Recent zbMATH articles in MSC 39https://zbmath.org/atom/cc/392021-01-08T12:24:00+00:00WerkzeugAsymptotic stability of a class of second order difference equation.https://zbmath.org/1449.390222021-01-08T12:24:00+00:00"Quan, Weizhen"https://zbmath.org/authors/?q=ai:quan.weizhen"Li, Xiaopei"https://zbmath.org/authors/?q=ai:li.xiaopei"Ma, Xiuxian"https://zbmath.org/authors/?q=ai:ma.xiuxian"Zeng, Yongjun"https://zbmath.org/authors/?q=ai:zeng.yongjun"Liu, Linniang"https://zbmath.org/authors/?q=ai:liu.linniang"Liu, Qiuju"https://zbmath.org/authors/?q=ai:liu.qiujuSummary: In this paper, firstly, a new proof of the theorem of asymptotic stability of a second order difference equation \(x_{n + 1} = \frac{p + q{x_n}}{1 + {x_n} + rx_{n-1}}\) in reference is given. Secondly, the asymptotic stability of zero and positive equilibrium solutions of a second order difference equation \(x_{n + 1} = \frac{qx_n}{1+x_{n-1}+rx_n}\) is studied.Development and research of linearly parametric discrete model of amplitude-frequency characteristic of mechanical system with linearly viscous friction.https://zbmath.org/1449.741562021-01-08T12:24:00+00:00"Popova, D. N."https://zbmath.org/authors/?q=ai:popova.d-n"Zoteev, V. E."https://zbmath.org/authors/?q=ai:zoteev.vladimir-evgenevichSummary: We consider the construction of linearly parametric discrete model in form of stochastic difference equation which describes the sequence of measurements of amplitude-frequency characteristic of dissipative system with linearly viscous friction. The numerical iterative method of mean-quadratic estimation of coefficients of stochastic difference equation is suggested. The comparative analysis of introduced algorithm of calculation of system dissipative characteristic with known method of resonance curve is made.The Green's function of fourth-order difference equation with periodic boundary value problem.https://zbmath.org/1449.390172021-01-08T12:24:00+00:00"Jiang, Lingfang"https://zbmath.org/authors/?q=ai:jiang.lingfang"Liu, Aihua"https://zbmath.org/authors/?q=ai:liu.aihuaSummary: In this paper, we study the Green's function of fourth-order difference equation with periodic boundary value problem. We obtain some new results and generalize some results in a literature.Dynamical behavior of a difference competitive system incorporating harvesting.https://zbmath.org/1449.390132021-01-08T12:24:00+00:00"Su, Qianqian"https://zbmath.org/authors/?q=ai:su.qianqianSummary: In this paper, we propose and study a class of almost periodic discrete harvesting competition system. With the help of the theory of difference inequality and some calculation technique, sufficient conditions are obtained to ensure the permanence of the system. By applying the relevant theorems of discrete almost periodic system, sufficient conditions which ensure the existence, uniqueness and uniformly asymptotical stability of almost periodic solution of the system are established. The plausibility of the main results is demonstrated by some numerical simulations.Equality and homogeneity of generalized integral means.https://zbmath.org/1449.260522021-01-08T12:24:00+00:00"Páles, Zs."https://zbmath.org/authors/?q=ai:pales.zsolt"Zakaria, A."https://zbmath.org/authors/?q=ai:zakaria.amrThe authors mainly generalize the results of the papers by \textit{L. Losonczi} and \textit{Z. Páles} [Aequationes Math. 81, No. 1--2, 31--53 (2011; Zbl 1234.39003)] and \textit{L. Losonczi} [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 41, 103--117 (2013; Zbl 1289.39045)] and also many former results obtained in various particular cases of this problem.
As direct applications of the results obtained on the equality of generalized Bajraktarevic means, they consider and solve the homogeneity problem of these means under general conditions.
A section is entirely dedicated to characterize the equality of generalized quasi-arithmetic means in various settings and the next to the homogeneity of generalized Bajraktarevic means under 3 times differentiability assumptions.
Finally, in the last section, the authors consider particular cases of the characterization of the equality of generalized Bajraktarevic means under 3 times differentiability assumptions, precisely when all the means are generalized quasi-arithmetic.
