On the Behaviour of the Solutions of a Second-Order Difference Equation

Abstract

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.

General form of some rational recursive sequences

“…Some other related results can be found in [19][20][21][22][23][24][25][26][27][28]. In this note, by employing the transformation method suggested by Berenhaut and Stević and following the lines in their paper [13] (see also [7]) we give a new proof for the conjectures therein.…”

Semistability of two systems of difference equations using centre manifold theory

“…x nC1 D ax n C by n 1 e xn , y nC1 D cy n C dx n 1 e yn , and x nC1 D ay n C bx n 1 e yn , y nC1 D cx n C dy n 1 e xn respectively, where a, b, c, d are positive constants and the initial values x 1 , x 0 , y 1 , y 0 are also positive numbers. For the asymptotic behaviour of positive solutions of some scalar equations closely related to the previous systems, see [7], where the most interesting case when the sum of the coefficients is equal to one was considered (it corresponds to the case a C b D 1 here), by employing asymptotic methods (see, also [8] and [9] for the applications of the methods). Some of the equations related to the one in [4] appeared in mathematical biology (e.g.…”

“…For the asymptotic behaviour of positive solutions of some scalar equations closely related to the previous systems, see , where the most interesting case when the sum of the coefficients is equal to one was considered (it corresponds to the case a + b = 1 here), by employing asymptotic methods (see, also and for the applications of the methods). Some of the equations related to the one in appeared in mathematical biology (e.g.…”

“…Finally, we note that because difference equations have several applications in applied sciences, there exists a rich bibliography concerning theory and applications (, ).…”

Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in Hydrodynamics

“…We would like to point out that recently there has been a huge interest in studying the existence of monotonous and nontrivial solutions of nonlinear difference equations. For papers during last three years see, for example, [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] . A lot of other interesting references can be found therein.…”