scholarly journals Positive solutions for a second-order difference equation with summation boundary conditions

2017 ◽  
Vol 41 (2) ◽  
pp. 167-178 ◽  
Author(s):  
F. Bouchelaghem ◽  
A. Ardjouni ◽  
A. Djoudi
Keyword(s):  
Difference Equation ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Second Order ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

scholarly journals Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Difference Equation ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

AbstractIn this work, we establish optimal Lyapunov-type inequalities for the second-order difference equation with p-Laplacian $$\begin{aligned} \Delta (\left| \Delta u(k-1)\right| ^{p-2}\Delta u(k-1))+a(k)\left| u(k)\right| ^{p-2}u(k)=0 \end{aligned}$$ Δ ( Δ u ( k - 1 ) p - 2 Δ u ( k - 1 ) ) + a ( k ) u ( k ) p - 2 u ( k ) = 0 with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.

scholarly journals The Behavior of Positive Solutions of a Nonlinear Second-Order Difference Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
Stevo Stević ◽  
Kenneth S. Berenhaut
Keyword(s):  
Periodic Solution ◽  
Asymptotic Stability ◽  
Difference Equation ◽  
Positive Solutions ◽  
Global Asymptotic Stability ◽  
Second Order ◽  
Order Difference ◽  
All Solutions ◽  
Second Order Difference Equation ◽  
Order Difference Equation

This paper studies the boundedness, global asymptotic stability, and periodicity of positive solutions of the equationxn=f(xn−2)/g(xn−1),n∈ℕ0, wheref,g∈C[(0,∞),(0,∞)]. It is shown that iffandgare nondecreasing, then for every solution of the equation the subsequences{x2n}and{x2n−1}are eventually monotone. For the case whenf(x)=α+βxandgsatisfies the conditionsg(0)=1,gis nondecreasing, andx/g(x)is increasing, we prove that every prime periodic solution of the equation has period equal to one or two. We also investigate the global periodicity of the equation, showing that if all solutions of the equation are periodic with period three, thenf(x)=c1/xandg(x)=c2x, for some positivec1andc2.

scholarly journals Oscillation Criteria of Solution for a Second Order Difference Equation with Forced Term

2010 ◽  
Vol 2010 ◽  
pp. 1-6
Author(s):  
Chen Huiqin ◽  
Jin Zhen
Keyword(s):  
Difference Equation ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Second Order ◽  
Order Difference ◽  
Oscillation Criteria ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Forced Term

We will consider oscillation criteria for the second order difference equation with forced termΔ(anΔ(xn+λxn−τ))+qnxn−σ=rn(n≥0). We establish sufficient conditions which guarantee that every solution is oscillatory or eventually positive solutions converge to zero.

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