Recent zbMATH articles in MSC 39https://www.zbmath.org/atom/cc/392022-05-16T20:40:13.078697ZUnknown authorWerkzeugReport of meeting. The fifty-sixth international symposium on functional equations, Bildungshaus Mariatrost, Graz (Austria), June 17--24, 2018https://www.zbmath.org/1483.000352022-05-16T20:40:13.078697Z(no abstract)Report of meeting. The 57th international symposium on functional equations Dom Zdrojowy SPA Hotel, Jastarnia (Poland), June 2--9, 2019https://www.zbmath.org/1483.000362022-05-16T20:40:13.078697Z(no abstract)On rational and hypergeometric solutions of linear ordinary difference equations in \(\Pi\Sigma^\ast\)-field extensionshttps://www.zbmath.org/1483.120052022-05-16T20:40:13.078697Z"Abramov, Sergei A."https://www.zbmath.org/authors/?q=ai:abramov.sergei-a"Bronstein, Manuel"https://www.zbmath.org/authors/?q=ai:bronstein.manuel-eric"Petkovšek, Marko"https://www.zbmath.org/authors/?q=ai:petkovsek.marko"Schneider, Carsten"https://www.zbmath.org/authors/?q=ai:schneider.carsten.1|schneider.carstenIn this paper the authors develop an algorithmic framework to solve linear difference equations in the context of \(\Pi\Sigma^*\)-fields. That is, the coefficients of these equations are assumed to be indefinitely nested sums and products that can be represented as elements of a \(\Pi\Sigma^*\)-field. The two main results of this paper are algorithms for computing (1) hypergeometric solutions of homogeneous linear difference equations, and (2) rational solutions of parameterized (inhomogeneous) linear difference equations. These algorithms are therefore generalizations of classical algorithms (Abramov, Petkovšek, etc.). They can also be viewed as the difference analog of Singer's algorithm, based on Risch's algorithm, for finding all Liouvillian solutions of linear differential equations whose coefficients are given in terms of Liouvillian extensions.
Reviewer: Christoph Koutschan (Linz)Inverse ambiguous functions and automorphisms on finite groupshttps://www.zbmath.org/1483.200452022-05-16T20:40:13.078697Z"Toborg, Imke"https://www.zbmath.org/authors/?q=ai:toborg.imkeSummary: If \(G\) is a finite group, then a bijective function \(f\ :\ G \rightarrow G\) is inverse ambiguous if and only if \(f(x)-1\ =\ f^-1(x)\) for all \(x \in G\). We give a precise description when a finite group admits an inverse ambiguous function and when a finite group admits an inverse ambiguous automorphism.Welcome to real analysis. Continuity and calculus, distance and dynamicshttps://www.zbmath.org/1483.260012022-05-16T20:40:13.078697Z"Kennedy, Benjamin B."https://www.zbmath.org/authors/?q=ai:kennedy.benjamin-bPublisher's description: Welcome to Real Analysis is designed for use in an introductory undergraduate course in real analysis. Much of the development is in the setting of the general metric space. The book makes substantial use not only of the real line and \(n\)-dimensional Euclidean space, but also sequence and function spaces. Proving and extending results from single-variable calculus provides motivation throughout. The more abstract ideas come to life in meaningful and accessible applications. For example, the contraction mapping principle is used to prove an existence and uniqueness theorem for solutions of ordinary differential equations and the existence of certain fractals; the continuity of the integration operator on the space of continuous functions on a compact interval paves the way for some results about power series.
