ON BOUNDEDNESS OF THE SOLUTIONS OF THE DIFFERENCE EQUATION xn+1=xn-1/(p+xn)
نویسنده
چکیده
Theorem 1. (i) If p > 1, then the unique equilibrium 0 of (1) is globally asymptotically stable. (ii) If p = 1, then every positive solution of (1) converges to a period-two solution. (iii) If 0 < p < 1, then 0 and x = 1− p are the only equilibrium points of (1), and every positive solution {xn}n=−1 of (1) with (xN − x)(xN+1 − x) < 0 for some N ≥ −1 is unbounded. They proposed the following open problem.
منابع مشابه
On Boundedness of Solutions of the Difference Equation xn+1=(pxn+qxn-1)/(1+xn) for q>1+p>1
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...
متن کاملOn the boundedness of positive solutions of the reciprocal max-type difference equation xn=maxAn-11xn-1, An-12xn-2, ⋯, An-1txn-t with periodic parameters
We investigate the boundedness of positive solutions of the reciprocal max-type difference equation xn = max { An−1 xn−1 , An−1 xn−2 , . . . , An−1 xn−t } , n = 1, 2, . . . , where, for each value of i, the sequence {An}n=0 of positive numbers is periodic with period pi. We give both sufficient conditions on the pi’s for the boundedness of all solutions and sufficient conditions for all solutio...
متن کاملThe Behaviour of the Positive Solutions of the Difference Equation x n = A + ( x n − 2 x n − 1 ) p Kenneth
This paper studies the boundedness, global asymptotic stability and periodicity for solutions of the equation xn = A + ( xn−2 xn−1 )p , n = 0, 1, . . . , with p, A ∈ (0,∞), p 6= 1, and x−2, x−1 ∈ (0,∞). It is shown that: (a) all solutions converge to the unique equilibrium, x̄ = A+1, whenever p ≤ min{1, (A+1)/2}, (b) all solutions converge to period two solutions whenever (A + 1)/2 < p < 1 and (...
متن کاملGlobal asymptotic behavior and boundedness of positive solutions to an odd-order rational difference equation
In this note we consider the following high-order rational difference equation xn = 1+ k ∏ i=1 (1− xn−i ) k ∑ i=1 xn−i , n = 0, 1, . . . , where k ≥ 3 is odd number, x−k , x−k+1, x−k+2, . . . , x−1 is positive numbers. We obtain the boundedness of positive solutions for the above equation, and with the perturbation of initial values, we mainly use the transformation method to prove that the pos...
متن کاملOn the Recursive Sequence
The paper considers the boundedness character of positive solutions of the difference equation xn+1 = A+ x n /x n−1, n ∈ N0, where A, p, and r are positive real numbers. It is shown that (a) If p2 ≥ 4r > 4, or p ≥ 1 + r, r ≤ 1, then this equation has positive unbounded solutions; (b) if p2 < 4r, or 2 √ r ≤ p < 1+ r, r ∈ (0,1), then all positive solutions of the equation are bounded. Also, an an...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2006