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In this paper, we solve the difference equation xn+1 = α xnxn-1 - 1 , n = 0, 1, . . . , where α > 0 and the initial values x-1, x0 are real numbers. We find some invariant sets and discuss the global behavior of the solutions of that equation. We show that when α > 2 3 √ 3 , under certain conditions there exist solutions, that are either periodic or converging to periodic solutions. We show also the existence of dense solutions in the real line. Finally, we show that when α < 2 3 √ 3 , one of the negative equilibrium points attracts all orbits with initials outside a set of Lebesgue measure zero.

In this paper, we show that the system of difference equations and the initial values x y z j    j j j,,, 1,2,3   are non-zero real numbers, can be solved in the closed form. Also, we determine the forbidden set of the initial values by using the obtained formulas. Finally, we obtain periodic solutions of aforementioned system. Keywords: periodicity; system of difference equations; forbidden set.In this paper, we show that the system of difference equations       2 3 2 3 2 3 0 1 2 3 1 2 3 1 2 3,,,, n n n n n n n n n n n n n n n n n n n n n n n n x z y x z y x y z n z a b x z x y x y A B z y                         where the sequences   0, n n a    0, n n b    0, n n     0, n n     0, n n A    0, n n B  and the initial values x y z j    j j j,,, 1,2,3   are non-zero real numbers, can be solved in the closed form. Also, we determine the forbidden set of the initial values by using the obtained formulas. Finally, we obtain periodic solutions of aforementioned system. [ABSTRACT FROM AUTHOR]

In this paper, we show that the following three-dimensional rational system of difference equations xn = zn−1zn−3 /bxn−2 + azn−3, yn = xn−1xn−3 /dyn−2 + cxn−3, zn = yn−1yn−3/ fzn−2 + eyn−3, n ∈ N0, where the parameters a, b, c, d, e, f and the initial values x−i, y−i, z−i, i ∈ {1, 2, 3}, are real numbers, can be solved in explicit form. In addition, the solutions of aforementioned systems according to the special cases of the parameters are given in closed form. Later, the forbidden set of the initial values for aforementioned system is described. Finally, an application and numerical examples to support our results are given. [ABSTRACT FROM AUTHOR]

In this paper, we study the following two-dimesional system of difference equations ... where the parameters a;b; c;d and the initial values x i; y i, i 2 f1;2;3;4;5;6g, are real numbers. We show that some subclasses of nonlinear two-dimensional system of difference equations are solvable in closed form. We also describe the forbidden set of solutions of the system of differ- ence equations. Some numerical examples are given to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we present that the following system of difference equations ... where n ∈ ℕ0, k, l ∈ ℕ0, the initial values x-i,y-i, z-i are real numbers, for i ∈ 1,k +1, and sequences (an)... (bn)..., (cn)..., (dn)..., (en)... and (fn)... are non-zero real numbers, for all n e No, which can be solved in closed form. We describe the forbidden set of the initial values using the obtained formulas and also determine the asymptotic behavior of solutions for the case k = 3, l = l, and the sequences (an)..., (bn)..., (cn)..., (dn)..., (en)... and (fn)... are constant. Our results considerably extend and improve some recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we show that the following three-dimensional system of difference equations xn+1 = ynxn−2/axn−2 + bzn−1, yn+1 = znyn−2/cyn−2 + dxn−1, zn+1 = xnzn−2/ezn−2 + ƒyn−1, n ∈ N0, where the parameters a, b, c, d, e, f and the initial values x−i, y−i, z−i, i ∈ {0, 1, 2}, are complex numbers, can be solved, extending further some results in the literature. Also, we determine the forbidden set of the initial values by using the obtained formulas. Finally, an application concerning a three-dimensional system of difference equations are given. [ABSTRACT FROM AUTHOR]

In this paper, we solve open problem (5) submitted by Sedaghat in his paper, On third order rational difference equations with quadratic terms, J. Differ. Equ. Appl., 14(8) (2008), pp. 889–897. We also confirm conjecture (6) in the mentioned paper. [ABSTRACT FROM PUBLISHER]

The concepts of permitted and forbidden sets enable a new perspective of the memory in neural networks. Such concepts exhibit interesting dynamics in recurrent neural networks. This paper studies the basic theories of permitted and forbidden sets of the linear threshold discrete-time recurrent neural networks. The linear threshold transfer function has been regarded as an adequate transfer function for recurrent neural networks. Networks with this transfer function form a class of hybrid analog and digital networks which are especially useful for perceptual computations. Networks in discrete time can directly provide algorithms for efficient implementation in digital hardware. The main contribution of this paper is to establish foundations of permitted and forbidden sets. Necessary and sufficient conditions for the linear threshold discrete-time recurrent neural networks are obtained for complete convergence, existence of permitted and forbidden sets, as well as conditionally multiattractivity, respectively. Simulation studies explore some possible interesting practical applications. [ABSTRACT FROM AUTHOR]

