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The oscillation property of third-order differential equations with non-positive neutral coefficients is discussed. New sufficient conditions are provided to guarantee that every solution of the considered equation is almost oscillatory. Both the canonical and non-canonical cases are considered. Illustrative examples are introduced to support the obtained results. [ABSTRACT FROM AUTHOR]

This paper introduced novel multidimensional Hardy-type inequalities with general kernels on time scales, extending existing results in the literature. We established generalized inequalities involving a general Hardy operator with multiple variables and kernels on arbitrary time scales. Our findings not only encompassed known results in the realm of real numbers (T = R), but also provided refinements and generalizations thereof. The proposed inequalities offered versatile applications in mathematical analysis and beyond, contributing to the ongoing exploration of inequalities on diverse time scales. [ABSTRACT FROM AUTHOR]

The aim of this article is to get the forms of the solutions of the following nonlinear higher-order difference equations χ s + 1 = χ s − 3 k + 1 / ± 1 ± Π l = 1 k χ s − 3 l + 1 , s = 0 , 1 , 2 , ... , where the initial conditions χ − j , j = 0 , 1 , 2 , ... , 3 k − 1 and k ∈ 1 , 2 , ... are arbitrary real numbers. Also, we examine stability, boundedness, oscillation, and the periodic nature of these solutions. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new dynamic Hardy-type inequalities with negative parameters on time scales nabla calculus by applying the reverse Holder's inequality, integration by parts, and chain rule on time scales nabla calculus. As special cases of our results (when 핋 = ℝ), we get the continuous analouges of inequalities proven by Benaissa and Sarikaya, and when 핋 = ℕ0, the results to the best of the authors' knowledge are essentially new. [ABSTRACT FROM AUTHOR]