Linear Third-Order Difference Equations: Oscillatory and Asymptotic Behavior
Copyright policy )Linear Third-Order Difference Equations: Oscillatory and Asymptotic Behavior
A point of contact of the graph of \(U = \{U_ n\}\) satisfying (1) \(\Delta^ 3U_ n + P_{n+1}\Delta U_{n+2} + Q_ nU_{n+2} = 0\), with the real axis is a node. A solution of (1) is said to be oscillatory if it has arbitrarily large nodes. It is proved that (1) always has an oscillatory solution. A sufficient condition is also given in terms of \(P_ n\) and \(Q_ n\) so that (1) has a solution satisfying, \(\text{Sgn }U_ n = \text{Sgn }\Delta U_ n = \text{Sgn }\Delta^ 2U_ n\).
oscillatory solution, linear third-order difference equations, asymptotic behavior, graph, point of contact, Additive difference equations
oscillatory solution, linear third-order difference equations, asymptotic behavior, graph, point of contact, Additive difference equations
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