Research Article On the Difference Equation xn+1=j=0kajfj(xn-j)
نویسنده
چکیده
Recently, there has been a great interest in studying the behavior of rational and nonlinear difference equations; see, for example, [1–20]. One of the most intriguing properties of solutions of difference equations is their boundedness character. There are numerous papers devoted, among others, to this research area, see; for example, [1–6, 9–19], and related references therein. It is said that a function f is decreasing on an interval J if for all x, y ∈ J such that x < y, f (x) > f (y). Consider the nonlinear higher-order difference equation of the form
منابع مشابه
Research Article On Global Periodicity of a Class of Difference Equations
We show that the difference equation xn = f3(xn−1) f2(xn−2) f1(xn−3), n ∈ N0, where fi ∈ C[(0,∞),(0,∞)], i ∈ {1,2,3}, is periodic with period 4 if and only if fi(x) = ci/x for some positive constants ci, i ∈ {1,2,3} or if fi(x) = ci/x when i = 2 and fi(x) = cix if i ∈ {1,3}, with c1c2c3 = 1. Also, we prove that the difference equation xn = f4(xn−1) f3(xn−2) f2(xn−3) f1(xn−4), n ∈ N0, where fi ∈...
متن کاملThe difference equation xn + 1 = α + xn − k ∑ k − 1 i = 0 cixn − i has solutions converging to zero
The aim of this note is to show that the following difference equation: xn+1 = α+ xn−k ∑k−1 i=0 cixn−i , n= 0,1, . . . , where k ∈ N, ci 0, i = 0, . . . , k − 1, ∑k−1 i=0 ci = 1, and α < −1, has solutions which monotonically converge to zero. This result shows the existence of such solutions which was not shown in the recently accepted paper: A.E. Hamza, On the recursive sequence xn+1 = α+ xn−1...
متن کاملSolvability of Nonlinear Difference Equations of Fourth Order
In this article we show the existence of solutions to the nonlinear difference equation xn = xn−3xn−4 xn−1(an + bnxn−2xn−3xn−4) , n ∈ N0, where the sequences (an)n∈N0 and (bn)n∈N0 , and initial the values x−j , j = 1, 4, are real numbers. Also we find the set of initial values for which solutions are undefinable when an 6= 0 and bn 6= 0 for every n ∈ N0. When these two sequences are constant, w...
متن کاملOn the Rational Recursive Sequence xn+1=(α-βxn)/(γ-δxn-xn-k)
We study the global stability, the periodic character, and the boundedness character of the positive solutions of the difference equation xn 1 α − βxn / γ − δxn − xn−k , n 0, 1, 2, . . . , k ∈ {1, 2, . . .}, in the two cases: i δ ≥ 0, α > 0, γ > β > 0; ii δ ≥ 0, α 0, γ, β > 0, where the coefficients α, β, γ, and δ, and the initial conditions x−k, x−k 1, . . . , x−1, x0 are real numbers. We show...
متن کاملGlobal Behavior of a Higher-order Rational Difference Equation
We investigate in this paper the global behavior of the following difference equation: xn+1 = (Pk(xn i0 ,xn i1 , . . . ,xn i2k ) + b)/(Qk(xn i0 ,xn i1 , . . . ,xn i2k ) + b), n = 0,1, . . ., under appropriate assumptions, where b [0, ), k 1, i0, i1, . . . , i2k 0,1, . . . with i0 < i1 < < i2k, the initial conditions xi 2k ,xi 2k+1, . . . ,x0 (0, ). We prove that unique equilibrium x = 1 of that...
متن کامل