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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 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]

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]

The article focuses on an introduction to discrete-time in economics. In the recent literature on economic dynamics, discrete-time models have steadily been gaining in popularity. In economics, as in many other fields of dynamics, both discrete-time and continuous-time models have been adopted. Traditionally, many economists have preferred differential equations; perhaps, this is partly because of the convenience of differential calculus and partly because, by limiting the number of state variables to two, it is possible to fully characterize dynamics by means of a simple two-dimensional phase diagram.

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]

Our aim is to investigate the global asymptotic behaviour, oscillation, periodicity and bifurcation in discontinuous Beverton–Holt type difference equationwherex0>0, functionskandrare discontinuous piecewise constant functionssatisfyingandhis Heaviside function. [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]

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]

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]

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]

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]

There has been a considerable number of recent discussions on applications of different types of dynamic derivatives. The dynamic derivatives not only play a central role in time scales theory and methods, but also provide a remarkable way of approximations in mathematical modelling and computational applications. In this paper, we will explore connections between existing dynamic and conventional derivatives over the time scales used. We will show that while first order dynamic derivatives provide acceptable accuracies in approximating conventional derivatives, second order dynamic derivatives are in general inconsistent with conventional derivatives. It is evident that, before incorporating structures of particular time scales used into the dynamic derivative formulae properly, one needs to be prudential when bridging discrepancies between the differential and difference equations modelling similar applications by higher order dynamic derivatives. Numerical experiments are given to further illustrate our conclusions. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to investigate the asymptotic behavior of solutions of the following difference equation [This symbol cannot be presented in ASCII format] where {an} is a positive bounded sequence and k is a positive integer. Sufficient conditions for boundedness and attractivity are obtained. The results are applied to the special case when {an} is a periodic sequence with prime period p. [ABSTRACT FROM AUTHOR]

The boundedness, global attractivity, oscillatory and asymptotic periodicity of the nonnegative solutions of the difference equation in the title is investigated, where all the coefficients are nonnegative real numbers The paper is motivated by an open problem proposed by Ladas [Open problems and conjectures, J. Differ, Equations Appl., 5 (1999), 211-215]. [ABSTRACT FROM AUTHOR]

A previously reported bifurcation technique is applied to the construction of nonstandard finite difference representations of systems of nonlinear differential equations. This technique provides a measure of the deviation between bifurcation parameters obtained from fixed step representations of the nonlinear system and the values of the parameters determined from computational experiments. Since this deviation or 'error' is characteristic of a particular scheme, we have used this measure to construct low-error nonstandard representations. We present results from several nonlinear test models which show that such nonstandard schemes yield orbits that followed closely the expected dynamics and also provide a large reduction in the computational error in comparison to standard numerical integration schemes. Finally, we outline a criteria for controlling possible numerical overflow in fixed step-size schemes. [ABSTRACT FROM AUTHOR]

We show that the following system of difference equationswhere,,, and sequences,,andare real, can be solved in closed form. For the case when the sequences,,andare constant and, we apply obtained formulas in the investigation of the asymptotic behaviour of well-defined solutions of the system. We also find domain of undefinable solutions of the system. Our results considerably extend and improve some recent results in the literature. [ABSTRACT FROM AUTHOR]

We exhibit an alternative method for solving inhomogeneous second-order linear ordinary dynamic equations on time scales, based on reduction of order rather than variation of parameters. Our form extends recent (and long-standing) analysis on [image omitted] to a new form for difference equations, quantum equations and arbitrary dynamic equations on time scales. [ABSTRACT FROM AUTHOR]

Let p and q be arbitrary positive numbers. It is shown that if [image omitted] , then all solutions to the difference equation[image omitted] converge to the positive equilibrium [image omitted] . The above result, taken together with the 1993 result of Kocic and Ladas for equation (E) with [image omitted] , gives global attractivity of the positive equilibrium of (E) for all positive values of the parameters, thus completing the proof of a conjecture of Ladas. [ABSTRACT FROM AUTHOR]

