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Lyapunov inequalities and stability for discrete linear Hamiltonian systems. (English) Zbl 1260.39025
This paper is concerned with the discrete Hamiltonian system $$\alignat \Delta x(n)&=\alpha (n)x(n+1)+\beta(n)y(n),\\ \Delta y(n)&=-\gamma(n)x(n+1)-\alpha(n)y(n),\endalignat $$ where $\alpha(n), \beta(n)$ and $\gamma(n)$ are real-valued functions defined on ${\Bbb Z}$, the set of all integers, and $\Delta$ denotes the forward difference operator defined by $\Delta x(n)=x(n+1)-x(n)$. The authors established several new Lyapunov-type inequalities for the above discrete linear Hamiltonian system when the end-points are not necessarily usual zeros, but rather, generalized zeros. These inequalities generalize and improve almost all related existing ones. By using these inequalities, an optimal stability criterion for discrete periodic linear Hamiltonian system is obtained.

39A30Stability theory (difference equations)
37J05Relations of dynamical systems with symplectic geometry and topology
39A12Discrete version of topics in analysis
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