynomial with two distinct zeros at least. We show that such type of equations have no meromorphic solutions $f$ and $g$ when $P(z)$ has at least three distinct zeros. Moreover, for some polynomials $P(z)$ with two distinct zeros only, such type of equations possess transcendental meromorphic solutions which can be expressed by Weierstrass $\wp$ function.

Fixed point theorems for single-valued and multi-valued maps

Coincidence and fixed point theorems for single-valued and multi-valued maps generalizing recent results of Suzuki and Kikkawa are obtained. Various applications, including the existence of common solutions of certain functional equations are presented.

Remarks on Recent Fixed Point Theorems

AbstractCoincidence and fixed point theorems for a new class of contractive, nonexpansive and hybrid contractions are proved. Applications regarding the existence of common solutions of certain functional equations are also discussed.

Inverse problems for regular variation of linear filters, a cancellation property for σ-finite measures and identification of stable laws

In this paper, we consider certain σ-finite measures which can be interpreted as the output of a linear filter. We assume that these measures have regularly varying tails and study whether the input to the linear filter must have regularly varying tails as well. This turns out to be related to the presence of a particular cancellation property in σ-finite measures, which in turn, is related to the uniqueness of the solution of certain functional equations. The techniques we develop are applied to weighted sums of i.i.d. random variables, to products of independent random variables, and to stochastic integrals with respect to Lévy motions.

Functional equations involving means

In this paper, the functional equation $$ f(px + (1 - p)y) + f((1 - p)x + py) = f(x) + f(y), (x,y \in I) $$ is considered, where 0 < p < 1 is a fixed parameter and f: I → R is an unknown function. The equivalence of this and Jensen’s functional equation is completely characterized in terms of the algebraic properties of the parameter p. As an application, solutions of certain functional equations involving four weighted arithmetic means are also determined.

Walks in the quarter plane: Kreweras’ algebraic model

We consider planar lattice walks that start from (0,0), remain in the first quadrant i,j≥0, and are made of three types of steps: North-East, West and South. These walks are known to have remarkable enumerative and probabilistic properties: • they are counted by nice numbers [Kreweras, Cahiers du B.U.R.O 6 (1965) 5–105], • the generating function of these numbers is algebraic [Gessel, J. Statist. Plann. Inference 14 (1986) 49–58], • the stationary distribution of the corresponding Markov chain in the quadrant has an algebraic probability generating function [Flatto and Hahn, SIAM J. Appl. Math. 44 (1984) 1041–1053]. These results are not well understood, and have been established via complicated proofs. Here we give a uniform derivation of all of them, which is more elementary that those previously published. We then go further by computing the full law of the Markov chain. This helps to delimit the border of algebraicity: the associated probability generating function is no longer algebraic, unless a diagonal symmetry holds. Our proofs are based on the solution of certain functional equations, which are very simple to establish. Finding purely combinatorial proofs remains an open problem.

Arithmetic Properties of Solutions of Certain Functional Equations in Several Variables

We consider the linear independence of the values of solutions of certain functional equations in several variables. As an application we obtain a quantitative refinement of a certain result of Bézivin.

Arithmetic Properties of Solutions of Certain Functional Equations in Several Variables

We consider the linear independence of the values of solutions of certain functional equations in several variables. As an application we obtain a quantitative refinement of a certain result of Bézivin.

Sphere-preserving maps in inversive geometry

We give an extensive discussion of sphere-preserving maps defined on subdomains of Euclidean n n -space, and their relationship to Möbius maps and to the preservation of cross-ratios. In the case n = 2 n=2 (the complex plane) we also relate these ideas to the solutions of certain functional equations.

On the two-block H∞ problem for a class of unstable distributed systems

This paper deals with the two-block H ∞ control problem for distributed plants with finitely many unstable modes. We assume that weighting filters in the H ∞ mixed-sensitivity problem are finite-dimensional. Then the corresponding optimal two-block problem can be solved by finding the Schmidt pairs of a Hankel operator whose symbol is of the form m∗ 2(m∗ 1u + u ̂ ) where u ∈ R H ∞ , u ̂ ∈ H ∞ , and m 2 ∈ RH ∞ and m 1 ∈ H ∞ are inner; and the suboptimal two-block problem can be solved by finding the solutions of certain functional equations very similar to the ones satisfied by the Schmit pairs of the above-mentioned Hankel operator. In this paper a unified approach is proposed for solving both the optimal and suboptimal two-block problems. We obtain two systems of linear equations, expressed in terms of state-space realizations of u and m 2, whose solutions give the Schmidt pairs of the associated Hankel operator and the functions needed for the parametrization of all the suboptimal solutions, respectively.

Difference operators, distributions and functional equations

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.

General Analytic Solution of Certain Functional Equations of Addition Type

The general analytic solutions of the following functional equations are exhibited: \[ \begin{gathered} {{\alpha (x + y)} / {[\alpha (x)\alpha (y)]}} = 1 + \varphi (x)\varphi (y)\psi (x + y), \hfill \\ {{\beta (x + y)} / {[\beta (x)\beta (y)]}} = \gamma (x) + \gamma (y) + \chi (x + y). \hfill \\ \end{gathered} \] These solutions are expressed in terms of Weierstrass elliptic functions; the special cases in which these reduce to elementary functions are also exhibited. Moreover, several remarkable formulae satisfied by Weierstrass elliptic functions are reported.

Homological methods in the solution of certain functional equations

Homological methods in the solution of certain functional equations

The Classification of the Conjugacy Classes of the Full Group of Homeomorphisms of an Open Interval and the General Solution of Certain Functional Equations

The Classification of the Conjugacy Classes of the Full Group of Homeomorphisms of an Open Interval and the General Solution of Certain Functional Equations