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Let $ \beta (z) $ β (z) be the Dirichlet beta function. Series for $ \beta (4) $ β (4) are very rare in the historical development of mathematics. With the help of the operator method and a transformation formula for hypergeometric series, we prove a surprising conjectural series of Z.-W Sun for the mathematical constant $ \beta (4) $ β (4). Furthermore, we find five new series for the same constant in this paper. [ABSTRACT FROM AUTHOR]

The k-Fibonacci sequence { F n (k) } n starts with the values 0 , ... , 0 , 1 (a total of k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all integers c having at least two representations as a difference between a k-Fibonacci number and a Pell number. This paper continues and extends the previous work of [J.J. Bravo, C.A. Gómez, and J.L. Herrera, On the intersection of k-Fibonacci and Pell numbers, Bull. Korean Math. Soc. 56(2) (2019), pp. 535–547; S. Hernández, F. Luca, and L.M. Rivera, On Pillai's problem with the Fibonacci and Pell sequences, Soc. Mat. Mex. 25 (2019), pp. 495–507 and M.O. Hernane, F. Luca, S.E. Rihane, and A. Togbé, On Pillai's problem with Pell numbers and powers of 2, Hardy- Ramanujan J. 41 (2018), pp. 22–31]. [ABSTRACT FROM AUTHOR]

Moothathu [Colloq. Math. 120 (2010), pp. 127–138] proved that multi-transitivity, weakly mixing and Δ-transitivity are equivalent for minimal homeomorphisms. In this paper, we extend this result to Abelian group actions. Moreover, we point out that a multi-transitive system of Abelian group action is Li–Yorke ε-chaotic for some ε > 0. [ABSTRACT FROM AUTHOR]

For PM functions of height 1, the existence of continuous iterative roots of any order was obtained under the characteristic endpoints condition. A natural open question about iterative roots without that condition was raised. This question was answered partially in the case that the function is increasing on its characteristic interval. In this paper, to the opposite, we consider the decreasing case and give the existence and nonexistence results for their iterative roots. [ABSTRACT FROM AUTHOR]

The aim of this paper is to note that properties of being elementary in differential and difference senses are different as a whole. In this paper, we introduce the notion of finitelyn-reducible difference field extensions and study the irreducibility of algebraic difference equations of order 1. As a result, we find that the exponential functionand the Weierstrass functioncannot be built up by using the variablezand constants, together with repeated algebraic operations and the taking of solutions of linear difference equations on the multiplicationand solutions of such algebraic difference equations as difference Riccati equations on. [ABSTRACT FROM PUBLISHER]

This paper studies the behavior under iteration of the maps Tab(x, y) = (Fab(x) - y, x) of the plane R², in which Fab(x) = ax if x ≥ 0 and bx if x « 0. The orbits under iteration correspond to solutions of the nonlinear difference equation xn+2 = 1/2(a - b)[xn+1] + 1/2(a + b)xn+1 - xn. This family of piecewise-linear maps has the parameter space (a, b] ϵ R². These maps are area-preserving homeomorphisms of R² that map rays from the origin into rays from the origin. The action on rays gives an auxiliary map Sab: S¹ ↵ S¹ of the circle, which has a well-defined rotation number This paper characterizes the possible dynamics under iteration of Tab when the auxiliary map Sab has rational rotation number, It characterizes cases where the map Tab is a periodic map. [ABSTRACT FROM AUTHOR]

Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Although the numerical methods for stochastic differential equations (SDEs) have been well studied, there are only a few results on the numerical solutions for SDEwMSs. By this manuscript we continue our series of papers on numerical solutions of SDEwMSs. The main aim of this paper is to investigate the stationary distributions of Euler-Maruyama-type stochastic difference equations with Markovian switching (EMSDEwMSs) and discuss their convergence. [ABSTRACT FROM AUTHOR]

In this paper we study the periodicity of higher order nonlinear equations. They are defined by a recursion which is generated by a mapping, whereXis a state set. Our main objective is to prove sharp conditions for the global periodicity of our equations assuming the weakest possible assumptions on the state setX. As an application of our general algebraic-like conditions we prove a new linearized global periodicity theorem assuming thatXis a normed space. We needed a new proof-technique since in the infinite dimensional case the Jacobian does not exist. We give new necessary and/or sufficient conditions as well as new examples for global periodicity, for instance whenever the state setXis a group. [ABSTRACT FROM AUTHOR]

