The nonlinear difference equation (E) xn+1 − xn = anϕn(xσ(n) ) + bn, where (an), (bn) are real sequences, ϕn : −→ , (σ(n)) is a sequence of integers and lim n−→∞ σ(n) = ∞, is investigated. Sufficient conditions for the existence of solutions of this equation asymptotically equivalent to the solutions of the equation yn+1 − yn = bn are given. Sufficient conditions under which for every real constant there exists a solution of equation (E) convergent to this constant are also obtained.
Asymptotic properties of solutions of the difference equation of the form ∆ mxn = anϕ(xτ1(n) , . . . , xτk(n) ) + bn are studied. Conditions under which every (every bounded) solution of the equation ∆myn = bn is asymptotically equivalent to some solution of the above equation are obtained.
In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation xn+1 = axnxn−1 ⁄−bxn + cxn−2 , n ∈ N0 where a, b, c are positive real numbers and the initial conditions x−2, x−1, x0 are real numbers. We show that every admissible solution of that equation converges to zero if either a < c or a > c with (a − c)/b < 1. When a > c with (a − c)/b > 1, we prove that every admissible solution is unbounded. Finally, when a = c, we prove that every admissible solution converges to zero.
We study the solutions and attractivity of the difference equation xn+1 = xn−3/(−1 + xnxn−1xn−2xn−3) for n = 0, 1, 2, . . . where x−3, x−2, x−1 and x0 are real numbers such that x0x−1x−2x−3 ≠ 1.
In this paper we investigate the global convergence result, boundedness and periodicity of solutions of the recursive sequence xn+1 = a0xn + a1xn−1 + . . . + akxn−k ⁄ b0xn + b1xn−1 + . . . + bkxn−k , n = 0, 1, . . . where the parameters ai and bi for i = 0, 1, . . . , k are positive real numbers and the initial conditions x−k, x−k+1, . . . , x0 are arbitrary positive numbers.
The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability oThe main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation xn+1 = α0xn + α1xn−l + α2xn−k ⁄ β0xn + β1xn−l + β2xn−k , n = 0, 1, 2, . . . where the coefficients αi , βi ∈ (0,∞) for i = 0, 1, 2, and l, k are positive integers. The initial conditions x−k, . . . , x−l , . . . , x−1, x0 are arbitrary positive real numbers such that l < k. Some numerical experiments are presented.
Some new oscillation and nonoscillation criteria for the second order neutral delay difference equation \[ \Delta (c_n\Delta (y_n+p_ny_{n-k}))+q_ny_{n+1-m}^\beta =0,\quad n\ge n_0 \] where $k$, $m$ are positive integers and $\beta $ is a ratio of odd positive integers are established, under the condition $\sum _{n=n_0}^{\infty }\frac{1}{c_n}<{\infty }.$.
Consider the difference equation ∆x(n) +∑m i=1 pi(n)x(τi(n)) = 0, n ≥ 0 [ ∇x(n) − ∑m i=1 pi(n)x(σi(n)) = 0, n ≥ 1 ] , where (pi(n)), 1 6 i 6 m are sequences of nonnegative real numbers, τi(n) [σi(n)], 1 6 i 6 m are general retarded (advanced) arguments and ∆ [∇] denotes the forward (backward) difference operator ∆x(n) = x(n + 1) − x(n) [∇x(n) = x(n) − x(n − 1)]. New oscillation criteria are established when the well-known oscillation conditions lim sup n→∞ ∑m i=1 ∑n j=τ(n) pi(j) > 1 [ lim sup n→∞ ∑m i=1 σ∑ (n) j=n pi(j) > 1 ] and lim inf n→∞ ∑m i=1 n∑−1 j=τi(n) pi(j) > 1⁄e [ lim inf n→∞ ∑m i=1 σ∑i(n) j=n+1 pi(j) > 1⁄e ] are not satisfied. Here τ (n) = max 1≤i≤m τi(n) [σ(n) = min 1≤i≤m σi(n)]. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.
In this paper the three-dimensional nonlinear difference system ∆xn = anf(yn−l ), ∆yn = bng(zn−m), ∆zn = δcnh(xn−k), is investigated. Sufficient conditions under which the system is oscillatory or almost oscillatory are presented.
The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type ∆x(n) + ∑ q k=−p ak(n)x(n + k) = 0, n > n0, where ∆x(n) = x(n + 1) − x(n) is the difference operator and {ak(n)} are sequences of real numbers for k = −p, . . . , q, and p > 0, q > 0. We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced.