Oscillation and nonoscillation in nonlinear third order difference equations.
Some new oscillation and nonoscillation criteria for the second order neutral delay difference equation where , are positive integers and is a ratio of odd positive integers are established, under the condition
Consider the difference equation where , are sequences of nonnegative real numbers, [], are general retarded (advanced) arguments and [] denotes the forward (backward) difference operator []. New oscillation criteria are established when the well-known oscillation conditions and are not satisfied. Here
2000 Mathematics Subject Classification: 39A10.The oscillatory and nonoscillatory behaviour of solutions of the second order quasi linear neutral delay difference equation Δ(an | Δ(xn+pnxn-τ)|α-1 Δ(xn+pnxn-τ) + qnf(xn-σ)g(Δxn) = 0 where n ∈ N(n0), α > 0, τ, σ are fixed non negative integers, {an}, {pn}, {qn} are real sequences and f and g real valued continuous functions are studied. Our results generalize and improve some known results of neutral delay difference equations.
Some new criteria for the oscillation of difference equations of the form and are established.
One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor's Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality...
Necessary and sufficient conditions are obtained for every solution of to oscillate or tend to zero as , where , and are sequences of real numbers such that . Different ranges for are considered.