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Relations between limit-point and Dirichlet properties of second-order difference operators. (English) Zbl 1156.39011

The author uses complex value sequences \(x= (x_n)_n\), \(y= (y_n)_n\), \(n\in\{-1\}\cup\mathbb N\), the second-order formally symmetric difference expression
\[ Mx_n= \begin{cases} w^{-1}_n(-\Delta(p_{n-1} \Delta x_{n-1})+ q_nx_n),\quad & n\geq 0,\\ -w^{-1}_{-1} p_{-1}\Delta x_n,\quad & n= -1,\end{cases} \]
where \(\Delta x_n= x_{n+1}- x_n\) and \(p= (p_n)_n\), \(q= (q_n)_n\), \(w= (w_n)_n\) are complex valued sequences such that \(q_{-1}= 0\) and
\[ (\forall n\in \{-1\}\cup\mathbb N)(w_n> 0\wedge p_n\neq 0), \]
the space \(l^1\) of absolutely summable complex sequences and the Hilbert spaces \[ l^2= \Biggl\{(x_n)_n;\;\sum^{+\infty}_{n=-1} |x_n|^2< +\infty\Biggr\},\;l^2_w= \Biggl\{(x_n)_n;\;\sum^{+\infty}_{n=-1} |x_n|^2 w_n<+ \infty\Biggr\} \] with scalar products \((x,y)= \sum^{+\infty}_{n=-1} x_n\overline y_n\) is \(l^2\) and \((x,y)= \sum^{+\infty}_{n=-1} x_n\overline y_n w_n\) in \(l^2_w\). Consider the second-order difference equation
\[ Mx_n=\lambda x_n,\quad n\in\mathbb N,\quad\lambda\in\mathbb C\tag{1} \]
and the function \(T(M): D_{T(M)}\to l^2_w\), where
\[ (\forall n\ni \{-1\}\cup\mathbb N)((T(M)x)_n= T(M) x_n= Mx_n) \]
and
\[ D_{T(M)}= \Biggl\{(x_n)_n\in l^2_w;\;\sum^{+\infty}_{n=-1} |T(M) x_n|^2 w_n< +\infty\Biggr\}. \]
The function \(M\) is said to be in the limit point case if there is one \(l^2_w\)-solution of (1) for \(\text{Im\,}\lambda\neq 0\); otherwise, if all solutions of (1) are in \(l^2_w\) for all \(\lambda\in C\) then \(M\) is said to be in the limit circle.
Moreover, for all \(x,y\) in \(D_{T(M)}\), \(M\) is said:
strong limit-point on \(D_{T(M)}\) if \(\lim_{m\to+\infty} p_m\Delta y_m\overline x_{m+1}= 0\),
conditional Dirichlet on \(D_{T(M)}\) if \((\sqrt{|p_n|}\Delta x_n)_n\in l^2\wedge\sum^{+\infty}_{n= 0} q_n|x_n|^2<+\infty\),
weak Dirichlet on \(D_{T(M)}\) if \(\sum^{+\infty}_{n=0} (p_n\overline{\Delta x}_n\Delta y_n+ q_n\overline x_n y_n)< +\infty\).
The author proves that if
\(p^{-1}\not\in l^1\) or \(p^{-1}\in l^1\) and \(\sum^{+\infty}_{n=0} q_n\) is not convergent then \(M\) is conditional Dirichlet and implies \(M\) strong limit-point on \(D_{T(M)}\),
\(w\in l^1\), \(p^{-1}\in l^1\), \(q\in l^1\) then \(M\) is both Dirichlet and limit-circle on \(D_{T(M)}\),
\((\forall n\in\mathbb{N})(p_n> 0)\), \((w_m\sum^{+\infty}_{n=-1} p^{-1}_n)_m\not\in l^1\) or \((q_n)_n\not\in l^1\) then \(M\) is Dirichlet on \(D_{T(M)}\) if and only if \((\sqrt{|q_n|}x_n)_n\in l^1\),
\((\forall n\in\mathbb{N})(p_n> 0)\), \(wp^{-1}\not\in l^1\) and \((w_n w^{-1}_{n+1})_n\) is bounded above then \(M\) is strong limit-point on \(D_{T(M)}\) if and only if \(M\) is weak Dirichlet on \(D_{T(M)}\).
Reviewer: D. M. Bors (Iaşi)

MSC:

39A70 Difference operators
39A12 Discrete version of topics in analysis
46B45 Banach sequence spaces
47B39 Linear difference operators
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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References:

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