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Research on how to pose good problems in mathematical science is rarely touched. Inspired by Kilpatrick's "Where do good problems come from?", the current research investigates the problem of the specific problem posed by mathematicians in mathematical sciences. We select a recent mathematical conjecture of Yang related to periodic functions in the field of functions of one complex variable. These problems are extended to complex differential equations, difference equations, differential-difference equations, etc. Through mathematical analysis, we try to reproduce the effective strategies or techniques used by mathematicians in posing these new problems. The results show that mathematicians often use generalization, constraint manipulation, and specialization when they pose new mathematical problems. Conversely, goal manipulation and targeting a particular solution are rarely used. The results of the study may have a potential impact and promotion on implementing problem-posing teaching in primary and secondary schools. Accordingly, teachers and students can be encouraged to think like mathematicians, posing better problems and learning mathematics better. Then, we give some examples of mathematical teaching at the high school level using problem-posing strategies, which are frequently employed by mathematicians or mathematical researchers, and demonstrate how these strategies work. Therefore, this is a pioneering research that integrates the mathematical problem posing by mathematicians and the mathematical problem posing by elementary and secondary school math teachers and students. In addition, applying constraint manipulation and analogical reasoning, we present four unsolved mathematical problems, including three problems of complex difference-related periodic functions and one problem with complex difference equations. [ABSTRACT FROM AUTHOR]

The distributed recursive filtering problem is investigated in this paper for discrete time-delayed nonlinear stochastic systems, where the well-known Takagi–Sugeno (T-S) fuzzy model is used to approximate the nonlinearities. According to obtain the system dynamics, a novel structure of distributed filters is developed, where the difference of estimated states from neighboring sensors is exploited to improve the one-step prediction, and the desired estimation is obtained by fusing the estimation under different rules. Attention is focused on the design of a distributed recursive filter such that, in the presence of time-delays and defuzzifying operations, an upper bound of the filtering error covariance is obtained and then minimized by properly designing filter parameters via elaborate mathematical analysis. With the exception of the desired gains with the online recursive form are dependent on the solutions of two Riccati-type difference equations, and the upper bound is further optimized via the introduced parameters. As a final point, a simulation examples is exploited to show the applicability of the developed filtering scheme. [ABSTRACT FROM AUTHOR]

The present paper constitutes a theoretical review on the relationship between debt and deficit. It is described using mathematical analysis and using a difference equation that relates the current debt stock to the past one and the fiscal deficit to show the conditions regarding the stability and dynamic sustainability of the debt. [ABSTRACT FROM AUTHOR]

A range of new fractional-order love dynamics is modelled in this paper using the Caputo (power-law), Caputo–Fabrizio (exponential decay-law) and Atangana–Baleanu (Mittag-Leffler-law) fractional operators. We utilize the new fractional Adams–Bashforth schemes for the approximation of these derivatives. This method was developed with the standard differentiation technique by applying the fundamental theorem of calculus and taking the difference of two times levels at t = (t n , t n + 1). The stability analysis applies the theory of classical order differential equations and dynamical system. The Existence and uniqueness of solution of fractional dynamics is proved by adopting the fixed-point theorem. Numerical results presented for various α − values justify the theoretical findings. It was observed that modelling of interpersonal and romantic love affairs with fractional derivative could exhibit some strange emotional attractors. • Fractional two-step Adams–Bashforth methods. • Numerical approximation and simulation. • Existence and uniqueness of solutions via fixed-point theorem. • The Caputo, Caputo–Fabrizio and Atangana–Baleanu fractional derivatives. [ABSTRACT FROM AUTHOR]

