On Boundedness of Solutions of the Difference Equation xn+1=(pxn+qxn-1)/(1+xn) for q>1+p>1
نویسندگان
چکیده
We study the boundedness of the difference equation xn 1 pxn qxn−1 / 1 xn , n 0, 1, . . . , where q > 1 p > 1 and the initial values x−1, x0 ∈ 0, ∞ . We show that the solution {xn}n −1 of this equation converges to x q p − 1 if xn ≥ x or xn ≤ x for all n ≥ −1; otherwise {xn}n −1 is unbounded. Besides, we obtain the set of all initial values x−1, x0 ∈ 0, ∞ × 0, ∞ such that the positive solutions {xn}n −1 of this equation are bounded, which answers the open problem 6.10.12 proposed by Kulenović and Ladas 2002 .
منابع مشابه
Global Behavior of the Difference Equation xn+1=(p+xn-1)/(qxn+xn-1)
and Applied Analysis 3 In the sequel, let q > 1 4p and . . . , φ, ψ, φ, ψ, . . . the unique prime period-two solution of 1.1 with φ < ψ. Define f ∈ C φ, ψ × φ, ψ , φ, ψ by f ( x, y ) p y qx y 2.2 for any x, y ∈ φ, ψ and g ∈ C φ, ψ , φ, ψ by y∗ g ( y ) p y − y2 qy 2.3 for any y ∈ φ, ψ . Then
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