Reviewer: Maria Alessandra Ragusa (Catania)Anti-periodic solutions for discrete systems with exponential dichotomy.https://zbmath.org/1449.390102021-01-08T12:24:00+00:00"Meng, Xin"https://zbmath.org/authors/?q=ai:meng.xinSummary: The author considered the anti-periodic solutions for a class of nonlinear discrete systems with exponential dichotomy. Firstly, it was proved that if the homogeneous linear system had exponential dichotomy, then the corresponding nonhomogeneous linear system had an anti-periodic solution. Secondly, by means of this conclusion and the Banach fixed point theorem, a sufficient condition for the existence and uniqueness of anti-periodic solutions for nonlinear discrete systems was given. Finally, an application example was given.Global attractivity of a discrete commensalism system with infinite delays.https://zbmath.org/1449.390262021-01-08T12:24:00+00:00"Xue, Yalong"https://zbmath.org/authors/?q=ai:xue.yalong"Xie, Xiangdong"https://zbmath.org/authors/?q=ai:xie.xiangdong"Lin, Qifa"https://zbmath.org/authors/?q=ai:lin.qifa"Chen, Fengde"https://zbmath.org/authors/?q=ai:chen.fengdeSummary: A two species discrete commensalism system with infinite delays is proposed and studied. Based on the discrete comparison theorem, the permanence of the system is obtained. Then, by constructing discrete Lyapunov functional, a set of sufficient conditions which guarantee the system global attractivity are obtained. An example together with its numeric simulation is given to illustrate the feasibility of our main result.Gelfand pairs over hypergroup joins.https://zbmath.org/1449.200702021-01-08T12:24:00+00:00"Vati, K."https://zbmath.org/authors/?q=ai:vati.kedumetseGelfand pairs over a locally compact hypergroup are characterized by the commutativity of double coset hypergroups.
Reviewer: Wiesław A. Dudek (Wrocław)Difference characteristic method and exact solution for a class of biological model equation.https://zbmath.org/1449.390232021-01-08T12:24:00+00:00"Jiang, Kun"https://zbmath.org/authors/?q=ai:jiang.kun"Wang, Zhike"https://zbmath.org/authors/?q=ai:wang.zhike"Li, Wenting"https://zbmath.org/authors/?q=ai:li.wentingSummary: In this paper, the difference characteristic set method is used to study a class of nonlinear differential equations I and II which have special biological properties. First of all, the relevant definitions and important theorems of the difference characteristic set method are proposed. After that, the Z-transformation method is introduced by the definitions and properties. And then, in the third part of the paper, two differential equations I, II with biological properties are studied by the difference characteristic set method step by step, such as changing the difference equations, finding the characteristic sets and zero sets, judging the consistency and irreducibility of the sets, etc. In addition, combing with the given initial conditions, the Z-transformation method is used to solve the zero sets of the differential equations I, II. At last, two groups of the exact solution of equations I, II are obtained respectively.On a class of difference equations involving a linear map with two dimensional kernel.https://zbmath.org/1449.390092021-01-08T12:24:00+00:00"Ferreira, Luis Simão"https://zbmath.org/authors/?q=ai:ferreira.luis-simao"Sanchez Rodrigues, Luis"https://zbmath.org/authors/?q=ai:sanchez-rodrigues.luisSummary: We establish necessary and sufficient conditions for the existence of periodic solutions to second-order nonlinear difference equations of the form \(\Delta^2x_i+\lambda x_i+\Delta f(x_i)=e_i\), \(i\in{\mathbb{N}}\), and for a simpler equation with difference-free nonlinearity. The linear part of the equation has two-dimensional kernel.On the generalized Hyers-Ulam stability of an \(n\)-dimensional quadratic and additive type functional equation.https://zbmath.org/1449.390282021-01-08T12:24:00+00:00"Lee, Yang-Hi"https://zbmath.org/authors/?q=ai:lee.yang-hiSummary: We investigate the generalized Hyers-Ulam stability of a functional equation \(f \left(\sum_{j=1}^n x_j\right) +(n - 2) \sum_{j = 1}^nf(x_j) - \sum_{1 \leq i < j \leq n} f(x_i + x_j) = 0\).On quasi-periodic solutions of forced higher order nonlinear difference equations.https://zbmath.org/1449.390122021-01-08T12:24:00+00:00"Qian, Chuanxi"https://zbmath.org/authors/?q=ai:qian.chuanxi"Smith, Justin"https://zbmath.org/authors/?q=ai:smith.justin-w|smith.justin-rSummary: Consider the following higher order difference equation \[ x(n+1)= f(n,x(n))+g(n,x(n-k))+b(n), \qquad n=0, 1, \dots \] where \(f(n,x), g(n,x): \{0, 1, \dots \}\times [0, \infty) \rightarrow [0,\infty)\) are continuous functions in \(x\) and periodic functions with period \(\omega\) in \(n\), \(\{b(n)\}\) is a real sequence, and \(k\) is a nonnegative integer. We show that under proper conditions, every nonnegative solution of the equation is quasi-periodic with period \(\omega\). Applications to some other difference equations derived from mathematical biology are also given.A note on sums of a class of series.https://zbmath.org/1449.330062021-01-08T12:24:00+00:00"Jun, Sungtae"https://zbmath.org/authors/?q=ai:jun.sungtae"Milovanović, Gradimir V."https://zbmath.org/authors/?q=ai:milovanovic.gradimir-v"Kim, Insuk"https://zbmath.org/authors/?q=ai:kim.insuk"Rathie, Arjun K."https://zbmath.org/authors/?q=ai:rathie.arjun-kumarSummary: The aim of this note is to provide sums of a unified class of series of the form \[S_i(a)=\sum_{k=0}^{\infty} (-1)^{k} \binom{a-i}{k} \frac{1}{2^{k}(a+k+1)}\] in the most general form for any \(i\in\mathbb{Z}\). For each \(\nu\in\mathbb{N}\), in four cases when \(i=\pm 2\nu\) and \(i=\pm(2\nu-1)\), simple explicit expressions for \(S_i(a)\) are obtained, e.g. \[S_{2\nu}(a)=\frac{2^{2\nu-1-a}}{(a-2\nu+1)_\nu}\left[\frac{\sqrt{\pi}\, \Gamma (a+1)}{\Gamma \left(a+\frac{3}{2}-\nu\right)}-P_{\nu-1}(a)\right],\] where \(P_\nu(a)\) is an algebraic polynomial in \(a\) of degree \(\nu\).
