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EN
This paper studies the solution space of systems of algebraic and difference equations, given as auto-regressive (AR) representations A(σ) β (k) = 0, where σ denotes the shift forward operator and A(σ) is a regular polynomial matrix. The solution space of such systems consists of forward and backward propagating solutions, over a finite time horizon. This solution space can be constructed from knowledge of the finite and infinite elementary divisor structure of A(σ) . This work deals with the inverse problem of constructing a family of polynomial matrices A(σ) such that the system A(σ) β (k) = 0 satisfies some given forward and backward behavior. Initially, the connection between the backward behavior of an AR representation and the forward behavior of its dual system is showcased. This result is used to construct a system satisfying a certain backward behavior. By combining this result with the method provided by Gohberg et al. (2009) for constructing a system with a forward behavior, an algorithm is proposed for computing a system satisfying the prescribed forward and backward behavior.
PL
W pracy rozważa się zagadnienie początkowe dla liniowego równania różnicowego o stałych współczynnikach określonego w przestrzeni ciągów dwustronnych C (Z) . Problem ten przedstawiono w ujęciu nieklasycznego rachunku operatorów Bittnera. Wykorzystując model nabla tego rachunku z pochodną rozumianą jako różnica wsteczna, rozpatrywane zagadnienie rozwiązano w tzw. przestrzeni wyników. Wyniki powstają przez dzielenie elementów przestrzeni C (Z) przez injekcyjne endomorfizmy tej przestrzeni. Przedstawione rozważania dają początek nabla-rachunkowi, który może być konkurencyjny w stosunku do rachunku opartego na dwustronnym przekształceniu.
EN
The paper analyses the initial value problem for a linear difference equation with constant coefficients, defined in the space of two-sided sequences. The above problem has been C (Z) presented using the non-classical Bittner operational calculus approach. Using the -model of that calculus with its derivative understood as a backward difference, the issue in question has been solved in a so-called results' space. The results are obtained by dividing the elements of the C (Z) space by the injective endomorphisms of that space. The described analysis gives rise to a nabla-calculus that can be considered competitive to the calculus based on the bilateral -transform.
EN
In this paper sufficient conditions for oscillation of all bounded solutions of the equation ...[wzór] where m ≥ 2, (pn) is an oscillatory sequence of real numbers, limn→∞ pn = 0, τ and σ are positive integers, f : N×R×R → R are established.
4
Content available remote On the solutions of the recursive sequence xx+1 = ...[wzór]
EN
In this paper we study the solutions of the difference equation x + ...[wzór] for n = 0,1,2, ... where a, x-(2k+1), x-(2k), x-(2k-1), ..., x0 are the real numbers such that x0x-(k+1) ≠ a, x-1x-(k+2) ≠ a, x-2x-(k+3) ≠ a. ..., x-kx-(2k+1) ≠ a and k is a natural number.
5
Content available remote On the solutions of the difference equation xn+1 = max ...[wzór]
EN
We study the solutions of the following difference equation ...[wzór] where initial conditions x-1and x0 are nonzero real numbers. In most of the cases we determine the solutions in function of the initial conditions x-1 and x0.
6
Content available remote On the solutions of a rational system of difference equations
EN
In this paper we deal with the solutions of the system of the difference equations xn+1 = ...[wzór], yn+1 = ...[wzór], with a nonzero real numbers initial conditions.
EN
The paper contains connections between oscillation of solutions of iterative functional equations, difference equations and differential equations with advanced or delayed arguments. New oscillatory criteria for these equations are given.
8
Content available remote On asymptotically periodic solutions of linear discrete Volterra equations
EN
We show that a class of linear nonconvolution discrete Volterra equations has asymptotically periodic solutions. We also examine an example for which the calculations can be done explicitly. The results are established using theorems on the boundedness and convergence to a finite limit of solutions of linear discrete Volterra equations.
9
Content available remote On a linear difference equation with several infinite lags
EN
This paper deals with asymptotic properties of the solutions of a variable order linear difference equation. As the main result, we derive the effective asymptotic estimate valid for all solutions of this equation. Moreover, we are going to discuss some consequences of this theoretical result, especially with respect to the numerical analysis of the multi-pantograph differential equation.
10
Content available remote On a class of rational difference equations
EN
In this paper we study the behavior of the positive solutions of the following nonlinear difference equation ...[wzór], n = 0, 1, 2, ... where the initial values ...[wzór] and k = 0, 1, 2, . . ..
11
Content available remote Further instances of periodicity in May's host parasitoid equation
EN
May’s host parasitoid equation is the difference equation (1) ...[wzór] We show that for each &slpha; there is a number k such that, whenever n > k, equation (1) has a one cycle periodic solution of period n. We also give some results on two cycle periodic solutions.
12
EN
In this paper three-dimensional nonlinear difference system with delays ...[wzór] is investigated. The classification of nonoscillatory solutions of the considered system are presented. Next, the sufficient conditions under which nonoscillatory solution of considered system is bounded or is unbounded are given. Key words: difference equation, nonlinear system, nonoscillatory, bounded, unbounded solution.
13
EN
The mean return time of a discrete Markov chain to a point x is the reciprocal of the invariant probability π(x). We revisit this classical theme to investigate certain exit times for stochastic difference equations of autoregressive type. More specifically, we will discuss the asymptotics, as 0, of the first time that the n-dimensional process ...[wzór] (where ξ1, ξ2, . . . is a sequence of i.i.d. random n-vectors) leaves a given neighborhood of the fixed point of the contraction f.
14
Content available remote On the solution of the recursive sequence
EN
We obtain in this paper the solution of the following recursive sequence where the initial conditions x-2, x-1are arbitrary non zero real numbers.
15
Content available remote On the dynamics of the difference equation xn+1 = ...[wzór]
EN
This paper studies the global behavior of the difference equation xn+1 = ...[wzór], n = 0, 1, 2, ... with non-negative parameters and non-negative initial conditions where k is an odd number.
16
Content available remote On the difference equation xn+1 = α + ...[wzór]
EN
In this paper, we investigate the global behavior of the difference equation of order three xn+1 = α + ...[wzór], n = 0, 1,… where the parameters α, k ∈ (0, ∞) and the initial values x-2, x-1 and x0 are arbitrary positive real numbers.
17
Content available remote Some nonlinear delay discrete inequalities and their applications
EN
In this paper, we establish some nonlinear delay discrete inequalities which provide explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain delay difference equations.
18
Content available remote On the discrete version of generalized Kiguradze's lemma
EN
The Kiguaradze's lemma for quasi-differeuces of a real sequence is presented. Some examples illustrating the result are included.
EN
In this paper necessary and sufficient conditions have been obtained so that every solution of the Neutral Delay Difference Equation (NDDE) oscillates or tends to zero as n —> &infin for different ranges of This paper improves and generalizes some recent work [2. 6, 8]. The results of this paper hold for linear, sublinear and superlinear equations and also for homogeneous equations, i.e. when fn ≡ 0.
20
Content available remote Global attractivity in a non-autonomous logistic type model with unbounded delay
EN
Consider the non-autonomous logistic model where {pn} is a sequence of positive real numbers, {kn} a sequence of nonnegative integers satisfying lim n (n - kn) = &isin, Lamda &isin [0, 1). We obtain new sufficient conditions for the attractivity of equilibrium x=1 of the model, which improve and generalize some recent results established by Chen and Yu [4], Zhou and Zhaug [6].
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