In this paper, we determine the forbidden set, introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equation x n + 1 = a x n x n - k + 1 b x n - k + 1 + c x n - k , n = 0 , 1 , ... , where a, b, c are positive real numbers, the initial conditions x - k , x - k + 1 , ... , x - 1 , x 0 are real numbers and k is a positive integer. [ABSTRACT FROM AUTHOR]

In this paper, we determine the forbidden set, introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equation xn+1 = axnxn-k/bxn - cxn-k-1 , n = 0, 1, . . . , where a, b, c are positive real numbers and the initial conditions x-k-1, x-k , . . . , x-1, x0 are real numbers. We show that when a = b = c, the behavior of the solutions depends on whether k is even or odd. [ABSTRACT FROM AUTHOR]

In this paper, we solve the difference equation x n + 1 = α 1 - x n x n - 1 , n = 0 , 1 , ... , where α > 0 and the initial values x - 1 , x 0 are real numbers. We find invariant sets and discuss the global behavior of the solutions of that equation. We show that when α < 2 3 3 , one of the positive equilibrium points attracts all orbits with initials outside a set of Lebesgue measure zero. Also, when α = 2 3 3 , the unique positive equilibrium points attracts all orbits with initials outside a set of Lebesgue measure zero. Finally, we show that when α > 2 3 3 , under certain conditions there exist solutions that are either periodic or converging to periodic solutions and give some examples. We show also the existence of dense solutions in the real line. [ABSTRACT FROM AUTHOR]

In this paper, we determine the forbidden sets, introduce an explicit formula for the solutions and discuss the global behaviors of solutions of the difference equations x n + 1 = a x n x n − 1 b x n − 1 + c x n − 2 , n = 0 , 1 , ... $$\begin{array}{} \displaystyle x_{n+1}=\frac{ax_{n}x_{n-1}}{bx_{n-1}+ cx_{n-2}},\quad n=0,1,\ldots \end{array} $$ where a,b,c are positive real numbers and the initial conditions x−2,x−1,x0 are real numbers. [ABSTRACT FROM AUTHOR]

In this paper we consider the following higher-order nonlinear difference equation ..., where k and l are fixed natural numbers, and the parameters α, β, γ, δ and the initial values x-i, ..., are real numbers such that β² + γ² ≠ 0. We solve the above-mentioned equation in closed form and considerably extend some results in the literature. We also determine the asymptotic behavior of solutions and the forbidden set of the initial values using the obtained formulae for the case l = 1. [ABSTRACT FROM AUTHOR]

In this paper, we find additional conditions to be satisfied by a globally periodic discrete dynamical system, so that its good set (the set of initial conditions providing well-defined solutions) is an open set of k or k. We will pay especial attention to the rational case and several examples will be given. [ABSTRACT FROM AUTHOR]

In this paper, we solve and study the global behavior of the admissible solutions of the difference equation where a, b > 0 and the initial values x−2, x−1, x0 are real numbers. We study also the oscillation of the admissible solutions of the aforementioned difference equation and give some illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the global behavior of all solutions of the difference equation xn+1 = xnxn-1/ axn + bxn-1 ; n ∈ N0 where a; b are real numbers and the initial conditions x-1, x0 are real numbers. We determine the forbidden set and give an explicit formula for the solutions. We show the existence of periodic solutions, under certain conditions. [ABSTRACT FROM AUTHOR]

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In recent literature there are an increasing number of papers where the forbidden sets (FSs) of difference equations are computed. We review and complete different attempts to describe the FS and propose new perspectives for further research and a list of open problems in this field. [ABSTRACT FROM AUTHOR]

Multimedia broadcasting is a popular application in an ad hoc wireless network, itself composed of battery-operated nodes. Hence, energy conservation and avoidance of frequent re-construction of broadcast paths are crucial to ensure robust and uninterrupted service of multimedia broadcasting applications. This paper introduces a class of distributed broadcast algorithms based on variations of Relative Neighborhood Graphs (RNG). In contrast to the original RNG-based algorithms, the proposed algorithms consider the remaining battery energy of nodes and the distance between nodes as criteria for determining the relative neighborhood of a node. This approach is intended to boost the resiliency of the broadcast path by avoiding the choice of nodes with low remaining battery capacity as rebroadcast nodes. Extensive simulations are conducted, demonstrating that the proposed algorithms improve over the original RNG in several aspects, including the reduction of broadcast storms, longer path lifetime, and shorter broadcast latency. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigate the dynamics of the rational difference equation xn+1 =xn-1/A + Bxnxn-1 with arbitrary initial conditions A and B as nonzero real numbers. The solution is obtained and analytical study and asymptotic behavior are investigated. The forbidden set is determined. The existence of periodic and oscillatory solutions are discussed. Our results are illustrated with numerical simulations. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigate the dynamics of the rational difference equation xn+1 =xn-1/A + Bxnxn-1 with arbitrary initial conditions A and B as nonzero real numbers. The solution is obtained and analytical study and asymptotic behavior are investigated. The forbidden set is determined. The existence of periodic and oscillatory solutions are discussed. Our results are illustrated with numerical simulations. [ABSTRACT FROM AUTHOR]

