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Search: MSC category 39 ( Difference and functional equations )

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1. CMB Online first

Oubbi, Lahbib
 On Ulam stability of a functional equation in Banach modules Let $X$ and $Y$ be Banach spaces and $f : X \to Y$ an odd mapping. For any rational number $r \ne 2$, C. Baak, D. H. Boo, and Th. M. Rassias have proved the Hyers-Ulam stability of the following functional equation: \begin{align*} r f \left(\frac{\sum_{j=1}^d x_j}{r} \right) & + \sum_{\substack{i(j) \in \{0,1\} \\ \sum_{j=1}^d i(j)=\ell}} r f \left( \frac{\sum_{j=1}^d (-1)^{i(j)}x_j}{r} \right) = (C^\ell_{d-1} - C^{\ell -1}_{d-1} + 1) \sum_{j=1}^d f(x_j) \end{align*} where $d$ and $\ell$ are positive integers so that $1 \lt \ell \lt \frac{d}{2}$, and $C^p_q := \frac{q!}{(q-p)!p!}$, $p, q \in \mathbb{N}$ with $p \le q$. In this note we solve this equation for arbitrary nonzero scalar $r$ and show that it is actually Hyers-Ulam stable. We thus extend and generalize Baak et al.'s result. Different questions concerning the *-homomorphisms and the multipliers between C*-algebras are also considered. Keywords:linear functional equation, Hyers-Ulam stability, Banach modules, C*-algebra homomorphisms.Categories:39A30, 39B10, 39A06, 46Hxx

2. CMB Online first

Chung, Jaeyoung; Ju, Yumin; Rassias, John
 Cubic functional equations on restricted domains of Lebesgue measure zero Let $X$ be a real normed space, $Y$ a Bancch space and $f:X \to Y$. We prove the Ulam-Hyers stability theorem for the cubic functional equation \begin{align*} f(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-12f(x)=0 \end{align*} in restricted domains. As an application we consider a measure zero stability problem of the inequality \begin{align*} \|f(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-12f(x)\|\le \epsilon \end{align*} for all $(x, y)$ in $\Gamma\subset\mathbb R^2$ of Lebesgue measure 0. Keywords:Baire category theorem, cubic functional equation, first category, Lebesgue measure, Ulam-Hyers stabilityCategory:39B82

3. CMB 2016 (vol 59 pp. 225)

Atıcı, Ferhan M.; Yaldız, Hatice
 Convex Functions on Discrete Time Domains In this paper, we introduce the definition of a convex real valued function $f$ defined on the set of integers, ${\mathbb{Z}}$. We prove that $f$ is convex on ${\mathbb{Z}}$ if and only if $\Delta^{2}f \geq 0$ on ${\mathbb{Z}}$. As a first application of this new concept, we state and prove discrete Hermite-Hadamard inequality using the basics of discrete calculus (i.e. the calculus on ${\mathbb{Z}}$). Second, we state and prove the discrete fractional Hermite-Hadamard inequality using the basics of discrete fractional calculus. We close the paper by defining the convexity of a real valued function on any time scale. Keywords:discrete calculus, discrete fractional calculus, convex functions, discrete Hermite-Hadamard inequalityCategories:26B25, 26A33, 39A12, 39A70, 26E70, 26D07, 26D10, 26D15

4. CMB 2014 (vol 58 pp. 30)

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, stabilityCategories:46F99, 39B82

5. 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 PrÃ©kopa-Leindler inequality and applies also to the reversed Brascamp-Lieb inequality, due to Barthe. Keywords:functional inequalities, Brownian motionCategories:39B62, 60J65

6. 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 groupsCategories:39B52, 22C05, 43A30

7. 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 equationCategories:39A10, 34B15

8. 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 convergenceCategories:39B12, 41A25, 42C40 9. 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, superlinearCategories:34K11, 39A10, 39A99 10. 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 inequalityCategory:39A10 11. 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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