Criteria for \(C\)-disfocality of a selfadjoint vector difference equation. (English) Zbl 0802.39003

The paper is concerned with the vector difference equation \(\Delta [P(t - 1) \Delta y (t-1)] + Q (t)y (t) = 0\), \(t \in [a+1, b+1]\) where the \(n \times n\) matrices \(P(t)\), \(Q (t)\) are Hermitian and \(P (t) > 0\) on \([a,b+1]\). This difference equation is said to be \(C\)-disfocal on \([a, b+2]\) provided that if \(y\) is a nontrivial solution with \(\Delta y (b+1) = 0\), then \(y\) has no generalized zeros in \([a, b+2]\).
One case of the main result is that the above difference equation is \(C\)- disfocal if and only if a certain quadratic functional is positive definite. Further, a necessary and sufficient condition in terms of the coefficients of the difference equation for \(C\)-disfocality is given.


39A10 Additive difference equations
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