In this paper, we give two generalized homogeneousq-operator identities, which are obtained byq-difference equations. In addition, we generalize certain generating functions from Chen et al. [W.Y.C. Chen, A. M. Fu, and B. Zhang,The homogeneousq-difference operator, Adv. Appl. Math. 31 (2003), pp. 659–668] and Saad and Sukhi [H. L. Saad and A.A. Sukhi,Another homogeneousq-difference operator, Appl. Math. Comput. 215 (2010), pp. 4332–4339] by the method ofq-difference equations. Moreover, we gainU(n+1) generalizations of generating functions for Verma–Jain polynomials byq-difference equations. Finally, we obtain two transformational identities byq-difference equations. [ABSTRACT FROM AUTHOR]

The boundedness character of positive solutions of the following cyclic system of difference equationswhereA,pandqare positive numbers, and where if, for someand, thendenotes, is completely described in this paper. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behaviour of the positive solutions of the system of two difference equationswhereare positive constants and the initial valuesare positive numbers. [ABSTRACT FROM AUTHOR]

In this paper we study the boundedness and the persistence of the positive solutions, the existence, the attractivity and the global asymptotic stability of the unique positive equilibrium and the existence of periodic solutions concerning the biological model given bywhere,are positive constants andis a positive real number. [ABSTRACT FROM AUTHOR]

In this paper, we consider an extension of some results to dynamic equations on discrete timescales originally due to Butler for second-order superlinear differential equations. As an application, we get that the superlinear difference equationis oscillatory, if. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of analytic invariant curves of an iterative equationwhich is from Pielou's equation. By reducing the equation with the Schröder transformation to an auxiliary equation, the author discusses not only that the constantat resonance, i.e. at a root of the unity, but also thosenear resonance under the Brjuno condition. [ABSTRACT FROM AUTHOR]

The original notion of distributional chaos introduced in 1994 by Schweizer and Smítal for continuous maps of the interval was later generalized to arbitrary compact metric space, and three types, DC1–DC3, are now considered. However, most of the results concern the case when the scrambled set consists of two points,. In this paper, we consider stronger versions of distributional chaos,, where uncountable scrambled set is required. We show, among others, that these types andare mutually non-equivalent, even in the class of triangular maps of the square. [ABSTRACT FROM PUBLISHER]

This paper is devoted to show that Hirsch's results on the existence of a carrying simplex are a powerful tool to understand the dynamics of Kolmogorov models. For two and three species, we prove that there is exclusion for our models if and only if there are no coexistence states. The proof of this result is based on a result in planar topology due to Campos, Ortega and Tineo. For an arbitrary number of species, we will obtain dominance criteria following the notions of Franke and Yakubu. In this scenario, the crucial fact will be that the carrying simplex is an unordered manifold. Applications in concrete models are given. [ABSTRACT FROM PUBLISHER]

In this paper, we study an inverse optimal problem in discrete-time stochastic control. We give necessary and sufficient conditions for a solution to a system of stochastic difference equations to be the solution of a certain optimal control problem. Our results extend to the stochastic case the work of Dechert. In particular, we present a stochastic version of an important principle in welfare economics. [ABSTRACT FROM PUBLISHER]

The purpose of this paper was to introduce the symbolic formalism based on kneading theory, which allows us to study the renormalization of non-autonomous periodic dynamical systems. [ABSTRACT FROM PUBLISHER]

Iterated function systems (IFSs) are useful for creating fractals, interesting probability distributions and enable a unifying framework for analysing stochastic processes with Markovian properties. In this paper, we present a survey of some basic results within the theory of random iterations of functions from an IFS based on average contraction conditions. [ABSTRACT FROM AUTHOR]

Understanding under what conditions populations, whether they be plants, animals or viral particles, persist is an issue of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt persistence. One approach to examining the interplay between these deterministic and stochastic forces is the construction and analysis of stochastic difference equations , where represents the state of the populations and is a sequence of random variables representing environmental stochasticity. In the analysis of these stochastic models, many theoretical population biologists are interested in whether the models are bounded and persistent. Here, boundedness asserts that asymptotically tends to remain in compact sets. In contrast, persistence requires that tends to be ‘repelled’ by some ‘extinction set’ . Here, results on both of these proprieties are reviewed for single species, multiple species and structured population models. The results are illustrated with applications to stochastic versions of the Hassell and Ricker single species models, Ricker, Beverton–Holt, lottery models of competition, and lottery models of rock–paper–scissor games. A variety of conjectures and suggestions for future research are presented. [ABSTRACT FROM AUTHOR]

