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Search: MSC category 39 ( Difference and functional equations )

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1. CMB Online first

Chung, Jaeyoung
On an Exponential Functional Inequality and its Distributional Version
Let $G$ be a group and $\mathbb K=\mathbb C$ or $\mathbb R$. In this article, as a generalization of the result of Albert and Baker, we investigate the behavior of bounded and unbounded functions $f\colon G\to \mathbb K$ satisfying the inequality $ \Bigl|f \Bigl(\sum_{k=1}^n x_k \Bigr)-\prod_{k=1}^n f(x_k) \Bigr|\le \phi(x_2, \dots, x_n),\quad \forall\, x_1, \dots, x_n\in G, $ where $\phi\colon G^{n-1}\to [0, \infty)$. Also, as a distributional version of the above inequality we consider the stability of the functional equation \begin{equation*} u\circ S - \overbrace{u\otimes \cdots \otimes u}^{n-\text {times}}=0, \end{equation*} where $u$ is a Schwartz distribution or Gelfand hyperfunction, $\circ$ and $\otimes$ are the pullback and tensor product of distributions, respectively, and $S(x_1, \dots, x_n)=x_1+ \dots +x_n$.

Keywords:distribution, exponential functional equation, Gelfand hyperfunction, stability
Categories:46F99, 39B82

2. CMB 2013 (vol 57 pp. 585)

Lehec, Joseph
Short Probabilistic Proof of the Brascamp-Lieb and Barthe Theorems
We give a short proof of the Brascamp-Lieb theorem, which asserts that a certain general form of Young's convolution inequality is saturated by Gaussian functions. The argument is inspired by Borell's stochastic proof of the Prékopa-Leindler inequality and applies also to the reversed Brascamp-Lieb inequality, due to Barthe.

Keywords:functional inequalities, Brownian motion
Categories:39B62, 60J65

3. CMB 2011 (vol 56 pp. 218)

Yang, Dilian
Functional Equations and Fourier Analysis
By exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations - the d'Alembert equation, the Wilson equation, and the d'Alembert long equation - on compact groups.

Keywords:functional equations, Fourier analysis, representation of compact groups
Categories:39B52, 22C05, 43A30

4. CMB 2011 (vol 55 pp. 214)

Wang, Da-Bin
Positive Solutions of Impulsive Dynamic System on Time Scales
In this paper, some criteria for the existence of positive solutions of a class of systems of impulsive dynamic equations on time scales are obtained by using a fixed point theorem in cones.

Keywords:time scale, positive solution, fixed point, impulsive dynamic equation
Categories:39A10, 34B15

5. CMB 2011 (vol 55 pp. 424)

Yang, Jianbin; Li, Song
Convergence Rates of Cascade Algorithms with Infinitely Supported Masks
We investigate the solutions of refinement equations of the form $$ \phi(x)=\sum_{\alpha\in\mathbb Z^s}a(\alpha)\:\phi(Mx-\alpha), $$ where the function $\phi$ is in $L_p(\mathbb R^s)$$(1\le p\le\infty)$, $a$ is an infinitely supported sequence on $\mathbb Z^s$ called a refinement mask, and $M$ is an $s\times s$ integer matrix such that $\lim_{n\to\infty}M^{-n}=0$. Associated with the mask $a$ and $M$ is a linear operator $Q_{a,M}$ defined on $L_p(\mathbb R^s)$ by $Q_{a,M} \phi_0:=\sum_{\alpha\in\mathbb Z^s}a(\alpha)\phi_0(M\cdot-\alpha)$. Main results of this paper are related to the convergence rates of $(Q_{a,M}^n \phi_0)_{n=1,2,\dots}$ in $L_p(\mathbb R^s)$ with mask $a$ being infinitely supported. It is proved that under some appropriate conditions on the initial function $\phi_0$, $Q_{a,M}^n \phi_0$ converges in $L_p(\mathbb R^s)$ with an exponential rate.

Keywords:refinement equations, infinitely supported mask, cascade algorithms, rates of convergence
Categories:39B12, 41A25, 42C40

6. CMB 2011 (vol 54 pp. 580)

Baoguo, Jia; Erbe, Lynn; Peterson, Allan
Kiguradze-type Oscillation Theorems for Second Order Superlinear Dynamic Equations on Time Scales
Consider the second order superlinear dynamic equation \begin{equation*} (*)\qquad x^{\Delta\Delta}(t)+p(t)f(x(\sigma(t)))=0\tag{$*$} \end{equation*} where $p\in C(\mathbb{T},\mathbb{R})$, $\mathbb{T}$ is a time scale, $f\colon\mathbb{R}\rightarrow\mathbb{R}$ is continuously differentiable and satisfies $f'(x)>0$, and $xf(x)>0$ for $x\neq 0$. Furthermore, $f(x)$ also satisfies a superlinear condition, which includes the nonlinear function $f(x)=x^\alpha$ with $\alpha>1$, commonly known as the Emden--Fowler case. Here the coefficient function $p(t)$ is allowed to be negative for arbitrarily large values of $t$. In addition to extending the result of Kiguradze for \eqref{star1} in the real case $\mathbb{T}=\mathbb{R}$, we obtain analogues in the difference equation and $q$-difference equation cases.

Keywords:Oscillation, Emden-Fowler equation, superlinear
Categories:34K11, 39A10, 39A99

7. CMB 2008 (vol 51 pp. 161)

Agarwal, Ravi P.; Otero-Espinar, Victoria; Perera, Kanishka; Vivero, Dolores R.
Wirtinger's Inequalities on Time Scales
This paper is devoted to the study of Wirtinger-type inequalities for the Lebesgue $\Delta$-integral on an arbitrary time scale $\T$. We prove a general inequality for a class of absolutely continuous functions on closed subintervals of an adequate subset of $\T$. By using this expression and by assuming that $\T$ is bounded, we deduce that a general inequality is valid for every absolutely continuous function on $\T$ such that its $\Delta$-derivative belongs to $L_\Delta^2([a,b)\cap\T)$ and at most it vanishes on the boundary of $\T$.

Keywords:time scales calculus, $\Delta$-integral, Wirtinger's inequality
Category:39A10

8. CMB 2005 (vol 48 pp. 505)

Bouikhalene, Belaid
On the Generalized d'Alembert's and Wilson's Functional Equations on a Compact group
Let $G$ be a compact group. Let $\sigma$ be a continuous involution of $G$. In this paper, we are concerned by the following functional equation $$\int_{G}f(xtyt^{-1})\,dt+\int_{G}f(xt\sigma(y)t^{-1})\,dt=2g(x)h(y), \quad x, y \in G,$$ where $f, g, h \colonG \mapsto \mathbb{C}$, to be determined, are complex continuous functions on $G$ such that $f$ is central. This equation generalizes d'Alembert's and Wilson's functional equations. We show that the solutions are expressed by means of characters of irreducible, continuous and unitary representations of the group $G$.

Keywords:Compact groups, Functional equations, Central functions, Lie, groups, Invariant differential operators.
Categories:39B32, 39B42, 22D10, 22D12, 22D15

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