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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 equation xn+1 = αxn+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]

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]

In this paper we introduce a generalized form of the well known ToDD's difference equation and give the closed form expressions for this generalized form . In other words , we have the following nonlinear rational partial difference equation T ‹X1,X2,X3, …,Xn› = 1 + T ‹X1 - 1,X2 - 1, …,Xn - 1› + T ‹X1 - 2,X2 - 2, …,Xn - 2› T ‹X1 - 3,X2 - 3,X3 - 3, …,Xn - 3› where X1,X2, …:,Xn ∊ ℕ,and the initial values T ‹p1, p2, …:, pn› ,T ‹p2, p1, p3, p4, …, pn› ,T ‹p2, p3, p1, p4, …, pn›,… …,T ‹p2, p3, p4, …p1, pn›,T ‹p2 - 3, p3 - 3, p4 - 3, …pn - 3, p1› are real numbers with p1 ∊ {0,-1,-2} and p2, p3, …, pn ∊ ℕ such that T ‹p1, p2, …:, pn› ≠ 0 ,T ‹p2, p1, p3, p4, …, pn› ≠ 0 , T ‹p2, p3, p1, p4, …, pn› ≠ 0,…,T ‹p2 - 3, p3 - 3, p4 - 3, …pn - 3, p1› ≠ 0. We will use a novel technique to prove the results by using what we call 'piecewise n-dimensional mathematical induction' which we intro- duce here for the first time . We will obvious that this new concept represents generalized form for many types of mathematical induction . As a direct consequences , we investigate and drive the explicit solutions for the well known ordinary ToDD's difference Equation . [ABSTRACT FROM AUTHOR]

In this paper, some qualitative properties are discussed such as the boundedness, the periodicity and the global stability of the positive solutions of the nonlinear difference equation... where the coefficients A; αi; βi 2 (0,1), i = 1,...., 5; while the initial conditions y-5,y-4,y-3,y-2; y-1; y0 are arbitrary positive real numbers. Some numerical examples will be 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]