The main purpose of this manuscript is to show asymptotic properties of a class of differential equations with variable coefficients ${\left(r\left(\nu \right){\left(w^{‴}\left(\nu \right)\right)}^{\beta }\right)}^{\prime }+{\sum }_{i=1}^j{q}_i\left(\nu \right)y^{\kappa }\left(g_i\left(\nu \right)\right)=0$ , where $\nu \ge {\nu }_0$ and $w\left(\nu \right):=y\left(\nu \right)+p\left(\nu \right)y\left(\sigma \left(\nu \right)\right)$ . By using integral averaging technique, we get conditions to ensure oscillation of solutions of this equation. The obtained results improve and generalize the earlier ones; finally an example is given to illustrate the criteria.

In this work, we consider a type of second-order functional differential equations and establish qualitative properties of their solutions. These new results complement and improve a number of results reported in the literature. Finally, we provide an example that illustrates our results.

The aim of this paper is to show some regularity properties of solutions to \(\mathcal{L} u = f\), where \(\mathcal{L}\) is a second order nondivergence form operator and f belongs to a homogeneous Herz space \(\dot{K}^{\alpha,p(\cdot)}_{q(\cdot)}\) with two variable exponents.