Nonoscillatory half-linear difference equations and recessive solutions.

Cecchi, Mariella; Došlá, Zuzana; Marini, Mauro

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Applications of Langenhop inequality to difference equations: Lower bounds and oscillation.

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Asymptotic properties for half-linear difference equations

Mariella Cecchi, Zuzana Došlá, Mauro Marini, Ivo Vrkoč (2006)

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Asymptotic properties of the half-linear difference equation $Δ (a_{n} | Δ x_{n} |^{α} s g n Δ x_{n}) = b_{n} {| x_{n + 1} |}^{α} s g n x_{n + 1} (*)$ are investigated by means of some summation criteria. Recessive solutions and the Riccati difference equation associated to $(*)$ are considered too. Our approach is based on a classification of solutions of $(*)$ and on some summation inequalities for double series, which can be used also in other different contexts.

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Saker, S.H., Alzabut, J.O., Mukheimer, A. (2010)

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Oscillation of third-order half-linear neutral difference equations

Ethiraju Thandapani, S. Selvarangam (2013)

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Some new criteria for the oscillation of third order nonlinear neutral difference equations of the form $Δ (a_{n} {(Δ^{2} (x_{n} + b_{n} x_{n - δ}))}^{α}) + q_{n} x_{n + 1 - τ}^{α} = 0$ and $Δ (a_{n} {(Δ^{2} (x_{n} - b_{n} x_{n - δ}))}^{α}) + q_{n} x_{n + 1 - τ}^{α} = 0$ are established. Some examples are presented to illustrate the main results.

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