The exposition is exceedingly clear and well-motivated. There are a wide variety of exercises and many pedagogical innovations. For example, each chapter includes Reading Questions so that students can check their understanding. In addition to the standard material in a first real analysis course, the book contains two concluding chapters on dynamical systems and fractals as an illustration of the power of the theory developed.Non-isolated, non-strictly monotone points of iterates of continuous functionshttps://www.zbmath.org/1483.260032022-05-16T20:40:13.078697Z"Murugan, Veerapazham"https://www.zbmath.org/authors/?q=ai:murugan.veerapazham"Palanivel, Rajendran"https://www.zbmath.org/authors/?q=ai:palanivel.rajendranSummary: There are continuous functions with complicated yet interesting sets of non-isolated, non-strictly monotone points. This paper aims to characterize the sets of isolated and non-isolated, non-strictly monotone points of the composition of continuous functions. Consequently, an uncountable dense set of measure zero in the real line and whose complement is also uncountable and dense is obtained.Uniqueness of differential \(q\)-shift difference polynomials of entire functionshttps://www.zbmath.org/1483.300622022-05-16T20:40:13.078697Z"Mathai, Madhura M."https://www.zbmath.org/authors/?q=ai:mathai.madhura-m"Manjalapur, Vinayak V."https://www.zbmath.org/authors/?q=ai:manjalapur.vinayak-vSummary: In this paper, we prove the uniqueness theorems of differential \(q\)-shift difference polynomials of transcendental entire functions.Paired Hayman conjecture and uniqueness of complex delay-differential polynomialshttps://www.zbmath.org/1483.300642022-05-16T20:40:13.078697Z"Gao, Yingchun"https://www.zbmath.org/authors/?q=ai:gao.yingchun"Liu, Kai"https://www.zbmath.org/authors/?q=ai:liu.kai.4|liu.kai.1|liu.kai.2|liu.kai|liu.kai.3|liu.kai.5Summary: In this paper, the paired Hayman conjecture of different types are considered, namely, the zeros distribution of \(f(z)^nL(g)-a(z)\) and \(g(z)^nL(f)-a(z)\), where \(L(h)\) takes the derivatives \(h^{(k)}(z)\) or the shift \(h(z+c)\) or the difference \(h(z+c)-h(z)\) or the delay-differential \(h^{(k)}(z+c)\), where \(k\) is a positive integer, \(c\) is a non-zero constant and \(a(z)\) is a non-zero small function with respect to \(f(z)\) and \(g(z)\). The related uniqueness problems of complex delay-differential polynomials are also considered.Introduction to differential and difference equationshttps://www.zbmath.org/1483.340012022-05-16T20:40:13.078697Z"Lewintan, Alexander"https://www.zbmath.org/authors/?q=ai:lewintan.alexander"Lewintan, Peter"https://www.zbmath.org/authors/?q=ai:lewintan.peterPublisher's description: In einer stärker computerisierten Welt finden Differential- und Differenzengleichungen immer mehr Anwendung. Das vorliegende Lehrbuch ist insbesondere für Studierende der ingenieurwissenschaftlichen, der informatikorientierten und der ökonomischen Studiengänge geeignet. Ausgewählte Kapitel sind auch für Schülerinnen und Schüler aus der Oberstufe mit den Leistungskursen Mathematik/Physik/Informatik interessant.
Der präsentierte Stoff entspricht einer zweistündigen Vorlesung im Grundlagenbereich, wobei Basis-Kenntnisse aus der Analysis und der Linearen Algebra vorausgesetzt sind. Die Autoren zeigen Parallelen bei den Untersuchungen von linearen Differential- und linearen Differenzengleichungen auf, wobei die Vorgehensweisen anhand von vielen Beispielen ausführlich illustriert werden. Es werden lineare Differential- und lineare Differenzengleichungen erster und zweiter Ordnung betrachtet, sowie den Leserinnen und Leser alle Werkzeuge für die Betrachtungen von Gleichungen höherer Ordnung zur Verfügung gestellt.The Brunn-Minkowski inequality and a Minkowski problem for nonlinear capacityhttps://www.zbmath.org/1483.350932022-05-16T20:40:13.078697Z"Akman, Murat"https://www.zbmath.org/authors/?q=ai:akman.murat"Gong, Jasun"https://www.zbmath.org/authors/?q=ai:gong.jasun"Hineman, Jay"https://www.zbmath.org/authors/?q=ai:hineman.jay-lawrence"Lewis, John"https://www.zbmath.org/authors/?q=ai:lewis.john-l"Vogel, Andrew"https://www.zbmath.org/authors/?q=ai:vogel.andrew-lIn this interesting paper, the authors consider two potential-theoretic problems in convex geometry.
The first main result of the paper proves a Brunn-Minkowski inequality for a nonlinear capacity, \(\text{Cap}_{\mathcal{A}}\), where \(\mathcal{A}\)-capacity is associated with a nonlinear elliptic PDE whose structure is modeled by the \(p\)-Laplace equation. More precisely, let \(1<p<n\), \(0<\lambda<1\), and \(E_1\), \(E_2\) be convex compact sets with positive \(\mathcal{A}\)-capacity. Then
\[
[\text{Cap}_{\mathcal{A}}(\lambda E_1+(1-\lambda)E_2)]^{\frac{1}{n-p}}\ge\lambda[\text{Cap}_{\mathcal{A}}(E_1)]^{\frac{1}{n-p}}+(1-\lambda)[\text{Cap}_{\mathcal{A}}(E_2)]^{\frac{1}{n-p}}.
\]
Furthermore, the authors show that if equality holds for some \(E_1\) and \(E_2\), then under certain conditions on \(\mathcal{A}\), the two sets must be homothetic.
The key ingredients of the proof include the fact that \(\{x \ : \ u(x)>t\}\) is convex for \(0<t<1\), if \(u\) is a nontrivial \(\mathcal{A}\)-harmonic capacitary function for a compact, convex set \(E\). The authors also use a maximum principle argument of [\textit{R. M. Gabriel}, J. Lond. Math. Soc. 30, 388--401 (1955; Zbl 0068.08303)], and the proof of equality is inspired by ideas in [\textit{A. Colesanti} and \textit{P. Salani}, Math. Ann. 327, No. 3, 459--479 (2003; Zbl 1052.31005); \textit{M. Longinetti}, SIAM J. Math. Anal. 19, No. 2, 377--389 (1988; Zbl 0647.31001)].
