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April 2020 Oscillatory behavior of solutions of dynamic equations of higher order on time scales
Taher S. Hassan, Donal O’Regan
Rocky Mountain J. Math. 50(2): 599-617 (April 2020). DOI: 10.1216/rmj.2020.50.599

Abstract

We study the n -th-order nonlinear dynamic equations

x [ n ] ( t ) + p ( t ) ϕ α n 1 [ ( x [ n 2 ] ( t ) ) Δ σ ] + q ( t ) ϕ γ ( x ( g ( t ) ) ) = 0

on an unbounded time scale 𝕋 , where n 2 and for i = 1 , , n 1

x [ i ] ( t ) : = r i ( t ) ϕ α i [ ( x [ i 1 ] ( t ) ) Δ ] ,

with r n = α n = 1 and x [ 0 ] = x ; here the constants α i and the functions r i , i = 1 , , n 1 , are positive and p , q are nonnegative functions. Criteria are established for the oscillation of solutions for both even- and odd-order cases. The results improve several known results in the literature on second-order, third-order, and higher-order linear and nonlinear dynamic equations. In particular our results can be applied when g is not (delta) differentiable and the forward jump operator σ and g do not commute.

Citation

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Taher S. Hassan. Donal O’Regan. "Oscillatory behavior of solutions of dynamic equations of higher order on time scales." Rocky Mountain J. Math. 50 (2) 599 - 617, April 2020. https://doi.org/10.1216/rmj.2020.50.599

Information

Received: 22 May 2019; Revised: 20 September 2019; Accepted: 20 September 2019; Published: April 2020
First available in Project Euclid: 29 May 2020

zbMATH: 07210982
MathSciNet: MR4104397
Digital Object Identifier: 10.1216/rmj.2020.50.599

Subjects:
Primary: 34K11, 39A10, 39A99

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.50 • No. 2 • April 2020
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