For \(i=1\) and \(a=n\) \((\in \mathbb{N})\), we recover the well known sum of the series due to \textit{M. Vowe} and \textit{H.-J. Seiffert} [Elem. Math. 42, No. 4, 111--112 (1987; Zbl 1253.33007)]. Several other known results due to \textit{H. M. Srivastava} [Proc. Japan Acad., Ser. A 65, No. 1, 8--11 (1989; Zbl 0653.33004)] and \textit{Y. S. Kim} et al. [Commun. Korean Math. Soc. 27, No. 4, 745--751 (2012; Zbl 1253.33004)] can be considered as special cases of our result.Dynamical behavior of a discrete Leslie-Gower-type food chain model.https://zbmath.org/1449.370592021-01-08T12:24:00+00:00"Su, Qianqian"https://zbmath.org/authors/?q=ai:su.qianqianSummary: In this paper, we study the dynamics behavior of a discrete Leslie-Gower three-dimensional food chain model. By using differential inequality, we get the conclusion that under some conditions, the species \({x_1}\) and \({x_3}\) are permanent and the species \({x_2}\) will be driven to extinction. Then, by constructing a suitable Lyapunov function, sufficient conditions are obtained to ensure the global attractivity of the system, which promotes the results of a literature.Upper and lower solutions and topological degree in difference equations boundary value problems.https://zbmath.org/1449.390212021-01-08T12:24:00+00:00"Zheng, Ying"https://zbmath.org/authors/?q=ai:zheng.ying"Wang, Faxing"https://zbmath.org/authors/?q=ai:wang.faxing"Gao, Guanghua"https://zbmath.org/authors/?q=ai:gao.guanghuaSummary: This paper deals with second order nonlinear difference equation \({\Delta^2}u (t-1) = f (t,u (t))\), \(t \in [1, T]\) with different boundary conditions, where \(f:[1, T] \times \mathbf{R} \to \mathbf{R}\) is continuous, \(T\ge 1\) is a fixed natural number. Firstly, we consider the case of well order lower and upper solutions. Secondly, we investigate the case of upper and lower solutions having the opposite orders. We prove the relation between the topological degree and strict upper and lower solutions in both cases and using this we get the existence results for the discrete boundary value problems under consideration.Nonconstant periodic solutions of discrete \(p\)-Laplacian system via Clark duality and computations of the critical groups.https://zbmath.org/1449.390112021-01-08T12:24:00+00:00"Zheng, Bo"https://zbmath.org/authors/?q=ai:zheng.bo|zheng.bo.1Summary: We study the existence of periodic solutions to a discrete \(p\)-Laplacian system. By using the Clark duality method and computing the critical groups, we find a simple condition that is sufficient to ensure the existence of nonconstant periodic solutions to the system.On the solutions and periodicity of some rational systems of difference equations.https://zbmath.org/1449.390042021-01-08T12:24:00+00:00"Elsayed, Elsayed M."https://zbmath.org/authors/?q=ai:elsayed.elsayed-mohammedSummary: In this paper we deal with the form of the solutions and the periodicity nature of the following systems of nonlinear difference equations \[ x_{n+1} =\frac{x_{n-3}y_{n-2}}{y_n(\pm 1\pm x_{n-1}y_{n-2}x_{n-3}},\quad y_{n+1} =\frac{x_{n-2}y_{n-3}}{x_n(\pm 1\pm y_{n-1}x_{n-2}y_{n-3}}, \] where the initial conditions \(x_{-3}, x_{-2}, x_{-1}, x_0, y_{-3}, y_{-2}, y_{-1}\), and \(y_0\) are nonzero real numbers.Stability and numerical approximation for a spacial class of semilinear parabolic equations on the Lipschitz bounded regions: Sivashinsky equation.https://zbmath.org/1449.130162021-01-08T12:24:00+00:00"Mesrizadeh, Mehdi"https://zbmath.org/authors/?q=ai:mesrizadeh.mehdi"Shanazari, Kamal"https://zbmath.org/authors/?q=ai:shanazari.kamalSummary: This paper aims to investigate the stability and numerical approximation of the Sivashinsky equations. We apply the Galerkin meshfree method based on the radial basis functions (RBFs) to discretize the spatial variables and use a group presenting scheme for the time discretization. Because the RBFs do not generally vanish on the boundary, they can not directly approximate a Dirichlet boundary problem by Galerkin method. To avoid this difficulty, an auxiliary parametrized technique is used to convert a Dirichlet boundary condition to a Robin one. In addition, we extend a stability theorem on the higher order elliptic equations such as the biharmonic equation by the eigenfunction expansion. Some experimental results will be presented to show the performance of the proposed method.On growth of meromorphic solutions of some kind of non-homogeneous linear difference equations.https://zbmath.org/1449.300772021-01-08T12:24:00+00:00"Zheng, Xiu-Min"https://zbmath.org/authors/?q=ai:zheng.xiumin"Zhou, Yan-Ping"https://zbmath.org/authors/?q=ai:zhou.yanpingSummary: In this paper, we investigate the growth of meromorphic solutions of some kind of non-homogeneous linear difference equations with special meromorphic coefficients. When there are more than one coefficient having the same maximal