 
Summary: Oscillation of secondorder forced functional
dynamic equations with oscillatory potentials
DOUGLAS R. ANDERSON*
Department of Mathematics and Computer Science, Concordia College, Moorhead, MN 56562, USA
(Received 27 July 2006; revised 12 November 2006; in final form 12 November 2006)
Oscillation criteria are established for a secondorder forced dynamic equation on time scales containing
both delay and advance arguments. Moreover, the potentials are allowed to change sign. Several
nontrivial examples from difference equations are provided to illustrate the easy application of the results.
The theory can be applied to secondorder dynamic equations regardless of the choice of delta or nabla
derivatives.
Keywords: Time scales; Oscillation; Riccati substitution; Second order
2000 Mathematics Subject Classification: 34K11; 34C10; 39A11; 39A13
1. Functional dynamic equation with forcing term
Following Hilger's landmark paper [11], a rapidly expanding body of literature has sought to
unify, extend and generalize ideas from discrete calculus, quantum calculus and continuous
calculus to arbitrary timescale calculus, where a time scale is simply any nonempty closed
set of real numbers. This paper illustrates this new understanding by extending some
continuous results from differential equations to dynamic equations on arbitrary time scales
(unbounded above), thus including as corollaries difference equations and qdifference
equations. Throughout this work we consider the secondorder forced nonlinear dynamic
