Criteria for C-Disfocality of a Self-Adjoint Vector Difference Equation

We are concerned with the vector difference equation Δ[P(t − 1) Δy(t − 1)] + Q(t) y(t) = 0, t ∈ [a + 1, h + 1], where the n × 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 Δy(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. Our results will imply several comparison theorems. Further, a necessary and sufficient condition in terms of the coefficients of the difference equation for C-disfocality will be given.