order and the same maximal type, the estimates on the lower bound of the order of meromorphic solutions of the involved equations are obtained. Meanwhile, the above estimates are sharpened by combining the relative results of the corresponding homogeneous linear difference equations.On a system of difference equations of second order solved in closed form.https://zbmath.org/1449.390032021-01-08T12:24:00+00:00"Akrour, Youssouf"https://zbmath.org/authors/?q=ai:akrour.youssouf"Touafek, Nouressadat"https://zbmath.org/authors/?q=ai:touafek.nouressadat"Halim, Yacine"https://zbmath.org/authors/?q=ai:halim.yacineSummary: In this work we solve in closed form the system of difference equations \[ x_{n+1}=\dfrac{ay_nx_{n-1}+bx_{n-1}+c}{y_nx_{n-1}},\; y_{n+1}=\dfrac{ax_ny_{n-1}+by_{n-1}+c}{x_ny_{n-1}},\;n=0,1,\dots,\] where the initial values \(x_{-1}\), \(x_0\), \(y_{-1}\) and \(y_0\) are arbitrary nonzero real numbers and the parameters \(a\), \(b\) and \(c\) are arbitrary real numbers with \(c\ne 0\). In particular we represent the solutions of some particular cases of this system in terms of Tribonacci and Padovan numbers and we prove the global stability of the corresponding positive equilibrium points. The results obtained here extend those obtained in some recent papers.Common fixed point theorem for generalized nonexpansive mappings on ordered orbitally complete metric spaces and application.https://zbmath.org/1449.540842021-01-08T12:24:00+00:00"Nashine, Hemant Kumar"https://zbmath.org/authors/?q=ai:nashine.hemant-kumar"Agarwal, Ravi P."https://zbmath.org/authors/?q=ai:agarwal.ravi-pSummary: We propose a common fixed point theorem for a new notion of generalized nonexpansive mappings for two pairs of maps in an ordered orbitally complete metric space. To illustrate our result, we give throughout the paper two examples. Existence of solutions for a certain system of functional equations arising in dynamic programming is also presented as application.Existence and uniqueness of solutions to anti-periodic boundary value problems of fractional difference equations.https://zbmath.org/1449.390162021-01-08T12:24:00+00:00"Hu, Weimin"https://zbmath.org/authors/?q=ai:hu.weimin"Zhen, Jianfang"https://zbmath.org/authors/?q=ai:zhen.jianfangSummary: The paper studied the existence and uniqueness of the solutions to the Caputo fractional difference equation with anti-periodic boundary value problems. First, the Green's function was given by using the Caputo fractional difference equation and anti-periodic boundary value conditions. By applying the Banach contraction mapping principle, some existence and uniqueness of solutions were obtained.On the stability of \( (\alpha, \beta)\)-derivations in cone Banach spaces.https://zbmath.org/1449.390292021-01-08T12:24:00+00:00"Liu, Jianhua"https://zbmath.org/authors/?q=ai:liu.jianhua"Meng, Qing"https://zbmath.org/authors/?q=ai:meng.qingSummary: The Hyers-Ulam stability of \( (\alpha, \beta)\)-derivations from a unital ring \(R\) to a \(R\)-bimodule which is a cone Banach space with the cone norm \(\|\cdot\|_P\), where \(P\) is a normal cone in a real Banach space \(E\), is investigated associated with the following functional equation \[f ((a + b)c) = f (a)\alpha (c) + \beta (a)f (c) + f (b)\alpha (c) + \beta (b)f (c)\] using the fixed point method and the direct method.A generalization of Elsayed's solution to the difference equation \(X_{n+1}=\frac{ X_{n-5}}{-1 + X_{n-2}X_{n-5}}\).https://zbmath.org/1449.390052021-01-08T12:24:00+00:00"Folly-Gbetoula, Mensah"https://zbmath.org/authors/?q=ai:folly-gbetoula.mensah-k"Nyirenda, Darlison"https://zbmath.org/authors/?q=ai:nyirenda.darlisonSummary: In this paper, we obtain solutions to difference equations of the form \[ x_{n+1}=\frac{ x_{n-5}}{a_n+b_n x_{n-2}x_{n-5}},\] where \((a_{n})\) and \((b_{n})\) are sequences of real numbers. Consequently, a result of \textit{E. M. Elsayed} [Eur. J. Pure Appl. Math. 4, No. 3, 287--303 (2011; Zbl 1389.39003)]
is generalized. To achieve this, we use Lie symmetry analysis.Quasimonotonicity and functional inequalities.https://zbmath.org/1449.354642021-01-08T12:24:00+00:00"Herzog, Gerd"https://zbmath.org/authors/?q=ai:herzog.gerd"Volkmann, Peter"https://zbmath.org/authors/?q=ai:volkmann.peterSummary: A comparison theorem for functional equations in ordered topological vector spaces will be given, which generalizes the results from \textit{P. Volkmann} [ISNM, Int. Ser. Numer. Math. 161, 269--273 (2012; Zbl 1253.26043); Ein Vergleichssatz für Integralgleichungen, KITopen, 3 p. (2016; \url{doi:10.5445/IR/1000061837})]. Quasimonotonicity is fundamental for these investigations.Growth, zeros and fixed points of differences of meromorphic solutions of difference equations.https://zbmath.org/1449.300682021-01-08T12:24:00+00:00"Lan, Shuang-ting"https://zbmath.org/authors/?q=ai:lan.shuangting"Chen, Zong-xuan"https://zbmath.org/authors/?q=ai:chen.zongxuanSummary: In this paper, we study the difference equation
