A disconjugacy criterion of W. T. Reid for difference equations

Abstract

Our main result is a disconjugacy criterion for the selfadjoint vector difference equation L y ( t ) ≡ Δ [ P ( t − 1 ) Δ y ( t − 1 ) ] + Q ( t ) y ( t ) = 0 Ly(t) \equiv \Delta [P(t - 1)\Delta y(t - 1)] + Q(t)y(t) = 0 . This result is the analogue of a famous result of W. T. Reid for the corresponding differential equations case. Unlike the differential equations case we will see there is an exceptional case in which, as we will show by counterexample, the conclusion of the main result is no longer valid. A disfocality criterion is also given. We believe these results are new even in the scalar case.