For non-autonomous difference equations of the form[image omitted] we consider homoclinic trajectories. These are pairs of trajectories that converge in both time directions towards each other. Assuming hyperbolicity, we derive a numerical method to compute homoclinic trajectories in two steps. In the first step, one trajectory is approximated by the solution of a boundary value problem and precise error estimates are given. In particular, influences of parameters [image omitted] with |n| large are discussed in detail. A second trajectory that is homoclinic to the first one is computed in a subsequent step as follows. We transform the original system into a topologically equivalent form having 0 as an n-independent fixed point. Applying the boundary value ansatz to the transformed system, we obtain a non-autonomous homoclinic orbit, converging towards the origin (T. Huls, J. Difference Equ. Appl. 12(11) (2006), pp. 1103-1126). Transforming back to the original coordinates leads to the desired homoclinic trajectories. The numerical method and the validity of the error estimates are illustrated by examples. [ABSTRACT FROM AUTHOR]

We study the set of periods of a p-periodic nonautonomous difference equation xn+1 = fn(xn), where fn: I → I are continuous interval maps satisfying the generic condition: Card{x ∈ I: fi(x) = fj(x)} < ∞, for all i ≠ j(mod p). [ABSTRACT FROM AUTHOR]

The Fredholm property of some linear infinite dimensional difference operators is studied. In the case corresponding to discretization of differential equations on the real axis, the index of the corresponding operators is computed and solvability conditions for the nonhomogeneous problem are established. In the multi-dimensional case, conditions of the normal solvability of the corresponding discrete operators are formulated in terms of limiting problems. The results on the location of the spectrum and the solvability conditions allow various applications to linear and nonlinear problems. [ABSTRACT FROM AUTHOR]

Leslie's method to construct a discrete two dimensional dynamical system dynamically consistent with the Lotka-Volterra type of competing two species ordinary differential equations is applied in a newly extended manner for the Lotka-Volterra prey-predator system which is structurally unstable. We show that, independently of the time step size, the derived discrete prey-predator system is dynamically consistent with the continuous counterpart, keeping the nature of neutrally stable periodic orbit. Further, we show that the extended method to construct the discrete prey-predator system can provide a dynamically consistent model also for the logistic Lotka-Volterra one. [ABSTRACT FROM AUTHOR]

We construct nonstandard finite-difference (NSFD) schemes that provide exact numerical methods for a first-order differential equation having three distinct fixed-points. An explicit, but also nonexact, NSFD scheme is also constructed. It has the feature of preserving the critical properties of the original differential equation such as the positivity of the solutions and the stability behavior of the three fixed-points. [ABSTRACT FROM AUTHOR]

We show that the condition [image omitted] is not necessary, though sufficient, for the asymptotic stability of [image omitted] . We prove the existence of a class of Volterra difference equations (VDEs) that violate this condition but whose zero solutions are asymptotically stable. [ABSTRACT FROM AUTHOR]

The authors consider the fourth order quasilinear difference equation[image omitted] where α and β are positive constants and {pn} and {qn} are positive real sequences. They classify the nonoscillatory solutions according to their asymptotic behavior for large n and then give necessary and sufficient conditions for existence of solutions of these various types. The results are illustrated with examples. [ABSTRACT FROM AUTHOR]

This article analyzes a nonlinear system of first-order difference equations with periodic and non-periodic boundary conditions. Some sufficient conditions are presented under which: potential solutions to the equations will satisfy certain a priori bounds; and the equations will admit at least one solution. The methods involve new dynamic inequalities and use of Brouwer degree theory. The new results are compared with those featuring in the theory of solutions to boundary value problems for differential equations. [ABSTRACT FROM AUTHOR]

We present a general procedure for finding linear recurrence relations for the solutions of the second order difference equation of hypergeometric type. Applications to wave functions of certain discrete system are also given. [ABSTRACT FROM AUTHOR]

The described algorithms enable one to find all solutions of parameterized linear difference equations within ΠΣ-fields, a very general class of difference fields. These algorithms can be applied to a very general class of multisums, for instance, for proving identities and simplifying expressions. [ABSTRACT FROM AUTHOR]

