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The symmetrical properties of dynamic and/or differential equations are kind of oscillation properties that allow us to conclude the character of solutions for dynamic equations. In this paper, we obtain some symmetrical properties of solutions to an even-order nonlinear dynamic equations with superlinear and sublinear neutral terms on time scales. Our approach is based on linearizing the considered equation in the sense that we would deduce the properties of the considered equation from that of the linear form and provide new oscillation results via comparing with first order as well as n th order non-neutral delay dynamic inequalities. The new obtained results outfit a general podium that enables to analyse the oscillatory behaviour for many types of even-order nonlinear dynamic equations. An example is provided to demonstrate the validity of the theoretical outcomes. [ABSTRACT FROM AUTHOR]

This paper is concerned with the oscillation and asymptotic behavior of solutions of third-order nonlinear neutral differential equations with a middle term and mixed nonlinear neutral terms in the case of the canonical operator. We establish several oscillation criteria that guarantee that all solutions are oscillatory or converge to zero. The given results are obtained by applying the comparison method, the Riccati transformation and the integral averaging technique. The results improve significantly and extend existing ones in the literature. Finally, illustrative examples are given. [ABSTRACT FROM AUTHOR]

The authors study the oscillatory behaviors of solutions of higher-order nonlinear differential equations with a nonlinear neutral term. The right hand side of their equation contains both an advanced and a delay term, and either (or both) of them can be sublinear or superlinear. The influence of these terms on the oscillatory and asymptotic behaviors of solutions is investigated by using a comparison to first-order advanced and delay differential equations. New oscillation criteria are presented that improve and extend many known oscillation criteria in the literature. An example is provided to illustrate the results. [ABSTRACT FROM AUTHOR]

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In the paper, we provide sufficient conditions for the oscillatory and asymptotic behavior of a new type of third-order nonlinear dynamic equations with mixed nonlinear neutral terms. Our theorems not only improve and extend existing theorems in the literature but also provide a new approach as far as the nonlinear neutral terms are concerned. The main results are illustrated by some particular examples.

In the paper, we provide sufficient conditions for the oscillatory and asymptotic behavior of a new type of third-order nonlinear dynamic equations with mixed nonlinear neutral terms. Our theorems not only improve and extend existing theorems in the literature but also provide a new approach as far as the nonlinear neutral terms are concerned. The main results are illustrated by some particular examples. [ABSTRACT FROM AUTHOR]