In this paper we introduce a generalized form of the well known ToDD's difference equation and give the closed form expressions for this generalized form . In other words , we have the following nonlinear rational partial difference equation T ‹X1,X2,X3, …,Xn› = 1 + T ‹X1 - 1,X2 - 1, …,Xn - 1› + T ‹X1 - 2,X2 - 2, …,Xn - 2› T ‹X1 - 3,X2 - 3,X3 - 3, …,Xn - 3› where X1,X2, …:,Xn ∊ ℕ,and the initial values T ‹p1, p2, …:, pn› ,T ‹p2, p1, p3, p4, …, pn› ,T ‹p2, p3, p1, p4, …, pn›,… …,T ‹p2, p3, p4, …p1, pn›,T ‹p2 - 3, p3 - 3, p4 - 3, …pn - 3, p1› are real numbers with p1 ∊ {0,-1,-2} and p2, p3, …, pn ∊ ℕ such that T ‹p1, p2, …:, pn› ≠ 0 ,T ‹p2, p1, p3, p4, …, pn› ≠ 0 , T ‹p2, p3, p1, p4, …, pn› ≠ 0,…,T ‹p2 - 3, p3 - 3, p4 - 3, …pn - 3, p1› ≠ 0. We will use a novel technique to prove the results by using what we call 'piecewise n-dimensional mathematical induction' which we intro- duce here for the first time . We will obvious that this new concept represents generalized form for many types of mathematical induction . As a direct consequences , we investigate and drive the explicit solutions for the well known ordinary ToDD's difference Equation . [ABSTRACT FROM AUTHOR]

In this article, we investigate the dynamics of the solutions of the following non-linear di¤erence equation xn+1 = xn-2xn-3 - 1; n ∞ N0 with arbitrary initial conditions x-2; x-1; x0. Besides, we have studied periodic behaviours of related di¤erence equation especially asymptotic periodicity and eventually periodicity. Then, we have researched unbounded solutions of di¤erence equation. [ABSTRACT FROM AUTHOR]

In this paper, we give some necessary conditions on the existence of meromorphic solutions on Fermat type difference equations. We also consider the properties of transcendental entire solutions on the systems of Fermat type differential-difference equations. [ABSTRACT FROM AUTHOR]

Closed form formulas for well-defined complex-valued solutions to a product-type system of difference equations of interest with six parameters are presented. The form of the solutions is described in detail in terms of the parameters and initial values. [ABSTRACT FROM AUTHOR]

In this study, we consider two different inequivalent formulations of the logistic difference equation xn+i = βxn (1 - xn), n = 0,1,..., where xn is a sequence of fuzzy numbers and β is a positive fuzzy number. The major contribution of this paper is to study theIn existence, uniqueness and global behavior of the solutions for two corresponding equations, using the concept of Hukuhara difference for fuzzy numbers. Finally, some examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

Abstract A criterion for the existence of a birational embedding of an algebraic curve into a projective plane with two Galois points is presented. Several novel examples of plane curves with two inner Galois points as an application are described. [ABSTRACT FROM AUTHOR]

In this work, sufficient conditions for oscillation of solutions of three dimensional difference systems of the form X(n + 1) = A(n)X(n) are established. Here X(n) = [x1(n), x2(n), x3(n)]T , A(n) = [aij (n)] is a given 3 × 3 matrix, xi : N → R, aij : N → R for i, j ∈ {1, 2, 3}. [ABSTRACT FROM AUTHOR]

We make a case for including difference equations, in particular the discrete logistic equation, in basic differential equations courses. Contrasting the behavior of discrete and continuous models enriches students’ understanding of both modeling and differential equations. To facilitate sharing discrete population models with students, some materials including class handouts and an “exploratory” computer assignment are provided. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new modelling approach for the dynamic granular matter formation process in the form of a system of difference equations, directly tailored to the physical nature of the process at hand. Respectively, the dynamic 1D and 2D discrete models, proposed in this paper, are not constructed as numerical schemes approximating some partial differential equations (PDEs).We propose here to look for the functions describing the standing and the rolling layers of the granular matter as the limits of discrete solutions to the aforementioned model equations as the size of the mesh tends to zero. In particular, this approach allows us to differentiate between the influx of the rolling layer coming down from different directions to the corner points of the standing layer. Such points are difficult to adequately describe by means of PDEs and their straightforward numerical approximations, typically 'ignoring' the system's behaviour on the sets of zero measure. However, these points are critical for understanding the dynamics of formation process when the standing layer is created by the moving front of the rolling matter or when the latter is filling a cavity and/or stops rolling. The existence of distributed (infinite-dimensional) limit solutions to our discrete models as the size of the mesh tends to zero is also discussed. We illustrate our findings by numerical examples which use our models as the direct algorithm. [ABSTRACT FROM AUTHOR]

We investigate global dynamics of the following second order rational difference equation xn+1... where the parameters α, β, a, b are positive real numbers and initial conditions x-1 and x0 are arbitrary positive real numbers. The map associated to the right-hand side of this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the corresponding parametric space. In most cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. [ABSTRACT FROM AUTHOR]

In this paper we analyze the existence of bounded solutions for a nonlinear second-order neutral difference equation, which is more general than other equations of this type studied recently. Moreover, our analysis is based under more general conditions than the required in other works, as we only assume the continuity of the involved functions. The main tool here are the so called measures of noncompactness, and more specifically, the celebrated Darbo fixed point theorem. Also we will state, in the specified sense, the stability of the solutions. [ABSTRACT FROM AUTHOR]

The solution form of the system of nonlinear difference equations [ABSTRACT FROM AUTHOR]

In this article, we study the oscillatory behavior of the third-order neutral type difference equation ∆(an(∆² (xn + pnxn-k))α) + qnƒ(xn-l) = 0 where α > 0, an > 0, qn ≥ 0 and 0 ≤ pn ≤ p < ∞. By using generalized Ricatti type transformation we present some new criteria which ensure that every solution is oscillatory. Also we provide examples that illustrate the importance of our results. [ABSTRACT FROM AUTHOR]

In this paper, we use a method based on orthogonal polynomials to give closed-form evaluations of the Hankel transform of sequences based on the Motzkin numbers. It includes linear combinations of consecutive two, three, and four Motzkin numbers. In some cases, we were able to derive the closed-form evaluation of the Hankel transform, while in the others we showed that the Hankel transform satisfies a particular difference equation. As a corollary, we reobtain known results and show some new results regarding the Hankel transform of Motzkin and shifted Motzkin numbers. Those evaluations also give an idea on how to apply the method based on orthogonal polynomials on the sequences having zero entries in their Hankel transform. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of the zero equilibria of the following systems of difference equations: [ABSTRACT FROM AUTHOR]

We obtain in this paper the expression of the solutions of the following recursive sequences: [ABSTRACT FROM AUTHOR]

A solvable class of product-type systems of difference equations with two dependent variables on the complex domain is presented. The main results complement some recent ones in the literature, while their proofs contain some refined methodological details. We provide closed form formulas for general solutions to the system or give procedures for how to get them. [ABSTRACT FROM AUTHOR]

In this paper, we study the following max-type system of difference equations { x n = max { 1 x n − m , min { 1 , A y n − r } } , y n = max { 1 y n − m , min { 1 , B x n − t } } , n ∈ N 0 where A , B ∈ ( 0 , + ∞ ) , m , r , t ∈ { 1 , 2 , … } with r ≠ m and t ≠ m . We show that every solution of this system with the initial values x − d , y − d , x − d + 1 , y − d + 1 , … , x − 1 , y − 1 ∈ ( 0 , + ∞ ) is eventually periodic with period 2 m , where d = max { m , r , t } . [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of S-asymptotically periodic solutions for an epidemic model with superlinear perturbation. It seems that this is a first result as regards such a model with superlinear perturbation. We give sufficient conditions to ensure the existence of S-asymptotically periodic solutions for the problem addressed and give an example to show that our sufficient conditions can be satisfied. [ABSTRACT FROM AUTHOR]

We prove Hyers-Ulam stability of the first-order difference equation of the form xi+1=F(i,xi), where F is a given function with some moderate features. Moreover, we introduce some conditions for the function F under which the difference equation is not stable in the sense of Hyers and Ulam. [ABSTRACT FROM AUTHOR]

A nonautonomous modified Leslie-Gower predator-prey model with Holling-type II schemes and a prey refuge is studied in this paper. Persistent property and stability property of the system are investigated. Some findings about the influence of prey refuge are obtained. [ABSTRACT FROM AUTHOR]

Recently, Kim, Kwon, and Seo (J. Nonlinear Sci. Appl. 9:2380-2392, ) studied the degenerate q-Changhee polynomials and numbers. In this paper, we consider the Appell-type degenerate q-Changhee polynomials and give some new and explicit identities related to these polynomials [ABSTRACT FROM AUTHOR]

T. Wanner We investigate global dynamics of the equation [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the dynamical behavior of positive solutions for a system of rational difference equations of the following form: un+1=αun-1/β+γvn-2p, vn+1=α1vn-1/β1+γ1un-2p, n=0,1,…, where the parameters α,β,γ,α1,β1,γ1,p and the initial values u-i,v-i for i=0,1,2 are positive real numbers. [ABSTRACT FROM AUTHOR]