In periodically varying environments, population models generate periodic dynamical systems. To understand the effects of unidirectional dispersal on local patch dynamics in fluctuating environments, dynamical systems theory is used to study the resulting periodic dynamical systems. In particular, a unidirectional dispersal linked two patch nonautonomous metapopulation model is constructed and used to explain the qualitative dynamics of linked versus unlinked independent patches. As in single-patch, single-species population models, unidirectional nonautonomous models support multiple attractors where local population models support single attractors. [ABSTRACT FROM AUTHOR]

We consider a difference equation where h?(0,h0],h0 is a fixed positive number, We obtain necessary and sufficient conditions under which assertions (I) and (II) hold together: (I) for a given p?[1,8), for any f?Lp(h),equation (1) has a unique solution y={yn(h)}n?Z?Lp(h) (regardless of h), and n?Z. (II) ?y?Lp(h)=c(p)??f??Lp(h) for any f?Lp(h). Here c(p) is an absolute positive constant, {Gn,m(h)}n,m?Z is the difference Green function corresponding to equation (1). [ABSTRACT FROM AUTHOR]

In this work, we investigate the boundedness and uniqueness of solutions to systems of dynamic equations on time scales. We define suitable Lyapunov-type functions and then formulate appropriate inequalities on these functions that guarantee all solutions to first-order initial value problems are uniformly bounded and/or unique. Several examples are given. [ABSTRACT FROM AUTHOR]

We consider a classical problem of free oscillations of the elastic weightless string with N + 1 beads which has been originally studied by Lagrange. It is proved that for N being prime or a power of 2, the maximal displacement of the bead from its equilibrium position increases logarithmically to infinity as N → ∞. [ABSTRACT FROM AUTHOR]

The asymptotic properties of the impulsive Beverton-Holt difference equation (This equation can not be captured) where p is fixed positive integer are considered. The results are applied to an impulsive logistic equation with non-constant coefficients (This equation can not be captured).In particular sufficient extinction and non-extinction condition are obtained for both equations. [ABSTRACT FROM AUTHOR]

The two-parameter absolute difference equation xn+1 = ∣axn-bxn-1∣ is studied. Based on the parameter values a, b and a pair of initial values, we consider the existence and bifurcations of solutions having one or more of the following properties: (i) unbounded, (ii) convergent (to zero or to a positive constant) (iii) monotonic, (iv) periodic and (v) non-periodic oscillatory. The semiconjugate first order equation satisfied by the ratios (xn/xn-1 is used to significant advantage for points (a, b) in certain regions of the parameter plane. Some open problems and conjectures are presented. [ABSTRACT FROM AUTHOR]

Existence of analytic solutions of an iterative functional differential equation is studied by reducing locally the equation to another functional differential equation without iteration of the unknown function. As done previously we first discuss the case that the constant f (0) is not on the unit circle in C and the case that the constant lies on the circle but fulfills the Diophantine condition. Furthermore, we investigate the case that the constant is a unit root, which violates the Diophantine condition. [ABSTRACT FROM AUTHOR]

We study the second-order difference equation [This equation can not be represented in ASCII.TXT] where f &ephiv; C[SUP1] ([0,∞), [0,∞)) and x[SUBn] &ephiv; Z. For the cases p ≤ 5, we find necessary and sufficient conditions on f for all solutions to be periodic with period p. We answer some questions and conjectures of Kulenović and Ladas. [ABSTRACT FROM AUTHOR]

Let T be a closed subset of R with inf T = ∞ and sup T = +∞. For the fourth order nonlinear dynamic equation, u[SUPΔ4] = f (t, u, u[SUPδ u[SUPδ2], u[SUPΔ3]). t ∊ T; where f : T × R[SUP4] → R is continuous, we assume solutions of initial value problems are unique and extend to T. We consider questions of the uniqueness of solutions implying the existence of solutions for conjugate boundary problems on T. [ABSTRACT FROM AUTHOR]