On the Global Stability of One Nonlinear Difference Equation

Abstract

We give exact sufficient conditions for the global stability of the zero solution of the difference equation xn+1 = qxn + fn(xn, . . . , x n−k ), n ∈ Z, where the nonlinear functions fn satisfy the conditions of negative feedback and sublinear growth.

A global attractivity criterion for nonlinear non-autonomous difference equations

“…This liberty in moving y j is allowed thanks to the non-constancy of a n . If a n ≡ a is constant, we should obtain a stronger stability result: for instance, the note [8] suggests the following condition improving (4):…”

“…Let now {x n } n −k be a solution to (1). Consider the initial value problem y(s) = ψ(s), s ∈ [−k − 1, 0] for (8) with continuous ψ such that ψ(j ) = x j for j = −k, . .…”

On the global stability of one nonlinear difference equation

“…Note that, in [8], sufficient conditions for the global stability of the trivial solution of the difference equation Note that, in [8], sufficient conditions for the global stability of the trivial solution of the difference equation…”

Global stability for nonlinear difference equations with variable coefficients

“…The above difference equation has been widely studied in the literature (see, for example, [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and the references therein).…”

“…, z m ), with min 0 i m z i s. (H2) There is a rational function r(x) = −x/(1 + bx) with b > 0 such that r M(z) f n (z) r −M(−z) , n∈ Z + , (1.5) where the first inequality holds for all z ∈ R m+1 , and the second one for all z ∈ R m+1 such that min 0 i m z i > −b −1 ∈ (−∞, 0). To obtain sharp stability conditions, Tkachenko and Trofimchuk [11] restricted the range of parameter q, and for the sublinear case (b = 0), Nenya, Tkachenko and Trofimchuk [9] extended in the following range such that q + q 2 + · · · + q m q m+1 1.…”