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On oscillatory behaviour of third-order half-linear dynamic equations on time scales

Grace Said R., Chhatria Gokula Nanda

Opuscula Mathematica

2022

Vol. 42, no. 6

849--865

In this work, we study the oscillation and asymptotic behaviour of third-order nonlinear dynamic equations on time scales. The findings are obtained using an integral criterion as well as a comparison theorem with the oscillatory properties of a first-order dynamic equation. As a consequence, we give conditions which guarantee that all solutions to the aforementioned problem are only oscillatory, different from any other result in the literature. We propose novel oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are associated with a numerical example. We point out that the results are new even for the case T = R or T = Z.

On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms

Graef John R., Grace Said R., Tune Ercan

Opuscula Mathematica

2020

Vol. 40, no. 2

227--239

This paper is concerned with the asymptotic behavior of the nonoscillatory solutions of the forced fractional differential equation with positive and negative terms of the form [formula] where t ≥ c ≥ α ∈(0, 1), η ≥ 1 is the ratio of positive odd integers, and [formula] denotes the Caputo fractional derivative of y of order α. The cases [formula] are considered. The approach taken here can be applied to other related fractional differential equations. Examples are provided to illustrate the relevance of the results obtained.

Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term

Graef John R., Grace Said R., Tunc Ercan

Opuscula Mathematica

2019

Vol. 39, no. 1

39--47

The authors present a new technique for the linearization of even-order nonlinear differential equations with a sublinear neutral term. They establish some new oscillation criteria via comparison with higher-order linear delay differential inequalities as well as with first-order linear delay differential equations whose oscillatory characters are known. Examples are provided to illustrate the theorems.