Subsequently, the authors consider a Minkowski problem for a measure associated with a compact convex set \(E\) with nonempty interior and its \(\mathcal{A}\)-harmonic capacitary function in the complement of \(E\). More precisely, denoting with \(\mu_E\) this measure, the authors consider the problem of, given a finite Borel measure \(\mu\) on \(\mathbb{S}^{n-1}\), find necessary and sufficient conditions for the existence of a set \(E\) as above, with \(\mu_E=\mu\). The authors prove that necessary and sufficient conditions for existence are the same conditions in the classical Minkowski problem for volume, and also in the work [\textit{D. Jerison}, Acta Math. 176, No. 1, 1--47 (1996; Zbl 0880.35041)], which addresses electrostatic capacity. Here, the authors are inspired by ideas from [\textit{A. Colesanti} et al., Adv. Math. 285, 1511--1588 (2015; Zbl 1327.31024); \textit{D. Jerison}, Acta Math. 176, No. 1, 1--47 (1996; Zbl 0880.35041); \textit{J. L. Lewis} and \textit{K. Nyström}, J. Eur. Math. Soc. (JEMS) 20, No. 7, 1689--1746 (2018; Zbl 1397.35088); \textit{M. Venouziou} and \textit{G. C. Verchota}, Proc. Symp. Pure Math. 79, 407--422 (2008; Zbl 1160.35022)]. Finally, using the Brunn-Minkowski inequality result from the first part of this paper, the authors prove that this problem has a unique solution, up to translation when \(p\neq n-1\), and translation and dilation when \(p=n-1\).
Reviewer: Mariana Vega Smit (Bellingham)Existence theorems and Hyers-Ulam stability for a coupled system of fractional differential equations with \(p\)-Laplacian operatorhttps://www.zbmath.org/1483.353202022-05-16T20:40:13.078697Z"Khan, Hasib"https://www.zbmath.org/authors/?q=ai:khan.hasib"Li, Yongjin"https://www.zbmath.org/authors/?q=ai:li.yongjin"Chen, Wen"https://www.zbmath.org/authors/?q=ai:chen.wen"Baleanu, Dumitru"https://www.zbmath.org/authors/?q=ai:baleanu.dumitru-i"Khan, Aziz"https://www.zbmath.org/authors/?q=ai:khan.azizSummary: In this paper, we study the existence and uniqueness of solution (EUS) as well as Hyers-Ulam stability for a coupled system of FDEs in Caputo's sense with nonlinear \(p\)-Laplacian operator. For this purpose, the suggested coupled system is transferred to an integral system with the help of four Green functions \(\mathcal{G}^{\alpha_{1}}(t,s)\), \(\mathcal{G}^{\beta_{1}}(t,s)\), \(\mathcal{G}^{\alpha_{2}}(t,s)\), \(\mathcal{G}^{\beta_{2}}(t,s)\). Then using topological degree theory and Leray-Schauder's-type fixed point theorem, existence and uniqueness results are proved. An illustrative and expressive example is given as an application of the results.Lax pair for a novel two-dimensional latticehttps://www.zbmath.org/1483.370932022-05-16T20:40:13.078697Z"Kuznetsova, Maria N."https://www.zbmath.org/authors/?q=ai:kuznetsova.maria-nSummary: In paper by \textit{I. Habibullin} [Phys. Scr. 87, No. 6, Article ID 065005, 5 p. (2013; Zbl 1275.17044)]
and our joint paper [\textit{I. T. Habibullin} and the author, Theor. Math. Phys. 203, No. 1, 569--581 (2020; Zbl 1452.37072); translation from Teor. Mat. Fiz. 203, No. 1, 161--173 (2020)] the algorithm for classification of integrable equations with three independent variables was proposed. This method is based on the requirement of the existence of an infinite set of Darboux integrable reductions and on the notion of the characteristic Lie-Rinehart algebras. The method was applied for the classification of integrable cases of different subclasses of equations \(u_{n,xy} = f(u_{n+1},u_n,u_{n-1}, u_{n,x},u_{n,y})\) of special forms. Under this approach the novel integrable chain was obtained. In present paper we construct Lax pair for the novel chain. To construct the Lax pair, we use the scheme suggested in papers by \textit{E. V. Ferapontov} [Theor. Math. Phys. 110, No. 1, 68--77 (1997; Zbl 0919.35132); translation from Teor. Mat. Fiz. 110, No. 1, 86--97 (1997); et al., J. Phys. A, Math. Theor. 