Neutral Emden–Fowler Differential Equation of Second Order: Oscillation Criteria of Coles Type

In this work, we study the asymptotic and oscillatory behavior of solutions to the second-order general neutral Emden–Fowler differential equation ${\left(a\left(v\right)\eta \left(x\left(v\right)\right)z^{\prime }\left(v\right)\right)}^{\prime }$+$q\left(v\right)F\left(x\left(g\left(v\right)\right)\right)$= 0, where $v\ge v_0$ and the corresponding function z=x+p$\left(x\circ h\right)$. Besides the importance of equations of the neutral type, studying the qualitative behavior of solutions to these equations is rich in analytical points and interesting issues. We begin by finding the monotonic features of positive solutions. The new properties contribute to obtaining new and improved relationships between x and z for use in studying oscillatory behavior. We present new conditions that exclude the existence of positive solutions to the examined equation, and then we establish oscillation criteria through the symmetry property between non-oscillatory solutions. We use the generalized Riccati substitution method, which enables us to apply the results to a larger area than the special cases of the considered equation. The new results essentially improve and extend previous results in the literature. We support this claim by applying the results to an example and comparing them with previous findings. Moreover, the reduction of our results to Euler’s differential equation introduces the well-known sharp oscillation criterion.