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In this paper we give a representation formula for the general solution to the following two-dimensional system of difference equations ... where parameters a,b and initial values x-2,x-1,x0,y-2,y-1,y0 are real numbers. We also give some theoretical explanations related to the representation. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the form of the solutions and the periodicity character of the following systems of nonlinear difference equations of order two Zn+1 = Zntn-1/±tn ± tn-1, tn+1 = tnZn-1/±Zn ± Zn-1, where the initial conditions Z-1, Z0, t-1 and t0 are nonzero real numbers. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to investigate the dynamics of a system of fourth-order rational difference equations... where the parameter A is arbitrary positive real number and the initial conditions x-3,x-2,x-1,x0,y-3,y-2,y-1,y0 are arbitrary nonnegative real numbers. By using new iteration method for the more general nonlinear difference equations and inequality skills, we establish some sufficient conditions which guarantee the existence, unstability and global asymptotic stability of the equilibriums for this nonlinear system. Numerical examples to the difference system are given to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this article we study the existence of solutions and some of their qualitative behavior of the following rational nonlinear difference equation xn+1 = axn-(2k+1)/ b + cxn-kxn-(2k+1) , n = 0, 1, ..., where a, b and c are real numbers, k is a non-negative integer number and the initial conditions x-2k-1, x-2k , ..., x-1, x0 are arbitrary non-negative real numbers. Also, the solutions of some special cases of the equation under consideration will be obtained. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the form of the solutions and the periodicity character of the following systems of nonlinear difference equations of order two zn+1=zn+tn-l/±tn±tn-1,tn+1=tnzn-l/±zn±zn-l where the initial conditions z-1, z0, t-1 and t0 are nonzero real numbers. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate the global stability of the solutions, the boundedness and the periodic character of the nonlinear difference equationxn+1 = βxn+βxn-l+γxn-k + axnxn-k/bxn + cxn-l + dxn-k, n = 0, 1, ..., where the parameters α, β, γ, a, b, c and d are positive real numbers and the initial conditions x-s, x-s+1, ..., x-1, x0 are positive real numbers where s = max{l, k}. Some numerical examples will be given to explicate our results. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to investigate the dynamics of a system of fourth-order rational difference equations xn+1=xn-3-yn-1/A+xn-3yn-1,yn+1=yn-3-xn-1/A+yn-3xn-1, n=0,1,..., where the parameter A is arbitrary positive real number and the initial conditions x-3,x-2,x-1,x0,y-3,y-2,y-1,y0 are arbitrary nonnegative real numbers. By using new iteration method for the more general nonlinear difference equations and inequality skills, we establish some sufficient conditions which guarantee the existence, unstability and global asymptotic stability of the equilibriums for this nonlinear system. Numerical examples to the difference system are given to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we will investigate the local stability of the equilibrium points, global attractor, boundedness and the form of the solutions for the following equations xn+1 = Axnxn-3/Bxn + Cxn-2 and xn+1 =Axnxn-3/ Bxn - Cxn-2 , where the coefficients A, B and C are real positive numbers, and the initial conditions x-3,x-2, x-1, x0 are arbitrary non zero real numbers. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the periodic character and the global stability of the positive solutions of the difference equation xn+1 = axn + bxn-k + cxn-l - dxn-s/exn-s - αxn-t, n = 0, 1, ..., where the parameters a, b, c, d, e and α are positive real numbers and the initial conditions x-σ, x-σ+1, ..., x-1, x0 are positive real numbers where σ = max{s, t, k,}. Some numerical examples were given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of the solutions of the following rational difference equation with big order xn+1 = axn-l + bxn-k + cxn-s + dxn-t/exn-s + fxn-t where the parameters a; b; c; d; e and f are positive real numbers and the initial conditions x-r, x-r+1, x-1 and x0 are positive real numbers where r = max(l, k, s, t). [ABSTRACT FROM AUTHOR]

Our aim in this paper is to investigate the behavior of the solution of the following max type difference equation of order three xn+1 = max{An/xn,xn-2} , n = 0, 1,. . ., where the initial conditions x-2, x-1, x0 are arbitrary positive real numbers and {An}n=0∝ is a periodic sequence of period two. [ABSTRACT FROM AUTHOR]

Hahn introduced the difference operator $D_{q,\omega}f(t)= (f(qt+\omega)-f(t) )/ (t(q-1)+\omega )$ in 1949, where $0< q<1$ and $\omega> 0$ are fixed real numbers. This operator extends the classical difference operator $\vartriangle_{\omega}f(t)= (f(t+\omega)-f(t) )/\omega$ as well as the Jackson q-difference operator $D_{q}f(t)= (f(qt)-f(t) )/ (t(q-1) )$. In this paper, we study the theory of abstract linear Hahn difference equations of the form where B and $A_{i}$ are mappings from an interval I into a Banach algebra $\mathbb{X}$, $i=1,\ldots,n$. We define the abstract exponential functions and the abstract trigonometric (hyperbolic) functions. We prove they are solutions of first and second order Hahn difference equations, respectively. Also, we obtain an integral equation corresponding to the second order linear Hahn difference equations which is known as the Volterra integral equation. Finally, we present the analogs of the variation of parameter technique and the annihilator method for the non-homogeneous case. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the periodic nature of the solution of the following max-type difference equation: [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of solutions of the higher-order neutral difference equation where τ ( n) is a general retarded argument, σ( n) is a general deviated argument, c ∈ $ \mathbb{R} $; ( p( n)) is a sequence of real numbers, ∆ denotes the forward difference operator ∆ x( n) = x( n+1) - x( n); and ∆ denotes the jth forward difference operator ∆ ( x( n) = ∆ (∆( x( n))) for j = 2, 3,..., m. Examples illustrating the results are also given. [ABSTRACT FROM AUTHOR]

We study the behavior of the well-defined solutions of the max type difference equation xn+1 = max {1/xn, Anxn-1}, n = 0, 1,..., where the initial conditions are arbitrary nonzero real numbers and {An} is a period-two sequence of real numbers with An ∈ [0,∞). [ABSTRACT FROM AUTHOR]

We show that the difference equation xn+1 = xnxn-k/xn-k+1(a + bxnxn-k), n ∈ ℕ0, where k ∈ ℕ, the parameters a, b and initial values x-i, i= 0, k are real numbers, can be solved in closed form considerably extending the results in the literature. By using obtained formulae, we investigate asymptotic behavior of well-defined solutions of the equation. [ABSTRACT FROM AUTHOR]

As an effective tool to unify discrete and continuous analysis, time scale calculus have been widely applied to study dynamic systems in both theoretical and practical aspects. In addition to such a classical role of unification, the dynamic equations on time scales have their own unique features which the difference and differential equations do not possess and these advantages have been highlighted in describing some complicated dynamical behavior in the hybrid time process. In this review article, we conduct a survey of abstract analysis and applied dynamic equations on hybrid time scales, some recent main results and the related developments on hybrid time scales will be reported and the future research related to this research field is discussed. The results presented in this article can be extended and generalized to study both pure mathematical analysis and real applications such as mathematical physics, biological dynamical models and neural networks, etc. [ABSTRACT FROM AUTHOR]