Oscillation and Asymptotic Properties of Differential Equations of Third-Order

The main purpose of this study is aimed at developing new criteria of the iterative nature to test the asymptotic and oscillation of nonlinear neutral delay differential equations of third order with noncanonical operator (a(ι)[(b(ι)x(ι)+p(ι)x(ι−τ)′)′]β)′+∫cdq(ι,μ)xβ(σ(ι,μ))dμ=0, where ι≥ι0 and w(ι):=x(ι)+p(ι)x(ι−τ). New oscillation results are established by using the generalized Riccati technique under the assumption of ∫ι0ιa−1/β(s)ds<∫ι0ι1b(s)ds=∞asι→∞. Our new results complement the related contributions to the subject. An example is given to prove the significance of new theorem.

New Asymptotic Properties of Positive Solutions of Delay Differential Equations and Their Application

In this study, new asymptotic properties of positive solutions of the even-order delay differential equation with the noncanonical operator are established. The new properties are of an iterative nature, which allows it to be applied several times. Moreover, we use these properties to obtain new criteria for the oscillation of the solutions of the studied equation using the principles of comparison.

Simplified and improved criteria for oscillation of delay differential equations of fourth order

An interesting point in studying the oscillatory behavior of solutions of delay differential equations is the abbreviation of the conditions that ensure the oscillation of all solutions, especially when studying the noncanonical case. Therefore, this study aims to reduce the oscillation conditions of the fourth-order delay differential equations with a noncanonical operator. Moreover, the approach used gives more accurate results when applied to some special cases, as we explained in the examples.

Asymptotic properties of noncanonical third order differential equations

The purpose of the paper is to show that noncanonical operator $$\begin{array}{} \displaystyle \mathcal {L}\,y=\left(r_2(t)\left(r_1(t)y'(t)\right)'\right)' \end{array}$$ can be easily written in essentially unique canonical form $$\begin{array}{} \displaystyle \mathcal {L}\,y = q_3(t)\left(q_2(t)\left(q_1(t)\left(q_0(t)y(t)\right)'\right)'\right)' \end{array}$$ such that $$\begin{array}{} \displaystyle \int\limits^\infty \frac{1}{q_i(s)}\,\text{d}{s}=\infty, \quad i=1,2. \end{array}$$ The canonical representation is applied for examination of the third order noncanonical equations $$\begin{array}{} \displaystyle \left(r_2(t)\left(r_1(t)y'(t)\right)'\right)'+p(t)y(\tau(t))=0. \end{array}$$