\[
a_1 (z) f(z+1) + a_0(z) f(z)=0,
\]
where \(a_1(z)\) and \(a_0(z)\) are entire functions of finite order. Under some conditions, we obtain some properties, such as fixed points, zeros etc., of the differences and forward differences of meromorphic solutions of the above equation.Chaotification schemes for first-order partial difference equations via sawtooth functions.https://zbmath.org/1449.390022021-01-08T12:24:00+00:00"Liang, Wei"https://zbmath.org/authors/?q=ai:liang.wei"Shi, Yuming"https://zbmath.org/authors/?q=ai:shi.yumingSummary: This paper is concerned with chaotification problems of first-order partial difference equations with non-period boundary conditions. Two chaotification schemes for the difference equations via sawtooth functions are established, and all the controlled systems are proved to be chaotic in the sense of both Devaney and Li-Yorke by applying the coupled-expansion theory of general discrete dynamical systems. At the end, one illustrative example is provided.Global dynamics of certain non-symmetric second order difference equation with quadratic term.https://zbmath.org/1449.390062021-01-08T12:24:00+00:00"Garić-Demirović, M."https://zbmath.org/authors/?q=ai:garic-demirovic.mirela"Hrustić, S."https://zbmath.org/authors/?q=ai:hrustic.sabina"Moranjkić, S."https://zbmath.org/authors/?q=ai:moranjkic.samraSummary: We investigate global dynamics of the equation \[ x_{n+1}=\frac{x_{n-1}+F}{ax^2_n+f},\quad n=0,1,2,\dots,\]
where the parameters \(a\), \(F\) and \(f\) are positive numbers and the initial conditions \(x_{-1}\), \(x_0\) are arbitrary nonnegative numbers such that \(x_{-1} + x_0 > 0\). The existence and local stability of the unique positive equilibrium are analyzed algebraically. We characterize the global dynamics of this equation with the basins of attraction of its equilibrium point and periodic solutions.Stability of a quintic functional equation in matrix paranormed spaces.https://zbmath.org/1449.390302021-01-08T12:24:00+00:00"Wang, Zhihua"https://zbmath.org/authors/?q=ai:wang.zhihuaSummary: The main goal of this paper is to prove the Hyers-Ulam stability of the quintic functional equation \(f (3x + y) - 5y (2x + y) + f (2x - y) + 10f (x + y) - 5f (x - y) = 10f (y) + f (3x) - 3f (2x) - 27f (x)\) in matrix paranormed spaces. In addition, we establish results concerning the Hyers-Ulam stability of the above functional equation in matrix Banach spaces, and then apply these results to prove the Hyers-Ulam stability of this functional equation in \({L^\infty}\)-normed Banach spaces.Global dynamics of certain mix monotone difference equation via center manifold theory and theory of monotone maps.https://zbmath.org/1449.390072021-01-08T12:24:00+00:00"Kulenović, Mustafa R. S."https://zbmath.org/authors/?q=ai:kulenovic.mustafa-r-s"Nurkanović, Mehmed"https://zbmath.org/authors/?q=ai:nurkanovic.mehmed"Nurkanović, Zehra"https://zbmath.org/authors/?q=ai:nurkanovic.zehraSummary: We investigate the global dynamics of the following rational difference equation of second order
\[ x_{n+1}=\frac{Ax^2_n+Ex_{n-1}}{x^2_n+f},\quad n=0,1,\dots,\]
where the parameters \(A\) and \(E\) are positive real numbers and the initial conditions \(x_{-1}\) and \(x_0\) are arbitrary non-negative real numbers such that \(x_{-1} + x_0 > 0\). The transition function associated with the right-hand side of this equation is always increasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric values. The unique feature of this equation is that the second iterate of the map associated with this transition function changes from strongly competitive to strongly cooperative. Our main tool for studying the global dynamics of this equation is the theory of monotone maps while the local stability is determined by using center manifold theory in the case of the nonhyperbolic equilibrium point.On the solution of Fermat-type differential-difference equations.https://zbmath.org/1449.300692021-01-08T12:24:00+00:00"Liu, Dan"https://zbmath.org/authors/?q=ai:liu.dan"Deng, Bingmao"https://zbmath.org/authors/?q=ai:deng.bingmao"Yang, Degui"https://zbmath.org/authors/?q=ai:yang.deguiSummary: In this paper, we mainly discuss the entire solutions of finite order of the following Fermat type differential-difference equation \[[f^{ (k)} (z)]^2 + [{\Delta_c}f (z)]^2 = 1,\] and the systems of differential-difference equations of the form \[\begin{cases}[{f_1^{ (k)}} (z)]^2 + [{\Delta_c}{f_2} (z)]^2 = 1, \\ [{f_2^{ (k)}} (z)]^2 + [{\Delta_c}{f_1} (z)]^2 = 1.