In this paper, we deal with the asymptotic and oscillatory behavior of quasi-linear delay differential equations of fourth order. We first find new properties for a class of positive solutions of the studied equation, ${\mathcal{N}}_a$. As an extension of the approach taken in ^[1], we establish a new criterion that guarantees that ${\mathcal{N}}_a=\mathrm{\varnothing }$. Then, we create a new oscillation criterion.

This article examines the oscillatory behaviour of solutions to a particular class of conformable elliptic partial differential equations of the Emden-Fowler type. Using the Riccati method, we create some new necessary conditions for the oscillation of all solutions. The previously discovered conclusions for the integer order equations are expanded upon by these additional findings. We provide an example to demonstrate the usefulness of our new finding.

In this article, we have derived a new method to study the oscillatory and asymptotic properties for third-order noncanonical functional differential equations with both positive and negative terms of the form

$$(p_2(t)(p_1(t)x^{\prime }(t){)}^{\prime }{)}^{\prime }+a(t)g(x(\tau (t)))-b(t)h(x(\sigma (t))=0$$

Firstly, we have converted the above equation of noncanonical type into the canonical type using the strongly noncanonical operator and obtained some new conditions for Property $\overline{A}$. We furnished illustrative examples to validate our main result.