Abstract
In this paper, we investigate some qualitative behavior of the solutions of the difference equation where the coefficients a, b and ci are positive real numbers, and where the initial conditions are arbitrary positive real numbers.
Highlights
Our aim in this paper is to study with some properties of the solutions of the difference equation xn+1 = a + kbxn−k
There is a class of nonlinear difference equations, known as the rational difference equations, each of which consists of the ratio of two polynomials in the sequence terms in the same form
There has been a lot of work concerning the global asymptotics of solutions of rational difference equations [1]-[8]
Summary
We will investigate the global behavior of (1.1) including the asymptotical stability of equilibrium points, the existence of bounded solution, the existence of period two solution of the recursive sequence of Equation (1). 1) An equilibrium point y of Equation (1.2) is called locally stable if for every ε > 0 there exists δ > 0 such that, if y−k , y−k+1, , y0 ∈ (0, ∞) with y−k − y + y−k+1 − y + + y0 − y < δ , yn − y < ε for all n ≥ −k . 5) An equilibrium point y of Equation (1.2) is called unstable if y is not locally stable. The following statements are true: 1) If all roots of Equation (1.4) lie in the open unit disk λ < 1, he equilibrium point y is locally asymptotically stable.
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