46, No. 24, Article ID 245207, 13 p. (2013; Zbl 1329.35263)]. We also study the periodic reduction of the chain.Almost reducibility for families of sequences of matriceshttps://www.zbmath.org/1483.390012022-05-16T20:40:13.078697Z"Barreira, Luis"https://www.zbmath.org/authors/?q=ai:barreira.luis-m"Valls, Claudia"https://www.zbmath.org/authors/?q=ai:valls.claudiaSummary: We consider the almost reducibility property of a nonautonomous dynamics with discrete time defined by a sequence of matrices. This corresponds to the reduction of the original nonautonomous dynamics to an autonomous dynamics via a coordinate change that preserves the Lyapunov exponents. In particular, we give a characterization of the almost reducibility of a sequence to a diagonal matrix and we use this result to characterize the class of matrices to which a given sequence is almost reducible. We also consider continuous \(1\)-parameter families of sequences of matrices and we show that the almost reducibility set of such a family is always an \(F_{\sigma\delta}\)-set. In addition, we show that for any \(F_{\sigma\delta}\)-set containing zero there exists a family with this set as its almost reducibility set.The Gronwall's inequality on the \((q,h)\)-time scalehttps://www.zbmath.org/1483.390022022-05-16T20:40:13.078697Z"Segi Rahmat, Rafi Mohamad"https://www.zbmath.org/authors/?q=ai:segi-rahmat.rafi-mohamadSummary: In this article, we present an analogue of Gronwall's inequality on the \((q,h)\)-time scale. Some particular cases are derived where discrete Mittag-Leffler functions are used. Using the inequality, we study the dependence of the solution on the order and initial condition of the fractional difference equation.On solvability of a system of three difference equationshttps://www.zbmath.org/1483.390032022-05-16T20:40:13.078697Z"Yalcinkaya, Ibrahim"https://www.zbmath.org/authors/?q=ai:yalcinkaya.ibrahim"Tollu, D. Turgut"https://www.zbmath.org/authors/?q=ai:tollu.durhasan-turgut"Sahinkaya, A. Furkan"https://www.zbmath.org/authors/?q=ai:sahinkaya.a-furkanSummary: In this paper, we show that the following system of nonlinear difference equations
\[
x_{n+1}=\frac{x_nz_n+a}{x_n+z_n},\quad y_{n+1}=\frac{y_nx_n+a}{y_n+x_n},\quad z_{n+1}=\frac{z_ny_n+a}{z_n+y_n}
\]
for \(n\in\mathbb{N}_0=\mathbb{N}\cup\{0\}\) where the parameter \(a\) and the initial values \(x_0\), \(y_0\), \(z_0\) are real numbers, can be solved in explicit form. Also, we investigate the asymptotic behavior of the well-defined solutions by using these formulas.Dynamics of a higher order nonlinear rational difference equationhttps://www.zbmath.org/1483.390042022-05-16T20:40:13.078697Z"Zayed, Elsayed M. E."https://www.zbmath.org/authors/?q=ai:zayed.elsayed-m-e"Alngar, Mohamed E. M."https://www.zbmath.org/authors/?q=ai:alngar.mohamed-e-mSummary: In this article, we investigate the periodicity, the blondeness and the global stability of the positive solutions of the following nonlinear rational difference equation in a higher order:
\[
x_{n+1}=\left(\prod\limits^N_{i=1}x_{n-k_i}\right)/\left(\sum\limits^N_{i=1}\alpha_ix_{n-k_i}\right),\quad n=0,1,2,\dots
\]
where the parameters \(\alpha_i\in(0,\infty)\), \((i=1,2,\dots,N)\), while \(k_i(i=1,2,\dots,N)\) are positive integers, such that \(k_1<k_2<\dots<k_N\), with \(k_1=0\). The initial conditions \(x_{-k_N},x_{-k_N+1},\dots,x_{-k_2},x_{-k_2+1},\dots,x_{-1},x_0\in(0,\infty)\).Oscillation criteria for linear difference equations with several variable delayshttps://www.zbmath.org/1483.390052022-05-16T20:40:13.078697Z"Benekas, Vasileios"https://www.zbmath.org/authors/?q=ai:benekas.vasileios"Garab, Ábel"https://www.zbmath.org/authors/?q=ai:garab.abel"Kashkynbayev, Ardak"https://www.zbmath.org/authors/?q=ai:kashkynbayev.ardak"Stavroulakis, Ioannis P."https://www.zbmath.org/authors/?q=ai:stavroulakis.ioannis-pSummary: We obtain new sufficient criteria for the oscillation of all solutions of linear delay difference equations with several (variable) finite delays. Our results relax numerous well-known limes inferior-type oscillation criteria from the literature by letting the limes inferior be replaced by the limes superior under some additional assumptions related to slow variation. On the other hand, our findings generalize an oscillation criterion recently given for the case of a constant, single delay.Kneser-type oscillation criteria for second-order half-linear advanced difference equationshttps://www.zbmath.org/1483.390062022-05-16T20:40:13.078697Z"Indrajith, N."https://www.zbmath.org/authors/?q=ai:indrajith.n"Graef, John R."https://www.zbmath.org/authors/?q=ai:graef.john-r"Thandapani, E."https://www.zbmath.org/authors/?q=ai:thandapani.ethirajuSummary: The authors present Kneser-type oscillation criteria for a class of advanced type second-order difference equations. The results obtained are new and they improve and complement known results in the literature. Two examples are provided to illustrate the importance of the main results.A non-consistent boundary value problem of a generalized linear discrete time systemhttps://www.zbmath.org/1483.390072022-05-16T20:40:13.078697Z"Ortega, Fernando"https://www.zbmath.org/authors/?q=ai:ortega.fernando"Cho, Sung"https://www.zbmath.org/authors/?q=ai:cho.sung"Barros, Maria Filomena"https://www.zbmath.org/authors/?q=ai:barros.maria-filomenaSummary: In this article we study a class of generalised linear systems of difference equations with given boundary conditions and assume that the boundary value problem is non-consistent, i.e. it has infinite many or no solutions. We take into consideration the case that the coefficients are square constant matrices with the leading coefficient singular and provide optimal solutions. Numerical examples are given to justify our theory.Trinition the complex number with two imaginary parts: fractal, chaos and fractional calculushttps://www.zbmath.org/1483.390082022-05-16T20:40:13.078697Z"Atangana, Abdon"https://www.zbmath.org/authors/?q=ai:atangana.abdon"Mekkaoui, Toufik"https://www.zbmath.org/authors/?q=ai:mekkaoui.toufikSummary: Human being live in three-dimensional space; they can accurately visualize processes taking place in one, two and three dimensions. Although the set of bi-complex numbers and quaternion have attracted attention of many researchers in physics and related branches, they do not really represent processes taking place in the space where human being are located. We suggested a new set of complex number called ``the Trinition''. The new set is comprised between complex number with one imaginary part and complex number with three imaginary parts called quaternion/bi-complex numbers. We established a bijection between the new set and the three-dimensional space. We presented some important properties of the new set. We showed that all chaotic attractors in three dimension are simply three-dimensional mapping in the new set. Fewer examples of mapping in such set were presented. A new methodology that can be used to obtain more strange attractors are equally suggested. The methodology combines fractional chaotic models and some fractal mapping within the new set. Some illustrative figures are presented.On the three-dimensional consistency of Hirota's discrete Korteweg-de Vries equationhttps://www.zbmath.org/1483.390092022-05-16T20:40:13.078697Z"Joshi, Nalini"https://www.zbmath.org/authors/?q=ai:joshi.nalini"Nakazono, Nobutaka"https://www.zbmath.org/authors/?q=ai:nakazono.nobutakaSummary: Hirota's discrete Korteweg-de Vries equation (dKdV) is an integrable partial difference equation on \(\mathbb{Z}^2\), which approaches the Korteweg-de Vries equation in a continuum limit. We find new transformations to other equations, including a second-degree second-order partial difference equation, which provide an unusual embedding into a three-dimensional lattice. The consistency of the resulting system extends a property that has been widely used to study partial difference equations on multidimensional lattices.Continuous solutions to two iterative functional equationshttps://www.zbmath.org/1483.390102022-05-16T20:40:13.078697Z"Baron, Karol"https://www.zbmath.org/authors/?q=ai:baron.karolLet \((\Omega,\mathcal{A},P)\) be a probability space and \((X,\rho)\) a separable metric space with the \(\sigma\)-algebra \(\mathcal{B}\) of all its Borel subsets. Let \(f:X\times \Omega \to X\) be a \(\mathcal{B}\otimes \mathcal{A}\) measurable function. The author's aim is to look for continuous solutions \(\varphi: X \to \mathbb{R}\) of the equations
\begin{align*}
\varphi(x)&=F(x)-\int_{\Omega} \varphi(f(x,\omega))P(d\omega), \tag{1} \\
\varphi(x)&=F(x)+\int_{\Omega} \varphi(f(x,\omega))P(d\omega). \tag{2}
\end{align*}
Define
\[
f^0(x,\omega_1,\omega_2,\dots)=x, \qquad f^n(x,\omega_1,\omega_2,\dots)=f(f^{n-1}(x,\omega_1,\omega_2,\dots),\omega_n),
\]
and
\[
\pi_n^f(x,B)=P^{\infty}(f^n(x,\cdot)\in B), \quad n\in \mathbb{N}\cup \{0\}, \ B\in \mathcal{B}.
