In this work, we obtain new sufficient conditions for the non-oscillatory solutions of forced nabla fractional difference equations with positive and negative terms. The results are developed in sense of Caputo nabla fractional difference operator and by the help of Young's inequality as well as an equivalent representation in form of a Volterra-type summation equation. The results improve some existing results in the literature. Further, two examples are presented to support and illustrate the applicability of the deduced results.

Impulsive differential equations of second-order appears in numerous applications such as fluid dynamics, electromagnetism, quantum mechanics, neural networks and the field of time symmetric electrodynamics. The aim of this work is to establish necessary and sufficient conditions for the oscillation of the solutions to a second-order neutral differential equation with impulses. Two examples are given and state an open problem.