This paper deals with a system of rational difference equations x n + 1 = a y n + b c y n + d , y n + 1 = a x n + b c x n + d , n = 0 , 1 , 2 , … , where a , b , c , d are real numbers with c ≠ 0 and a d − b c ≠ 0 . We establish a representation formula of solutions of the system and classify global behavior of solutions when no initial values belong to the forbidden set of the system. [ABSTRACT FROM AUTHOR]

In this paper, we solve the difference equation ... where a and b are positive real numbers and the initial values x-2, x-1 and x0 are real numbers. We find invariant sets and discuss the global behavior of the solutions of that equation. We show that when a > 4/27 b³, under certain conditions there exist solutions, either periodic or converge to periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, we derive the forbidden set and determine the solutions of the difference equation that contains a quadratic term \begin{equation*} x_{n+1}=\frac{x_{n}x_{n-p}}{ax_{n-(p-1)}+bx_{n-p}},\quad n\in\mathbb{N}_0, \end{equation*} where the parameters $a$ and $b$ are real numbers, $p$ is a positive integer and the initial conditions $x_{-p}$, $x_{-p+1}$, $\cdots$, $x_{-1}$, $x_{0}$ are real numbers.

In this study, we show that the system of difference equations xn = xn-2yn-3 yn-1 (a + bxn-2yn-3), yn = yn-2zn-3 zn-1 (c + dyn-2zn-3), n = N0, zn = zn-2xn-3 xn-1 (e + fzn-2xn-3), where the initial values x-i, y-i, z-i, i = 1, 3 and the parameters a, b, c, d, e, f are non-zero real numbers, can be solved in closed form. Moreover, we obtain the solutions of above system in explicit form according to the parameters a, c and e are equal 1 or not equal 1. In addition, we get periodic solutions of aforementioned system. Finally, we define the forbidden set of the initial conditions by using the acquired formulas. [ABSTRACT FROM AUTHOR]

In this paper, we solve and study the global behavior of the well defined solutions of the difference equation $$x_{n+1}=\frac{x_{n}x_{n-3}}{Ax_{n-2}+Bx_{n-3}}, \quad n=0,1,...,$$ where $A, B>0$ and the initial values $x_{-i}$, $i\in\{0,1,2,3\}$ are real numbers.

In this paper, we solve and study the global behavior of all admissible solutions of the two difference equations $$x_{n+1}=\frac{x_{n}x_{n-2}}{x_{n-1}-x_{n-2}}, \quad n=0,1,...,$$ and $$x_{n+1}=\frac{x_{n}x_{n-2}}{-x_{n-1}+x_{n-2}}, \quad n=0,1,...,$$ where the initial values $x_{-2}$, $x_{-1}$, $x_{0}$ are real numbers.\\ We show that every admissible solution for the first equation converges to zero. For the other equation, we show that every admissible solution is periodic with prime period six. Finally we give some illustrative examples.

We determine the forbidden set, introduce an explicit formula for the solutions, and discuss the global behavior of solutions of a fourth order difference equation. [ABSTRACT FROM AUTHOR]

We study the asymptotic behaviour and stability properties of solutions of the second-order Riccati difference equation for all values of parameters and any initial conditions . [ABSTRACT FROM PUBLISHER]

The second order rational difference equation[image omitted] is associated with a linear third order difference equation in the same way that the first order Riccati equation (c = 0) is associated with a linear second order equation. This association and other features are used to study the global behaviour of solutions. If [image omitted] and [image omitted] then the above equation has a unique positive fixed point that is stable and attracts all orbits with initial points outside a set M of Lebesgue measure zero in the plane. However, within M there is an invariant subset containing periodic orbits of all possible periods. [ABSTRACT FROM AUTHOR]

The problem of existence of solutions for difference equations is discussed here. Especially, we give sufficient conditions in order to obtain that the solution of an equation is well defined for almost every initial condition (except initial conditions in a set of Lebesgue measure zero). The obtained results will be generalized to discrete dynamical systems and several examples will be given. [ABSTRACT FROM AUTHOR]