\]
Under the following conditions
\[
\int_{\Omega} \rho(f(x,\omega),f(z,\omega))P(d\omega)\le \lambda \rho(x,z), \quad x,z \in X, \ \lambda \in (0,1), \tag{3}
\]
and
\[
\int_{\Omega} \rho(f(x,\omega),x)P(d\omega)< \infty,
\]
there exists a probability Borel measure \(\pi^f\) on \(X\) such that for every \(x\in X\) the sequence \((\pi_n^f(x,\cdot))\) converges weakly to \(\pi^f\). Assuming these conditions with a fixed \(\lambda \in (0,1)\), let \(\mathcal{F}(X)\) be defined as the set of all continuous functions \(F:X \to \mathbb{R}\) such that there are a sequence \((F_n)\) of real functions on \(X\) and constants \(\theta \in (0,1)\), \(L\in (0,1/\lambda)\) and \(\alpha, \beta \in (0, \infty)\) such that
\[
|F(x)-F_n(x)|\le \alpha \theta^n, \quad x\in X,\ n\in \mathbb{N},
\]
and
\[
|F_n(x)-F_n(z)|\le \beta L^n\rho(x,z), \quad x,z \in X,\ n\in \mathbb{N}.
\]
The first result can now be stated.
Theorem. Assume the previous conditions. If \(F \in \mathcal{F}(X)\) then
\[
\varphi(x)=F(x)-\frac{1}{2}\int_X F(z)\pi^f(dz)+\sum_{n=1}^{\infty} (-1)^n\Big(\int_X F(z)\pi_n^f(x,dz)-\int_X F(z)\pi^f(dz)\Big), \quad x\in X,
\]
defines a continuous solution of (1). If additionally the condition \(\int_X F(x)\pi^f(dx)=0\) holds true, then the formula
\[
\varphi_0(x)=F(x)+\sum_{n=1}^{\infty} \int_X F(z)\pi_n^f(x,dz), \quad x\in X,
\]
defines a continuous solution \(\varphi_0:X \to \mathbb{R}\) of (2).
Concerning the problem of uniqueness of solution the following is proved.
Theorem. Assume the previous conditions. Let \(F \in \mathcal{F}(X)\).
\begin{itemize}
\item[(i)] If \(\varphi_1, \varphi_2\in \mathcal{F}(X)\) are solutions of (1), then \(\varphi_1=\varphi_2\).
\item[(ii)] If \(\varphi_1, \varphi_2\in \mathcal{F}(X)\) are solutions of (2), then \(\varphi_1-\varphi_2\) is a constant function.
\end{itemize}
The last problem investigated in the paper is about the number of functions \(F\) for which (1) and (2) have continuous solutions.
Assume that \((X,\rho)\) is a compact metric space and that condition (3) holds true. Define
\begin{align*}
\mathcal{F}_1&=\{F\in C(X): \text{Eq. (1) has a continuous solution} \}, \\
\mathcal{F}_2&=\{F\in C_f: \text{Eq. (2) has a continuous solution} \},
\end{align*}
where
\[
C_f=\{F\in C(X): \int_X F(x)\pi^f(dx)=0 \}.
\]
Theorem. Under the previous assumptions, the following holds:
\begin{itemize}
\item[(i)] \(\mathcal{F}_1\) is a Borel and dense subset of \(C(X)\), and if \(\mathcal{F}_1\neq C(X)\), then \(\mathcal{F}_1\) is of first category in \(C(X)\) and a Haar zero subset of \(C(X)\).
\item[(ii)] \(\mathcal{F}_2\) is a Borel and dense subset of \(C_f\), and if \(\mathcal{F}_2\neq C_f\), then \(\mathcal{F}_2\) is of first category in \(C_f\) and a Haar zero subset of \(C_f\).
\end{itemize}
Reviewer: Gian Luigi Forti (Milano)On Young's convolution inequality for Heisenberg groupshttps://www.zbmath.org/1483.390112022-05-16T20:40:13.078697Z"Christ, Michael"https://www.zbmath.org/authors/?q=ai:christ.michaelThe author studies the Young's convolution inequality for the Heisenberg group and obtains a characterization of the ordered pairs of functions that nearly achieve equality. This characterization is carried out by means of approximate solutions of a certain class of functional equations and by taking into account some structural features of the Heisenberg group. The main result of the paper states that an ordered triple of functions nearly saturates the inequality if and only if it differs by a small amount, in the relevant norm, from the image of these special ordered triple of Gaussians under some elements of symmetry group.
For the entire collection see [Zbl 1470.42001].
Reviewer: James Adedayo Oguntuase (Abeokuta)Bi-additive s-functional inequalities and biderivation in modular spaceshttps://www.zbmath.org/1483.390122022-05-16T20:40:13.078697Z"Shateri, T. L."https://www.zbmath.org/authors/?q=ai:shateri.tayebe-laal|shateri.tayebeh-lalFor an algebra \(\mathcal{A}\) and a \(\rho\)-complete modular space \(\mathcal{X}_{\rho}\) (with a convex modular \(\rho\) satisfying some additional properties), the author considers a mapping
\[
d: \mathcal{A}\times \mathcal{A}\to\mathcal{X}_{\rho}
\]
which satisfies some conditions and inequalities involving \(d\), \(\rho\) and a control mapping \(\psi\). It is shown that under these conditions the mapping \(d\) can be approximated by a bi-additive mapping \(D\). If in addition some other conditions are satisfied, in particular if
\[
\rho(d(xy,z)-d(x,z)y-zd(y,z))\leq\psi(x,y)\psi(z,z),\quad x,y,z\in \mathcal{A},
\]
then \(d\) is a bi-derivation.
In the proofs, a fixed point method is used. Some particular forms of the control mapping \(\psi\) are considered as well.
Reviewer: Jacek Chmieliński (Kraków)Interactions between Hlawka type-1 and type-2 quantitieshttps://www.zbmath.org/1483.390132022-05-16T20:40:13.078697Z"Luo, Xin"https://www.zbmath.org/authors/?q=ai:luo.xinSummary: The classical Hlawka inequality possesses deep connections with zonotopes and zonoids in convex geometry, and has been related to Minkowski space. We introduce Hlawka Type-1 and Type-2 quantities, and establish a Hlawka-type relation between them, which connects a vast number of strikingly different variants of the Hlawka inequalities, such as Serre's reverse Hlawka inequality in the future cone of the Minkowski space, the Hlawka inequality for subadditive functions on abelian groups by \textit{P. Ressel} [J. Math. Inequal. 9, No. 3, 883--888 (2015; Zbl 1333.26009)], and the integral analogs by \textit{S.-E. Takahasi} et al. [Math. Inequal. Appl. 3, No. 1, 63--67 (2000; Zbl 0949.26012); Math. Inequal. Appl. 12, No. 1, 1--10 (2009; Zbl 1177.26050)]. Besides, we announce several enhanced results, such as the Hlawka inequality for the power of measure functions. Particularly, we give a complete study of the Hlawka inequality for quadratic forms which relates to a work of \textit{D. Serre} [C. R., Math., Acad. Sci. Paris 353, No. 7, 629--633 (2015; Zbl 1322.51002)].Stability property of functional equations in modular spaces: a fixed-point approachhttps://www.zbmath.org/1483.390142022-05-16T20:40:13.078697Z"Saha, P."https://www.zbmath.org/authors/?q=ai:saha.parbati"Mondal, Pratap"https://www.zbmath.org/authors/?q=ai:mondal.pratap"Choudhury, B. S."https://www.zbmath.org/authors/?q=ai:choudhury.binayak-samadderThis paper is devoted to the problem of stability of quadratic functional equations in modular metric spaces in the Hyers-Ulam-Rassias sense. More precisely, the functional equation here considered is the Pappus functional equation, which arises from some geometrical considerations. The obtained results generalize previously known findings about the considered equation.
Reviewer: Bilal Bilalov (Baku)Fine structure of the dichotomy spectrumhttps://www.zbmath.org/1483.470112022-05-16T20:40:13.078697Z"Pötzsche, Christian"https://www.zbmath.org/authors/?q=ai:potzsche.christianSummary: The dichotomy spectrum is a crucial notion in the theory of dynamical systems, since it contains information on stability and robustness properties. However, recent applications in nonautonomous bifurcation theory showed that a detailed insight into the fine structure of this spectral notion is necessary. On this basis, we explore a helpful connection between the dichotomy spectrum and operator theory. It relates the asymptotic behavior of linear nonautonomous difference equations to the point, surjectivity and Fredholm spectra of weighted shifts. This link yields several dynamically meaningful subsets of the dichotomy spectrum, which not only allows to classify and detect bifurcations, but also simplifies proofs for results on the long term behavior of difference equations with explicitly time-dependent right-hand side.On the Bari basis properties of the root functions of non-self adjoint \(q\)-Sturm-Liouville problemshttps://www.zbmath.org/1483.470322022-05-16T20:40:13.078697Z"Allahverdiev, B. P."https://www.zbmath.org/authors/?q=ai:allahverdiev.bilender-pasaoglu"Tuna, H."https://www.zbmath.org/authors/?q=ai:tuna.huseyin|tuna.huseinSummary: This paper deals with the dissipative regular \(q\)-Sturm-Liouville problem. We prove that the system of root functions of this operator forms a Bari bases in the space \(L_q^2(I)\) by using the asymptotic behavior at infinity for its eigenvalues.Spectral analysis of certain spherically homogeneous graphshttps://www.zbmath.org/1483.470532022-05-16T20:40:13.078697Z"Breuer, Jonathan"https://www.zbmath.org/authors/?q=ai:breuer.jonathan"Keller, Matthias"https://www.zbmath.org/authors/?q=ai:keller.matthiasSummary: We study operators on rooted graphs with a certain spherical homogeneity. These graphs are called path commuting and allow for a decomposition of the adjacency matrix and the Laplacian into a direct sum of Jacobi matrices which reflect the structure of the graph. Thus, the spectral properties of the adjacency matrix and the Laplacian can be analyzed by means of the elaborated theory of Jacobi matrices. For some examples which include antitrees, we derive the decomposition explicitly and present a zoo of spectral behavior induced by the geometry of the graph. In particular, these examples show that spectral types are not at all stable under rough isometries.From backward approximations to Lagrange polynomials in discrete advection-reaction operatorshttps://www.zbmath.org/1483.471262022-05-16T20:40:13.078697Z"Solis, Francisco J."https://www.zbmath.org/authors/?q=ai:solis.francisco-javier"Barradas, Ignacio"https://www.zbmath.org/authors/?q=ai:barradas.ignacio"Juarez, Daniel"https://www.zbmath.org/authors/?q=ai:juarez.danielSummary: In this work we introduce