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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]

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 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]

This paper is concerned with local stability, oscillatory character of positive solutions to the system of the two nonlinear difference equations xn+1=A+xn-1p/ynp and yn+1=A+yn-1p/xnp, n=0,1,…, where A∈(0,∞), p∈[1,∞), xi∈(0,∞), and yi∈(0,∞), i=-1,0. [ABSTRACT FROM AUTHOR]

We study boundary value problems for q-difference equations and inclusions with nonlocal and integral boundary conditions which have different quantum numbers. Some new existence and uniqueness results are obtained by using fixed point theorems. Examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the boundedness, persistence, and global asymptotic behavior of positive solution for a system of two rational difference equations xn+1 = A + (xn/ ∑i=1k yn-i), yn+1 = B + (yn/ ∑i=1k xn-i), n = 0,1,..., k ∈ {1,2,...}, where A,B ∈ (0, ∞), x-i ∈ (0, ∞), and y-i ∈ (0, ∞), i = 0,1,2,..., k. [ABSTRACT FROM AUTHOR]

We deal with the form of the solutions for the following systems of rational difference equations xn+1 = (xn-1yn/(±xn-1 ± yn-2)), yn+1 = (xnyn-1/(±yn-1 ± xn-2))> with nonzero real numbers initial conditions. Also we investigate some properties of the obtained solutions and present some numerical examples. [ABSTRACT FROM AUTHOR]

We focus on a system of a rational m-order difference equation xn+1 = xn-m+1/(A + yn yn-1 ... yn-m+1, yn+1 = (yn-m+1)/(B + xn xn-1... xn-m+1), n = 0,1, ..., where A, B, x0, x-1, ..., x-m+1, y0, y-1, ..., y-m+1 ∈ (0, ∞). We investigate the dynamical behavior of positive solution for the system. [ABSTRACT FROM AUTHOR]

Here we show that the main results in the papers by Yalcinkaya (2008), Yalcinkaya and Cinar (2010), and Yalcinkaya, Cinar, and Simsek (2008), as well as a conjecture from the last mentioned paper, follow from a slight modification of a result by G. Papaschinopoulos and C. J. Schinas. We also give some generalizations of these results. [ABSTRACT FROM AUTHOR]

We present sufficient conditions ensuring the lower and upper functions on the reversed-order for the periodic difference equations. This enables us to obtain the existence of positive periodic solutions of the second-order difference equation Δ²u(t-1) = g(t)/uμ(t)- h(t)/uλ(t) + f(t), t ∊ Z, where g, h : Z → [0,∞), and f : Z → R are T-periodic functions, λ, μ > 0. [ABSTRACT FROM AUTHOR]

The behavior of well-defined solutions of the difference equation xn = anxn-k/(bn + cnxn-1 · · · xn-k), n ∊ N0, where k ∊ N is fixed, the sequences an, bn and cn are real, (bn, cn) ≠(0, 0), n ∊ N0, and the initial values x-k, . . ., x-1 are real numbers, is described. [ABSTRACT FROM AUTHOR]

We are concerned with a kind of three-dimensional system of rational difference equations, given by Kurbanli (2011). A new expression of solution of the system is presented, and the asymptotical behavior is described. At the same time, we also consider a different system and obtain some results, which expand the study of such a kind of difference equations and the method can be applied to other systems. [ABSTRACT FROM AUTHOR]

We establish some new oscillation criteria for the second-order neutral delay dynamic equations of Emden-Fowler type, [a(t)(x(t) + r(t)x(τ(t)))Δ]Δ + p(t)xγ (δ(t)) = 0, on a time scale unbounded above. Here γ > 0 is a quotient of odd positive integers with a and p being real-valued positive functions defined on 픛. Our results in this paper not only extend and improve the results in the literature but also correct an error in one of the references. [ABSTRACT FROM AUTHOR]

The following difference equation Xn = Xn-1Xn-k-1, n ∈ ℕ0, where k, l ∈ ℕ, k < l, gcd(k, l) = 1, and the initial values x-1,…, x-2, x-1 are real numbers, has been investigated so far only for some particular values of k and l. To get any general result on the equation is turned out as a not so easy problem. In this paper, we give the first result on the behaviour of solutions of the difference equation of general character, by describing the long-term behavior of the solutions of the equation for all values of parameters k and l, where the initial values satisfy the following condition min {x-1,…, x-2, x-1}. [ABSTRACT FROM AUTHOR]

When an impulsive control is adopted for a stochastic delay difference system (SDDS), there are at least two situations that should be contemplated. If the SDDS is stable, then what kind of impulse can the original system tolerate to keep stable? If the SDDS is unstable, then what kind of impulsive strategy should be taken to make the system stable? Using the Lyapunov-Razumikhin technique, we establish criteria for the stability of impulsive stochastic delay difference equations and these criteria answer those questions. As for applications, we consider a kind of impulsive stochastic delay difference equation and present some corollaries to our main results. [ABSTRACT FROM AUTHOR]

The article investigates the oscillatory behavior, the global stability and periodic character of the solutions of a difference equation. It is stated that the interest in studying nonlinear and rational equations has been increased by the necessity for some techniques which can be used in investigating equations arising in mathematical models describing real-life situations in population biology and economics. It tries to answer several questions, including the monotone solution for the difference equation.

The article presents a study on the result of a global convergence for a higher order difference equation. It is noted that the function ƒ in the equation does not need to be monotonous in each variable, and the condition ƒ (x,....,x) ≤ x can be omitted as an irrelevant one. It is assumed that (xn) is a solution of the difference equation, for which there is an n0 ∈ ℕ. It is concluded that any function ƒ from the previously described class of functions satisfies the conditions from Theorem 1.4.

The article examines the behavior of the solutions of a second-order difference equation. The equation studied in this article is xn+1=α-xn/xn-1. It is proven that when α=3 the equilibrium &xtilde;=2 of the equation is not stable, which corrects a result due to X. X. Yan, W. T. Li and Z. Zhao. For the case α=1, the authors show that there is a strictly monotone solution of the equation, and they also find its asymptotics. An explicit formula for the solutions of the equation are presented for the case α=0.