Weighted Averaging Techniques in Oscillation Theory for Second Order Difference Equations

1992 ◽  
Vol 35 (1) ◽  
pp. 61-69 ◽  
Author(s):  
Lynn H. Erbe ◽  
Pengxiang Yan
Keyword(s):  
Oscillation Theory ◽  
Second Order ◽  
Weighted Averaging ◽  
Real Numbers ◽  
Order Difference ◽  
The Matrix ◽  
System 2 ◽  
Oscillation And Nonoscillation Criteria ◽  
Oscillation And Nonoscillation ◽  
Averaging Techniques

AbstractWe consider the self-adjoint second-order scalar difference equation (1) Δ(rnΔxn) +pnXn+1 = 0 and the matrix system (2) Δ(RnΔXn) + PnXn+1 = 0, where are seQuences of real numbers (d x d Hermitian matrices) with rn > 0(Rn > 0). The oscillation and nonoscillation criteria for solutions of (1) and (2), obtained in [3, 4, 10], are extended to a much wider class of equations by Riccati and averaging techniques.

scholarly journals Boundedness of solutions and stability of certain second-order difference equation with quadratic term

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Emin Bešo ◽  
Senada Kalabušić ◽  
Naida Mujić ◽  
Esmir Pilav
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Second Order ◽  
Quadratic Term ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Rational Difference Equation

AbstractWe consider the second-order rational difference equation $$ {x_{n+1}=\gamma +\delta \frac{x_{n}}{x^{2}_{n-1}}}, $$xn+1=γ+δxnxn−12, where γ, δ are positive real numbers and the initial conditions $x_{-1}$x−1 and $x_{0}$x0 are positive real numbers. Boundedness along with global attractivity and Neimark–Sacker bifurcation results are established. Furthermore, we give an asymptotic approximation of the invariant curve near the equilibrium point.

Oscillation and Nonoscillation Criteria for a Second Order Linear Equation

1999 ◽  
Vol 6 (5) ◽  
pp. 401-414
Author(s):  
T. Chantladze ◽  
N. Kandelaki ◽  
A. Lomtatidze
Keyword(s):  
Linear Equation ◽  
Integrable Function ◽  
Second Order ◽  
Order Linear ◽  
Locally Integrable Function ◽  
Oscillation And Nonoscillation Criteria ◽  
Oscillation And Nonoscillation

Abstract New oscillation and nonoscillation criteria are established for the equation 𝑢″ + 𝑝(𝑡)𝑢 = 0, where 𝑝 : ]1, + ∞[ → 𝑅 is the locally integrable function. These criteria generalize and complement the well known criteria of E. Hille, Z. Nehari, A. Wintner, and P. Hartman.

Oscillation and nonoscillation of nonlinear second order difference equations

2006 ◽  
Vol 21 (1-2) ◽  
pp. 203-214 ◽  
Author(s):  
M. M. A. El-Sheikh ◽  
M. H. Abd Alla ◽  
E. M. El-Maghrabi
Keyword(s):  
Difference Equations ◽  
Second Order ◽  
Order Difference ◽  
Oscillation And Nonoscillation

scholarly journals Dynamics of a second-order nonlinear difference system with exponents

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
D. S. Dilip ◽  
Smitha Mary Mathew
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Second Order ◽  
Difference System ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference

AbstractIn this paper, we study the persistence, boundedness, convergence, invariance and global asymptotic behavior of the positive solutions of the second-order difference system $$\begin{aligned} x_{n+1}&= \alpha _1 + a e ^{-x_{n-1}} + b y_{n} e ^{-y_{n-1}},\\ y_{n+1}&= \alpha _2 +c e ^{-y_{n-1}}+ d x_{n} e ^{-x_{n-1}} \quad n=0,1,2,\ldots \end{aligned}$$ x n + 1 = α 1 + a e - x n - 1 + b y n e - y n - 1 , y n + 1 = α 2 + c e - y n - 1 + d x n e - x n - 1 n = 0 , 1 , 2 , … where $$\alpha _1, \alpha _2, a, b , c,d$$ α 1 , α 2 , a , b , c , d are positive real numbers and the initial conditions $$x_{-1},x_0, y_{-1}, y_0$$ x - 1 , x 0 , y - 1 , y 0 are arbitrary nonnegative numbers.

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