a family of operators called discrete advection-reaction operators. These operators are important on their own right and can be used to efficiently analyze the asymptotic behavior of a finite differences discretization of variable coefficient advection-reaction-diffusion partial differential equations. They consist of linear bidimensional discrete dynamical systems defined in the space of real sequences. We calculate explicitly their asymptotic evolution by means of a matrix representation. Finally, we include the special case of matrices with different eigenvalues to show the connection between the operators evolution and interpolation theory.Regularized classical optimality conditions in iterative form for convex optimization problems for distributed Volterra-type systemshttps://www.zbmath.org/1483.490292022-05-16T20:40:13.078697Z"Sumin, Vladimir Iosifovich"https://www.zbmath.org/authors/?q=ai:sumin.v-i"Sumin, Mikhail Iosifovich"https://www.zbmath.org/authors/?q=ai:sumin.mikhail-iosifovichSummary: We consider the regularization of the \textit{classical optimality conditions} (COCs) -- the Lagrange principle and the Pontryagin maximum principle -- in a convex optimal control problem with functional constraints of equality and inequality type. The system to be controlled is given by a general linear functional-operator equation of the second kind in the space \(L^m_2\), the main operator of the right-hand side of the equation is assumed to be quasinilpotent. The objective functional of the problem is strongly convex. Obtaining regularized COCs in iterative form is based on the use of the iterative dual regularization method. The main purpose of the regularized Lagrange principle and the Pontryagin maximum principle obtained in the work in iterative form is stable generation of minimizing approximate solutions in the sense of J. Warga. Regularized COCs in iterative form are formulated as existence theorems in the original problem of minimizing approximate solutions. They ``overcome'' the ill-posedness properties of the COCs and are regularizing algorithms for solving optimization problems. As an illustrative example, we consider an optimal control problem associated with a hyperbolic system of first-order differential equations.Harnack and shift Harnack inequalities for degenerate (functional) stochastic partial differential equations with singular driftshttps://www.zbmath.org/1483.600842022-05-16T20:40:13.078697Z"Lv, Wujun"https://www.zbmath.org/authors/?q=ai:lv.wujun"Huang, Xing"https://www.zbmath.org/authors/?q=ai:huang.xingSummary: The existence and uniqueness of the mild solutions for a class of degenerate functional stochastic partial differential equations (SPDEs) are obtained, where the drift is assumed to be Hölder-Dini continuous. Moreover, the non-explosion of the solution is proved under some reasonable conditions. In addition, the Harnack inequality is derived by the method of coupling by change of measure. Finally, the shift Harnack inequality is obtained for the equations without delay, which is new even in the non-degenerate case. An example is presented in the final part of the paper.A lower bound for the spectral gap of the conjugate Kac process with 3 interacting particleshttps://www.zbmath.org/1483.820092022-05-16T20:40:13.078697Z"Ferreira, Luís Simão"https://www.zbmath.org/authors/?q=ai:ferreira.luis-simaoSummary: In this paper, we proceed as suggested in the final section of [\textit{E. Carlen} et al., Ann. Probab. 48, No. 6, 2807--2844 (2020; Zbl 1456.60252)] and prove a lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles. This bound turns out to be around \(0.02 \), which is already physically meaningful, and we perform Monte Carlo simulations to provide a better empirical estimate for this value via entropy production inequalities. This finishes a complete quantitative estimate of the spectral gap of the Kac process.A probabilistic point of view on peak effects in linear difference equationshttps://www.zbmath.org/1483.932592022-05-16T20:40:13.078697Z"Shcherbakov, Pavel"https://www.zbmath.org/authors/?q=ai:shcherbakov.pavel-s"Dabbene, Fabrizio"https://www.zbmath.org/authors/?q=ai:dabbene.fabrizioSummary: It is known from the literature that solutions of homogeneous linear stable difference equations may experience large deviations, or peaks, from the nonzero initial conditions at finite time instants. While the problem has been studied from a deterministic standpoint, not much is known about the probability of occurrence of such event when both the initial conditions and the coefficients of the equation have random nature. In this paper, by exploiting results on the volume of the Schur domain, we are able to compute the probability for deviations to occur. This turns out to be very close to unity, even for equations of low degree. Hence, we claim that \textit{``solutions of stable difference equations probably experience peak''}. Then, we make use of tools from statistical learning to address other issues such as evaluation of the mean magnitude and maximum value of peak.