We consider the inhomogeneous version of the fixed-point equation of the smoothing transformation, that is, the equation , where means equality in distribution, is a given sequence of non-negative random variables and is a sequence of i.i.d. copies of the non-negative random variable X independent of . In this situation, X (or, more precisely, the distribution of X) is said to be a fixed point of the (inhomogeneous) smoothing transform. In the present paper, we give a necessary and sufficient condition for the existence of a fixed point. Furthermore, we establish an explicit one-to-one correspondence with the solutions to the corresponding homogeneous equation with C = 0. Using this correspondence and the known theory on the homogeneous equation, we present a full characterization of the set of fixed points under mild assumptions. [ABSTRACT FROM AUTHOR]

This paper presents a proof of almost sure asymptotic stability of trivial solution of stochastic systems of linear difference equations driven by -integrable martingale-type noise. For this purpose, we exploit a convergence theorem for non-negative semi-martingale decompositions and verify a practical criteria based on non-random eigenvalues. Two 2D examples illustrate the applicability of obtained criteria. [ABSTRACT FROM AUTHOR]

Let , where is the -th cyclotomic polynomial. Let or , depending if the leading coefficient of the polynomial is ‘+1’ or ‘ − 1’, respectively. The rational function can be written as , where , the 's are positive integers, 's are integers and is a positive integer depending on . In the present paper, we study the set where the intersection is considered over all the possible decompositions of of the type mentioned above. Here, we describe the set in terms of the arithmetic properties of the integers . We also study the question: given S a finite subset of the natural numbers, does exists a , such that ? The set is called the minimal set of Lefschetz periods associated with q(t). The motivation of these problems comes from differentiable dynamics, when we are interested in describing the minimal set of periods for a class of differentiable maps on orientable surfaces. In this class of maps, the Morse–Smale diffeomorphisms are included (cf. Llibre and Sirvent, Houston J. Math. 35 (2009), pp. 835–855). [ABSTRACT FROM AUTHOR]

The aim of this paper is to study analytical and combinatorial methods to solve a special type of recurrence relation with two indices. It is shown that the recurrence relation enumerates a natural combinatorial object called a plane composition. In addition, further interesting recurrence relations arise in the study of statistics for these plane compositions. [ABSTRACT FROM AUTHOR]

In this paper, a discrete bidirectional associative memory network model consisting of three neurons is considered. We study the stability of the equilibria of the system by analysing the distribution of the eigenvalues. It is found that there are three kinds of bifurcations: Pitchfork bifurcation, Period-Doubling bifurcation and Neimark-Sacker bifurcation. The direction and stability of the Neimark-Sacker are determined by using normal forms and centre manifold theory. Some numerical simulations are carried out to illustrate the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider kth order difference equations of the form[image omitted] where every function [image omitted] is continuous and monotonic in each of its arguments. We generalize and improve upon some previous techniques to prove convergence using invariant intervals as the common theme. We apply our techniques to several different special cases of the equation above. [ABSTRACT FROM AUTHOR]

We consider a nonoscillatory second-order linear dynamic equation on a time scale together with a linear perturbation of this equation and improve recently given conditions on the perturbation that guarantee that the perturbed equation is also nonoscillatory and has solutions that behave asymptotically like a recessive and a dominant solution of the unperturbed equation. The presented results are time scales analogues of a 2002 paper on the differential equations case by Trench. The difference equations case has not been treated yet, but is contained in our study together with other cases of dynamic equations. [ABSTRACT FROM AUTHOR]

The paper presents theorems on the global asymptotic behaviour of non-autonomous population models which for the special case of periodically forced populations generalize known results. It is shown that a non-autonomous discrete dynamical system on an interval possesses the property of path stability, if the system is given by cave functions. From this it follows for a periodic system with strictly cave functions that there exists a globally asymptotically stable cycle. Old and new examples are discussed. [ABSTRACT FROM AUTHOR]

This paper studies solutions of some piecewise-linear difference equations. In two particular cases, a descent argument is used to show that all solutions are periodic with either prime period 3(2k - 1) or 6(2k - 1) for some k ≥ 1. The existence of solutions with such periods is also considered. [ABSTRACT FROM AUTHOR]

In this paper, using critical point theory, some sufficient conditions are obtained for the existence of periodic solutions of a class of second order difference equation. [ABSTRACT FROM AUTHOR]