We investigate the following difference equation xn = xn-3xn-4xn-5 xn-1xn-2(an + bnxn-3xn-4xn-5) ; n ∊ N0; where (an)n∊N0 and (bn)n∊N0 are two real sequences and the initial values x-5, . . ., x-1 are real numbers. The case when the sequences (an)n∊N0 and (bn)n∊N0 are constant is thoroughly studied. Our results considerably extend some results in the recent literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of solutions for a nonlocal boundary value problem of impulsive fractional $q_{k}$-difference equations involving a new $q_{k}$-shifting operator ${}_{a}\Phi_{q_{k}}(m) = q_{k} m + (1-q_{k})a$. Our main results rely on Banach's contraction mapping principle, Leray-Schauder nonlinear alternative, and Rothe fixed point theorem. Examples illustrating the obtained results are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the system that has solutions and the periodicity character of the following systems of rational difference equations with order three [ABSTRACT FROM AUTHOR]

We present a complete local dynamics and investigate the global dynamics of the following second-order difference equation: xn+1=Axn2+Exn-1+F/axn2+exn-1+f,  n=0,1,2,…, where the parameters A,E,F,a,e, and f are nonnegative numbers with condition A+E+F>0, a+e+f>0, and the initial conditions x-1, x0 are arbitrary nonnegative numbers such that axn2+exn-1+f>0,  n=0,1,2,…. [ABSTRACT FROM AUTHOR]

Although spectral analysis of stationary stochastic processes has solid mathematical foundations, this is not always so for some non-stationary cases. Here, we establish a rigorous mathematical extension of the classic Fourier spectrum to the case in which there are AR roots in the unit circle, i.e., the transfer function of the linear time-invariant filter has poles on the unit circle. To achieve it we: embed the classical problem in a wider framework, extend the Discrete Time Fourier Transform and defined a new Extended Fourier Transform pair pseudo-covariance function/pseudo-spectrum. Our approach is a proper extension of the classical spectral analysis, within which the Fourier Transform pair auto-covariance function/spectrum is a particular case. Consequently spectrum and pseudo-spectrum coincide when the first one is defined. [ABSTRACT FROM AUTHOR]

Some formulas for the "general solution" to the binomial partial difference equation cm,n = cm-1,n + cm-1,n-1, are known in the literature. However, it seems that there is no such a formula on the most natural domain connected to the equation, that is, on the set D = {(m, n) ∈ N²0: 0 ≤ n ≤ m. By using a connection with the scalar linear first order difference equation we show that the equation on the domain D n f(0, 0)g, can be solved in closed form by presenting a formula for the solution in terms of the "side" values ck,0, ck,k, k∈ℕ. [ABSTRACT FROM AUTHOR]

Consider the following ν-th order nabla and delta fractional difference equations ∇νρ(a) x(t) = c(t)x(t), t ∈ Na+1, x(a) > 0. (*) and ∆νa+ν-1x(t) = c(t)x(t + ν - 1), t ∈ Na, x(a + ν - 1) > 0. (* *) We establish comparison theorems by which we compare the solutions x(t) of (*) and (* *) with the solutions of the equations ∇νρ(a) x(t) = bx(t) and ∆νa+ν-1x(t) = bx(t + ν - 1), respectively, where b is a constant. We obtain four asymptotic results, one of them extends the recent result [F. These results show that the solutions of two fractional difference equations ∇νρ(a) x(t) = cx(t), 0 < ν < 1, and ∆νa+ν-1x(t) = cx(t + ν - 1), 0 < ν < 1, have similar asymptotic behavior with the solutions of the first order difference equations ∇x(t) = cx(t), |c| < 1 and ∆x(t) = cx(t), |c| < 1, respectively. [ABSTRACT FROM AUTHOR]

The dynamics of the maps associated to F 0 and d P 3 quivers is studied in detail. We show that the corresponding reduced symplectic maps are conjugate to globally periodic maps by providing explicit conjugations. The dynamics in R + N of the original maps is obtained by lifting the dynamics of these globally periodic maps and the solution of the discrete dynamical systems generated by each map is given. A better understanding of the dynamics is achieved by considering first integrals. The relationship between the complete integrability of the globally periodic maps and the dynamics of the original maps is explored. [ABSTRACT FROM AUTHOR]

This paper deals with the global attractivity of positive solutions of the second-order nonlinear difference equationxn+1 = axkn + bk-1Σj=1xjnxk-jn-1 + cxkn-1/Axkn + Bk-1Σj=1xjnxk-jn-1 + Cxkn-1, k = 3, 4; ..., n = 0, 1, ..., where the parameters a, b, c, A, B, C and the initial values x0, x-1 are arbitrary positive real numbers. [ABSTRACT FROM AUTHOR]

In this paper, limit periodic and almost periodic homogeneous linear difference systems are considered. We study the systems in which the coefficient matrices are taken from a given bounded group and the elements of the matrices are from an infinite field with an absolute value. We show a condition on limit periodic and almost periodic systems which ensures, that the considered systems can be transformed into new systems having certain properties. The new systems possess non-asymptotically almost periodic solutions. The transformation can be done by arbitrarily small changes. [ABSTRACT FROM AUTHOR]

The boundedness character of positive solutions of the following system of difference equations: $x_{n+1}=A+\frac{y^{p}_{n}}{x_{n-3}^{r}}$, $y_{n+1}=A+\frac {x^{p}_{n}}{y_{n-3}^{r}}$, $n\in{\mathbb{N}}_{0}$, when $\min\{A,r\}>0$ and $p\ge0$, is studied. [ABSTRACT FROM AUTHOR]

In this paper, a new Monod type chemostat model with delay and impulsive input on two substrates is considered. By using the global attractivity of a k times periodically pulsed input chemostat model, we obtain the bound of the system. By the means of a fixed point in a Poincaré map for the discrete dynamical system, we obtain a semi-trivial periodic solution; further, we establish the sufficient conditions for the global attractivity of the semi-trivial periodic solution. Using the theory on delay functional and impulsive differential equations, we obtain a sufficient condition with time delay for the permanence of the system. [ABSTRACT FROM AUTHOR]

This article discusses the control system of fractional endpoint variable variational problems. For this problem, we prove the Euler-Lagrange type necessary conditions which must be satisfied for the given functional to be extremum. Finally, one example is provided to show the application of our results. [ABSTRACT FROM AUTHOR]

The p-adic q-integral (sometimes called q-Volkenborn integration) was defined by Kim. From p-adic q-integral equations, we can derive various q-extensions of Bernoulli polynomials and numbers. DS Kim and T Kim studied Daehee polynomials and numbers and their applications. Kim et al. introduced the q-analogue of Daehee numbers and polynomials which are called q-Daehee numbers and polynomials. Lim considered the modified q-Daehee numbers and polynomials which are different from the q-Daehee numbers and polynomials of Kim et al. In this paper, we consider $(h,q)$-Daehee numbers and polynomials and give some interesting identities. In case $h=0$, we cover the q-analogue of Daehee numbers and polynomials of Kim et al. In case $h=1$, we modify q-Daehee numbers and polynomials. We can find out various $(h,q)$-related numbers and polynomials which are studied by many authors. [ABSTRACT FROM AUTHOR]

This paper is concerned with the notions of statistical limit and cluster points defined by Fridy. Following the concept of a Δ-density for a subset of a time scale, we established a generalization of these notions which are called Δ-limit and Δ-cluster points for a function defined on a time scale $\mathbb{T}$. [ABSTRACT FROM AUTHOR]

The existence of the nontrivial periodic solutions for the nonautonomous first order delay differential equation $x'(t)=-[f(t, x(t-1))+f(t, x(t-2))+\cdots+f(t, x(t-(2N-1)))]$ is investigated, where $f\in C({\mathbf{R}}\times{\mathbf{R}}, {\mathbf{R}})$ is 2 N-periodic in t and odd in x, N is a positive integer. We prove several new existence results by some recent critical point theorems. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the continuous dependence of mild solutions on initial values and orders for the initial value problem of fractional evolution equations in infinite dimensional spaces. The results obtained in this paper improve and extend some related conclusions on this topic. This paper can be considered as a contribution to this emerging field. [ABSTRACT FROM AUTHOR]