In [1] it is shown that every second-order linear difference equation with, in general, variable coefficients can be reduced to the formIt is also shown that (1) can be solved explicitly provided that a particular solution a (n) of the ‘auxiliary equation’can be found, and many cases in which a solution of (2) is available are discussed. However, [1] is defective in that no method of finding a solution of (2) in the general case is given.
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
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k
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1
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p
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2
Δ
u
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k
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1
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+
a
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k
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u
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k
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p
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2
u
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k
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=
0
with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.
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.
In this paper we will present the Julia set and the global behavior of a
quadratic second order difference equation of type xn+1 = axnxn-1 + ax2n-1
+ bxn-1 where a > 0 and 0 ? b < 1 with non-negative initial conditions.