We consider unimodal quasiperiodically forced maps, that is, skew products with irrational rotations of the circle in the base and unimodal interval maps in the fibres: the map in the fibre over θ is a unimodal map f of the interval [0, 1] onto itself multiplied by g(θ), where g is a continuous function from the circle to [0, 1]. Here we consider the 'pinched' case, when g attains the value 0. This case is similar to the one considered by Gerhard Keller, except that the function f in his case is increasing. Since in our case f is unimodal, the basic tools from the Keller's paper do not work in general. We prove that under some additional assumptions on the system there exists a strange nonchaotic attractor. It is the graph of a measurable function from the circle to [0,1], which is invariant, discontinuous almost everywhere and attracts almost all trajectories. Moreover, both Lyapunov exponents on this attractor are nonpositive. There are also cases when the dynamics is completely different, because one can apply the results of Jerome Buzzi implying the existence of an invariant measure absolutely continuous with respect to the Lebesgue measure (and then the attractor is some region in [image omitted] ), and the maximal Lyapunov exponent is positive. Finally, there are cases when we can only guess what the behaviour is by performing computer experiments. [ABSTRACT FROM AUTHOR]

In this paper, we will establish some new interval oscillation criteria for the forced second-order nonlinear dynamic equation[image omitted] on a time scale  where γ ≥ 1. As a special case when  =  our results not only include the oscillation results for second-order differential equations established by Wong (Oscillation criteria for a forced second-order linear differential equation, J. Math. Anal. Appl. 231 (1999), pp. 233-240) and Nasr (Sufficient condition for the oscillation of forced super-linear second order differential equations with oscillatory potential, Proc. Amer. math. Soc. 126 (1998), pp. 123-125) but also improve these results. When  = ,  = h or  = qN, i.e. for difference equations, generalized difference equations or q-difference equations our results are essentially new and also can be applied on different types of time scales, as illustrated in several examples. [ABSTRACT FROM AUTHOR]

The study on the existence of periodic solutions of higher order nonlinear functional difference equations with p-Laplacian is made. Sufficient conditions for the existence of at least one periodic solution of these equations are established, respectively. We give no restriction on the deviating function, which is the significance of the paper. Our result is based on Mawhin's continuation theorem. The methods we used to estimate a priori bound on periodic solutions are also new. [ABSTRACT FROM AUTHOR]

The article presents the first part of a paper about second-order rational difference equations. This study aims to demonstrate the results of the global character of solutions of a second-order rational difference equation. Moreover, a number of open problems and conjectures related to the topic will also be presented in the article.

This paper is devoted to the study of the existence of extremal solutions to a first-order initial value problem on an interval of an arbitrary time scale. We prove the existence of extremal solutions for problems satisfying Carathéodory's conditions. Moreover, they are approximated uniformly by a sequence of lower and upper solutions to this problem, respectively. We also can warrant the existence and approximation of extremal solutions for the problem by relaxing their continuity properties. [ABSTRACT FROM AUTHOR]

This paper studies the stability properties of a nonstandard finite difference (NSFD) scheme used to simulate the dynamics of a mouse population model in hantavirus epidemics. It is shown that this difference scheme and the underlying system of differential equations have the same dynamics. The proof uses the fact that the total population obeys the logistic equation, as well as techniques from calculus, graphical analysis, and dynamical systems. [ABSTRACT FROM AUTHOR]

Let X, Y be Banach modules over a [image omitted] -algebra and let [image omitted] be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital [image omitted] -algebra:[image omitted] We show that if [image omitted] and an odd mapping [image omitted] satisfies the functional equation (0.1) then the odd mapping [image omitted] is Cauchy additive. As an application, we show that every almost linear bijection [image omitted] of a unital [image omitted] -algebra A onto a unital [image omitted] -algebra B is a [image omitted] -algebra isomorphism when [image omitted] for all unitaries [image omitted] , all [image omitted] , and all [image omitted] . The concept of generalized Hyers-Ulam stability originated from Th.M. Rassias' stability Theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. [ABSTRACT FROM AUTHOR]

This paper studies the boundedness character of the positive solutions of the difference equation.[image omitted] where [image omitted] with [image omitted] . We prove that if [image omitted] , then every solution of the equation is bounded, and if [image omitted] and k is even, then there exist positive unbounded solutions. For the case [image omitted] and k odd, we consider the related equation [image omitted] and show that every integer solution is eventually periodic. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the periodicity character of some classes of difference equations. Among other results, we give a simpler proof of an Open Problem addressed in Csörnyei and Laczkovich, 2001, “Some periodic and non-periodic recursions”, Monath. Math. , 132 , 215–236. We also consider Open Problem 2.8 in Grove and Ladas, 2005, Periodicities in Nonlinear Difference Equations , (Chapman and Hall/CRC), and the case when a difference equation has periodic solutions depending on arbitrary parameters. [ABSTRACT FROM AUTHOR]

