Search

The present paper deals with Hyers–Ulam stability of the first-order linear difference equation Δ h x ( t ) − a x ( t ) = f ( t ) on h Z , where Δ h x ( t ) = ( x ( t + h ) − x ( t ) ) / h and h Z = { h k | k ∈ Z } for the constant stepsize h  > 0; a is a real number; f ( t ) is a real-valued function on h Z . The main purpose of this paper is to find the best HUS constant on h Z . Several relationships between solutions of two different perturbed difference equations are also given. [ABSTRACT FROM AUTHOR]

We show that the following system of difference equations x n = x n − 1 y n − 2 a y n − 2 + b y n − 1 , y n = y n − 1 x n − 2 c x n − 2 + d x n − 1 , n ∈ N 0 , where parameters a, b, c, d , as well as initial values x − 2 , x − 1 , y − 2 , y − 1 , are real numbers, is solvable in closed form and by using obtained formulas we give some results on the long term behavior of their solutions. We also give natural explanations and considerably extend some recent results in the literature. [ABSTRACT FROM AUTHOR]

Abstract: Let k be a positive integer and be non-negative real numbers with . We show that if then the rational Riccati difference equation of order k has a unique positive equilibrium point that is stable and attracts all solutions with initial points outside a set of zero Lebesgue measure. This holds in particular if . The case is studied in detail. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the boundedness character, the periodic character and the global behavior of positive solutions of the difference equation where and is a two periodic sequence of nonnegative real numbers and the initial conditions are arbitrary positive real numbers. [Copyright &y& Elsevier]

Abstract: We show that the system of difference equationswhere the sequences , , , , , , , and initial values , , are real numbers, such that , , , can be solved in closed form, and for the case when all the sequences , , , , and are constant we describe the asymptotic behavior of well-defined solutions of the system. [Copyright &y& Elsevier]

Abstract: We show that the system of difference equationswhere , the sequences , , and initial values are real numbers can be solved in closed form. By using obtained formulae we investigate asymptotic behavior of well-defined solutions of the system for the case when all the sequences , and are constant. [Copyright &y& Elsevier]

Abstract: We show that the system of difference equationswhere the parameters , and initial values , are real numbers, can be solved, developing further the results in the literature. [Copyright &y& Elsevier]

Abstract: We prove that the difference equationwhere and are sequences periodic with period two and the initial values x −2, x −1 are real numbers, can be solved explicitly. Some applications of obtained formulae are given. [Copyright &y& Elsevier]

Abstract: We show that the system of difference equationswhere the sequences a n , b n , c n , α n , β n , γ n are two-periodic and initial values x −1, x 0, y −1, y 0 are real numbers, can be solved. [Copyright &y& Elsevier]

Abstract: We show that the system of difference equationswhere the parameters a, b, c, α, β, γ and initial values x −1, x 0, y −1, y 0 are real numbers, can be solved, considerably improving the results in the literature. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the behaviour of the positive solutions of the system of rational difference equationwhere q >3 is a positive integer with 3∤ q, a and b are positive constants and the initial values x −q+1, x −q+2,…, x 0, y −q+1, y −q+2,…, y 0 are positive real numbers. [Copyright &y& Elsevier]

Abstract: We study the global stability, the boundedness character and the periodic nature of the positive solutions of the difference equation yn+1=α+yn-1β+yn+yn-1yn,n=0,1,2,…, where α, β∈(0,∞), α≠β, β≠2 and the initial conditions y−1, y0 are arbitrary positive real numbers. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the boundedness, periodic character, invariant intervals and the global asymptotic stability of all nonnegative solutions of the difference equationwhere a, b, A, B are positive real numbers, k ⩾1 is a positive integer, and the initial conditions x −k ,…, x −1, x 0 are nonnegative real numbers. It is shown that the zero equilibrium of this equation is globally asymptotically stable if b ⩽ A − a and the unique positive equilibrium is globally asymptotically stable if A − a < b < A + a. The results obtained solve a open problem proposed by Kulenovic and Ladas [M.R.S. Kulenovi, G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman & Hall/CRC, Boca Raton, FL, 2002, p. 129]. [Copyright &y& Elsevier]

Abstract: In this paper, we study the global behavior of the higher order nonlinear difference equationwhere A, B and α, β, γ are positive, k ∈{1,2,3,…}, and the initial conditions x −2k+1,…, x −1, x 0 are positive real numbers. We investigate the character of semicycles, periodicity, global stability and boundedness of above mentioned equation. [Copyright &y& Elsevier]

Abstract: Our aim in this paper is to investigate the global attractivity of nonnegative solution of the difference equationwhere α >0, β ⩾0 are real numbers, the initial conditions x −1, x 0 are nonnegative real numbers and with β and x −1 not both zero. We show that the positive equilibrium point of the equation is a global attractor. [Copyright &y& Elsevier]

Abstract: In this paper, we study the behavior of the positive solutions of the system of two difference equationswhere p ⩾1, r ⩾1, s ⩾1, A ⩾0, and x 1−r , x 2−r ,…, x 0, y 1−max{p,s}, y 2−max{p,s},…, y 0 are positive real numbers. [Copyright &y& Elsevier]

Abstract: In this paper the global asymptotic stability of the nonlinear difference equationis investigated, where α, β, A, B, C >0 are real numbers, and the initial conditions x −1 is nonnegative real numbers and x 0 is a positive real number. We show that the unique positive equilibrium of the equation is globally asymptotically stable. In particular, our results solve two conjectures proposed by Kulenovic and Ladas in their monograph [M.R.S. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002] and by Kulenovic et al. in their paper [M.R.S. Kulenovic, G. Ladas, L.F. Martins, I.W. Rodrigues, The dynamics of facts and conjectures, Comput. Math. Appl. 45 (2003) 1087–1099]. [Copyright &y& Elsevier]

We investigate the solutions of the difference equation xn+1=xn-1/-1+axnxn-1, n=0,1,2,…, where a>0, x-1 and x0 are real numbers such that ax-1x0≠1. [Copyright &y& Elsevier]

The positive solutions of the difference equation xn+1=xn-1/1+axnxn-1, n=0,1,2,…, are investigated, where x-1, x0 and a are the nonnegative real numbers. [Copyright &y& Elsevier]

In this paper, we investigate the solutions of the difference equation xn+1=xn-1/-1+xnxn-1, n=0,1,2,…, where x-1 and x0 are the real numbers such that x-1x0≠1. [Copyright &y& Elsevier]

We study the positive solutions of the difference equation xn+1=axn-1/1+bxnxn-1, n=0,1,2,… where a, b, x-1 and x0 are non-negative real numbers. [Copyright &y& Elsevier]

The positive solutions of the difference equation xn+1=(xn−1)/(1+xnxn−1), n=0,1,2,…, are investigated, where x−1 and x0 are the positive real numbers. [Copyright &y& Elsevier]

Our aim in this paper is to investigate the global attractivity of all positive solutions of the higher order nonlinear difference equationxn+1=a−bxn/A−∑ki=0bixn−i, n=0,1,…,where A,b,bk∈(0,∞), a,b0,…,bk−1∈[0,∞), k∈{1,2,…} and the initial conditions x−k,…,x0 are arbitrary real numbers. We show that one positive equilibrium of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients. [Copyright &y& Elsevier]