\end{cases}\] Our results can be proved to be the sufficient and necessary solutions to both the equation and the system of equations.Further results on meromorphic functions and their \(n\)th order exact differences with three shared values.https://zbmath.org/1449.300652021-01-08T12:24:00+00:00"Chen, Shengjiang"https://zbmath.org/authors/?q=ai:chen.shengjiang"Xu, Aizhu"https://zbmath.org/authors/?q=ai:xu.aizhu"Lin, Xiuqing"https://zbmath.org/authors/?q=ai:lin.xiuqingSummary: Let \(E (a,f)\) be the set of \(a\)-points of a meromorphic function \(f (z)\) counting multiplicities. We prove that if a transcendental meromorphic function \(f (z)\) of hyper order is strictly less than 1 and its \(n\)th exact difference \(\Delta_c^nf (z)\) satisfies \(E (1,f) = E (1,\Delta_c^nf)\), \(E (0,f) \subset E (0, \Delta_c^nf)\) and \(E (\infty, f) \supset E (\infty, \Delta_c^nf)\), then \(\Delta_c^nf (z) \equiv f (z)\). This result improves a more recent theorem by using a simple method.Existence of solutions to discrete and continuous second-order boundary value problems via Lyapunov functions and a priori bounds.https://zbmath.org/1449.390192021-01-08T12:24:00+00:00"Tisdell, Christopher"https://zbmath.org/authors/?q=ai:tisdell.christopher-c"Liu, Yongjian"https://zbmath.org/authors/?q=ai:liu.yongjian"Liu, Zhenhai"https://zbmath.org/authors/?q=ai:liu.zhenhaiSummary: This article analyzes nonlinear, second-order difference equations subject to ``left-focal'' two-point boundary conditions. Our research questions are: RQ1: What are new, sufficient conditions under which solutions to our ``discrete'' problem will exist?; RQ2: What, if any, is the relationship between solutions to the discrete problem and solutions of the ``continuous'', left-focal analogue involving second-order ordinary differential equations? Our approach involves obtaining new a priori bounds on solutions to the discrete problem, with the bounds being independent of the step size. We then apply these bounds, through the use of topological degree theory, to yield the existence of at least one solution to the discrete problem. Lastly, we show that solutions to the discrete problem will converge to solutions of the continuous problem.Hypergeometric type difference equations on nonuniform lattices: Rodrigues type representation for the second kind solution.https://zbmath.org/1449.330112021-01-08T12:24:00+00:00"Cheng, Jinfa"https://zbmath.org/authors/?q=ai:cheng.jinfa"Jia, Lukun"https://zbmath.org/authors/?q=ai:jia.lukunSummary: By building a second order adjoint equation, the Rodrigues type representation for the second kind solution of a second order difference equation of hypergeometric type on nonuniform lattices is given. The general solution of the equation in the form of a combination of a standard Rodrigues formula and a ``generalized'' Rodrigues formula is also established.Parametrical identification of creep's curve on the basis of stochastic difference equations.https://zbmath.org/1449.742042021-01-08T12:24:00+00:00"Zoteev, V. E."https://zbmath.org/authors/?q=ai:zoteev.vladimir-evgenevichSummary: The numerical method of parametrical identification of the creep's curve is considered, allowing to increase accuracy of forecasting of processes not elastic deformations in tasks of an estimation of individual behavior of a concrete element of a structure. The method is based on linear parametrical discrete model, describing in the form of stochastic difference equations results of supervision of creep's curve during experiment lays.Entire function solutions of two types of Fermat type \(q\)-difference differential equations.https://zbmath.org/1449.300532021-01-08T12:24:00+00:00"Fan, Bo"https://zbmath.org/authors/?q=ai:fan.bo"Ding, Jie"https://zbmath.org/authors/?q=ai:ding.jieSummary: In this paper, using Nevanlinna's value distribution theory and the complex differential equations theory, the existence of finite order transcendental entire function solutions for two types of Fermat type \(q\)-difference differential equations of the following form \[{f^2} (qz+c) + ({f^{ (k)}} (z))^2 = 1,\; [f (qz+c) - f (z)]^2 + ({f^{ (k)}} (z))^2 = 1\] is investigated. Moreover, the precise expression of the solutions is obtained under some assumptions.Some results on difference Riccati equations and delay differential equations.https://zbmath.org/1449.300492021-01-08T12:24:00+00:00"Wang, Qiong"https://zbmath.org/authors/?q=ai:wang.qiong"Long, Fang"https://zbmath.org/authors/?q=ai:long.fang"Wang, Jun"https://zbmath.org/authors/?q=ai:wang.jun.2Summary: We investigate difference Riccati equations with rational coefficients and delay differential equations with constant coefficients. For difference Riccati equations with some relation among coefficients, we prove that every transcendental meromorphic solution is of order no less than one. We also consider the rational solutions for delay differential equations.Permanence of a discrete logistic equation with pure time delays.https://zbmath.org/1449.390242021-01-08T12:24:00+00:00"Lv, Yangyang"https://zbmath.org/authors/?q=ai:lv.yangyang"Chen, Lijuan"https://zbmath.org/authors/?q=ai:chen.lijuan"Chen, Liujuan"https://zbmath.org/authors/?q=ai:chen.liujuanSummary: In this paper we propose a discrete logistic system with pure delays. By giving the detail analysis of the right-hand side functional of the system, we consider its permanence property which is one of the most important topic in the study of population dynamics. The results obtained in this paper are good extensions of the existing results to the discrete case. Also we give an example to show the feasibility of our main results.Dynamical behavior of first order nonlinear fuzzy difference equation.https://zbmath.org/1449.390152021-01-08T12:24:00+00:00"Zhang, Qianhong"https://zbmath.org/authors/?q=ai:zhang.qianhong"Wang, Guiying"https://zbmath.org/authors/?q=ai:wang.guiyingSummary: In this paper, using the \(g\)-division of fuzzy numbers, we study the existence, the boundedness and the asymptotic behavior of the positive solutions of the first order nonlinear fuzzy difference equation, \({x_{n + 1}} = M + \frac{{{x_n} + A}}{{{x_n} + B}}\), \(n= 0,1, \dots\), where \( ({x_n})\) is a sequence of positive fuzzy numbers, \(M, A, B\) and initial value \({x_0}\) are positive fuzzy numbers. Finally, a numerical simulation example is given to verify the validity of the results.Existence theorems of boundary value problems for fourth order nonlinear discrete systems.https://zbmath.org/1449.390202021-01-08T12:24:00+00:00"Yang, Lianwu"https://zbmath.org/authors/?q=ai:yang.lianwuSummary: In the manuscript, we concern with the existence of solutions of boundary value problems for fourth order nonlinear discrete systems. Some criteria for the existence of at least one nontrivial solution of the problem are obtained. The proof is mainly based upon the variational method and critical point theory. An example is presented to illustrate the main result.Dynamical behavior of high-order fuzzy difference equation.https://zbmath.org/1449.390142021-01-08T12:24:00+00:00"Zhang, Qianhong"https://zbmath.org/authors/?q=ai:zhang.qianhong"Lin, Fubiao"https://zbmath.org/authors/?q=ai:lin.fubiao"Zhong, Xiaoying"https://zbmath.org/authors/?q=ai:zhong.xiaoyingSummary: This paper studied the existence of positive solution and positive equilibrium, asymptotic behavior of the positive solutions of a high-order fuzzy nonlinear difference equation, where the parameters and initial values of the equation were positive fuzzy numbers. Finally, an illustrative example was given to verify the results obtained.Uniqueness of difference operators of entire functions.https://zbmath.org/1449.300542021-01-08T12:24:00+00:00"Li, Qian"https://zbmath.org/authors/?q=ai:li.qian"Wu, Xiaoying"https://zbmath.org/authors/?q=ai:wu.xiaoyingSummary: Under the assumption that a given entire equation has positive deficiency and with the method of complex difference equations, some results about the uniqueness of difference operators of finite order entire functions sharing values with the same multiplicities are presented. These findings can be seen as the difference analogue of differential cases.Approximate quadratic functional inequality in \(\beta\)-homogeneous normed spaces.https://zbmath.org/1449.390312021-01-08T12:24:00+00:00"Wang, Zhihua"https://zbmath.org/authors/?q=ai:wang.zhihuaSummary: Using the direct method, we investigate the generalized Hyers-Ulam stability of the following quadratic functional inequality \(\|f (x-y) + f (y-z) + f (x-z) - 3f (y) - 3f (z)\| \le \| f (x+y+z)\|\) in \(\beta\)-homogeneous complex Banach spaces.Generalizations of Askey-Wilson integral with multiple variables.https://zbmath.org/1449.330182021-01-08T12:24:00+00:00"Cai, Liping"https://zbmath.org/authors/?q=ai:cai.liping"Cao, Jian"https://zbmath.org/authors/?q=ai:cao.jianSummary: In this paper, \(q\)-difference equations and related problems in special functions, whose formal solutions are \(q\)-polynomials, are discussed. Multi-variable Askey-Wilson integral and its inverse integral are extended by the method of \(q\)-difference equation. In addition, Bailey \(_6\phi_6\) summation is generalized.Noise-proof method for determining the linear dynamic system parameters based on pulse-response characteristic.https://zbmath.org/1449.932562021-01-08T12:24:00+00:00"Zoteev, V. E."https://zbmath.org/authors/?q=ai:zoteev.vladimir-evgenevichSummary: We consider the method for parametric identification of the linear dynamic system based on the stochastic difference equation describing the sequence of the measurements of instantaneous values of system pulse-response characteristics. The algorithm of method includes the iteration procedure which allows almost completely removing the transition of estimates of stochastic difference equation coefficients and essentially increasing the accuracy of calculating of system parameters.Polynomial solutions of the polynomial-like iterative equation.https://zbmath.org/1449.390272021-01-08T12:24:00+00:00"Yu, Zhiheng"https://zbmath.org/authors/?q=ai:yu.zhiheng"Gong, Xiaobing"https://zbmath.org/authors/?q=ai:gong.xiaobingSummary: Most of the known results for the polynomial-like iterative equation were given for monotone functions. In this paper, we discuss this equation for a polynomial function, which is non-monotonic. In one-dimensional case, we apply the method of computer algebra system SINGULAR decomposing algebraic varieties to find a sufficient and necessary condition for the polynomial-like iterative equation of orders 2 and 3 having quadratic polynomial solutions respectively and give quadratic polynomial solutions of the two equations. Then we give a procedure for computing polynomial solutions of the polynomial-like equation. In two-dimensional case, applying the idea of one-dimensional case we obtain several sufficient and necessary conditions for second order polynomial-like iterative equation having quadratic degree-preserving polynomial solutions when the given function is a two-dimensional homogeneous polynomial mapping of degree 2.An ordering on Green's function and a Lyapunov-type inequality for a family of nabla fractional boundary value problems.https://zbmath.org/1449.390182021-01-08T12:24:00+00:00"Jonnalagadda, Jagan Mohan"https://zbmath.org/authors/?q=ai:jonnalagadda.jaganmohanSummary: In this article, we consider a family of two-point Riemann-Liouville type nabla fractional boundary value problems involving a fractional difference boundary condition. We construct the corresponding Green's function and deduce its ordering property. Then, we obtain a Lyapunov-type inequality using the properties of Green's function, and illustrate a few of its applications.Asymptotic properties of solutions to difference equations of Emden-Fowler type.https://zbmath.org/1449.390082021-01-08T12:24:00+00:00"Migda, Janusz"https://zbmath.org/authors/?q=ai:migda.januszSummary: We study the higher order difference equations of the following form \[ \Delta^m x_n=a_nf(x_{\sigma(n)})+b_n. \] We are interested in the asymptotic behavior of solutions \(x\) of the above equation. Assuming \(f\) is a power type function and \(\Delta^m y_n=b_n\), we present sufficient conditions that guarantee the existence of a solution \(x\) such that \[ x_n=y_n+o(n^s), \] where \(s\leq 0\) is fixed. We establish also conditions under which for a given solution \(x\) there exists a sequence \(y\) such that \(\Delta^m y_n=b_n\) and \(x\) has the above asymptotic behavior.General solutions to four classes of nonlinear difference equations and some of their representations.https://zbmath.org/1449.390012021-01-08T12:24:00+00:00"Stevic, Stevo"https://zbmath.org/authors/?q=ai:stevic.stevoSummary: We present general solutions to four classes of nonlinear difference equations, as well as some representations of the general solutions for two of the classes in terms of specially chosen solutions to linear homogeneous difference equations with constant coefficients which are naturally associated to the equations of the classes. Our main results considerably generalize some very special ones in recent literature, and present concrete methods for solving the equations.Permanence for a modified Leslie-Gower discrete predator-prey system.https://zbmath.org/1449.390252021-01-08T12:24:00+00:00"Wu, Liping"https://zbmath.org/authors/?q=ai:wu.lipingSummary: A discrete modified Leslie-Gower predator-prey delay model with Beddington-DeAngelis functional response and feedback control is studied. By applying the theory of difference inequality, sufficient conditions which guarantee the permanence of the system are obtained. The result shows that feedback control variables have no influence on the permanence of the system under some conditions.Mean invariance identity.https://zbmath.org/1449.330032021-01-08T12:24:00+00:00"Matkowski, Janusz"https://zbmath.org/authors/?q=ai:matkowski.januszSummary: For a continuous and increasing function \(f\) in a real interval \(I\), and a bivariable mean \(P\) defined in \(I^2\), we prescribe a pair of bivariable means \(M\) and \(N\) such that the quasiarithmetic mean \(A_f\) generated by \(f\) is invariant with respect to the mean-type mapping \((M,N)\). This allows to find effectively the limit of the iterates of the mean-type mapping \((M,N)\). The means \(M\) and \(N\) are equal iff \(P\) is the arithmetic mean \(A\); they are symmetric iff so is \(P\). Treating \(f\) and \(P\) as the parameters, we obtain the family of all pairs of means \((M,N)\) such that the quasiarithmetic mean \(A_f\) is invariant with respect to \((M,N)\). In particular, we indicate the function \(f\) and the mean \(P\) such that the invariance identity \(A_f\circ (M,N) = A_f\) coincides with the equality \(G\circ (H,A)\), where \(G\) and \(H\) are the geometric and harmonic means, equivalent to the classical Pythagorean harmony proportion.
Some examples and an application are also presented.