In this paper, we will study the max difference equation in the title when is a constant sequence or period-four sequence of real numbers greater than one. In the former case, we provide an alternative proof to that already given in Amleh, Hoag, and Ladas, 1998, A difference equation with eventually periodic solutions, Computers and Mathematics with Applications , 36, 401–404, and Grove and Ladas, 2005, Periodicities in Nonlinear Difference Equations (Boca Raton: Chapman & Hall/CRC), where we show that every solution is eventually periodic with period 4. We prove that every solution is eventually periodic with period 8 in the latter case. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the set of solutions of a linear difference equation of high-order. More precisely, we obtain a basis of such a linear subspace. We deduce integral representations for elements of this basis. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of multiple positive solutions of boundary value problems for second-order discrete equations ? 2 x ( n - 1) - p ? x ( n - 1) - qx ( n - 1)+ f ( n , x ( n )) = 0, n ? {1,2,...}, ax (0) - ß? x (0) = 0, x (8) = 0. The proofs are based on the fixed point theorem in Fréchet space (see Agarwal and O'Regan, 2001, Cone compression and expansion and fixed point theorems in Fréchet spaces with application, Journal of Differential Equations , 171, 412–42). [ABSTRACT FROM AUTHOR]

This paper studies the behavior of positive solutions of the recursive equationwith We prove that every positive solution { y i } converges to a period two solution or to the equilibrium . This result answers Open Problem 11.4.8 (a) in Kulenovic and Ladas, 2002 Dynamics of Second Order Rational Difference Equations. With Open Problems and Conjectures (Boca Raton:Chapman and Hall/CRC). [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear difference equation of the form xn+1 =f(pn,xn-2s,xn-2t-1), n = 0,1,…, under some certain assumptions, where s,t ∈ {0,1,2,…} with s ⩽ t and the initial values x-2t-1 ,x-2t,ߪ, x0 ∈ (0, + ∞) and pn is the period-two sequence. We give sufficient conditions under which every positive solution of this equation converges to the period-two solution. [ABSTRACT FROM AUTHOR]

In this paper, we first show that if x is a bounded solution of the difference equation x(n + 1) = Ax(n) +f(n), ∀n ∈ N and the sequence (x(n)}n∈N is totally ergodic, σΓ.(A): = σ(A) ∩ Γ is countable and the sequence {f(n)}n∈N is asymptotically almost periodic, then the sequence {x(n)}n∈N is asymptotically almost periodic. As an application, we consider the asymptotical periodicity of mild solutions to periodic evolution equations. [ABSTRACT FROM AUTHOR]

The Goulden-Jackson cluster method is a powerful method to find generating functions of pattern occurrences in random sequences [1]. The method is clearly explained, extended and implemented by Noonan and Zeilberger [2]. In this paper, we elaborate on one of the several extensions in [2], namely the extension from symmetrical Bernoulli sequences where the occurrences of each symbol have equal probability, to asymmetrical Bernoulli sequences with different probabilities of symbol generations. An explicit formula is derived for the extension, which is implicitly embedded in the treatment of [2]. The extended result is then compared with the method of Régnier-Szpankowski [3], a method which was developed independently to tackle the same problem. By manipulating some matrix inversions, we show that the Régnier-Szpankowski method can be simplified to the extended Goulden-Jackson method. [ABSTRACT FROM AUTHOR]

In section 1, we correct an error in the proof of Lemma 3.1 in L. Alsedà, M.A. del Río, and J.A. Rodríguez. Transitivity and dense periodicity for graph maps. J. Difference Equ. Appl. 9, 577–598, 2003. In section 2 we give a simple proof that weak mixing implies mixing for maps on topological graphs. The proofs can also be extended to (nut necessarily compact) intervals, so in particular, this paper shows that for one-dimensional manifolds, weak mixing implies mixing. [ABSTRACT FROM AUTHOR]

In this paper we develop some techniques to obtain global hyperbolicity for a certain class of endomorphisms of with This kind of endomorphisms are obtained from vectorial difference equations where the mapping defining these equations satisfy a circulant condition. In particular, we show that one-parameter families of these quadratic endomorphisms are hyperbolic for large values of the parameter. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behaviour and the stability of a class of linear neutral delay difference equations with variable coefficients and constant delays. Via an appropriate solution of the so-called generalized characteristic equation, an asymptotic criterion and a stability result are established. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of positive solutions of nonautonomous Beverton-Holt equationwhere { K n} is a positive persistent and bounded sequence. In our main result, we proved that all positive solutions { x n} satisfy the inequalityThe obtained result represents the extension of the Cushing-Hensons's conjecture. Applications to the case when { K n} is a positive periodic sequence are given and the alternative proof of the Cushing-Henson's conjecture is also presented. [ABSTRACT FROM AUTHOR]