Oscillation of third order Impulsive Differential Equations with delay

This paper deals with the oscillation of third order impulsive differential equations with delay. The results of this paper improve and extend some results for the differential equations without impulses. Some examples are givento illustrate the main results

Oscillation criteria of second order neutral delay dynamic equations with distributed deviating arguments

In this paper we establish some oscillation theorems for second order neutral dynamic equations with distributed deviating arguments. We use the Riccati transformation technique to obtain sufficient conditions for the oscillation of all solutions. Further, some examples are provided to illustrate the results

On the oscillation of third-order quasi-linear neutral functional differential equations

summary:The aim of this paper is to study asymptotic properties of the third-order quasi-linear neutral functional differential equation \begin{equation*} \big [a(t)\big ([x(t)+p(t)x(\delta (t))]^{\prime \prime }\big )^\alpha \big ]^{\prime }+q(t)x^\alpha (\tau (t))=0\,, E \end{equation*} where $\alpha >0$, $0\le p(t)\le p_0<\infty$ and $\delta (t)\le t$. By using Riccati transformation, we establish some sufficient conditions which ensure that every solution of () is either oscillatory or converges to zero. These results improve some known results in the literature. Two examples are given to illustrate the main results

Oscillation of second-order half-linear difference equations

AbstractSome new oscillation criteria are obtained for the second-order half-linear difference equation (−1)m+1δm yi(n) + σj=1Nqijyi(n−Tji)=0, m ⩾1, i=1,…N where α 0 is a ratio of odd positive integers. The method uses techniques based on a Riccati type difference inequality. Examples are inserted to illustrate the results

Oscillation Criteria of Second-Order Quasi-Linear Neutral Delay Difference Equations

2000 Mathematics Subject Classification: 39A10.The oscillatory and nonoscillatory behaviour of solutions of the second order quasi linear neutral delay difference equation Δ(an | Δ(xn+pnxn-τ)|α-1 Δ(xn+pnxn-τ) + qnf(xn-σ)g(Δxn) = 0 where n ∈ N(n0), α > 0, τ, σ are fixed non negative integers, {an}, {pn}, {qn} are real sequences and f and g real valued continuous functions are studied. Our results generalize and improve some known results of neutral delay difference equations

On the oscillation of third order half-linear neutral type difference equations

In this paper, the authors study the oscillatory properties of third order quasilinear neutral difference equation of the form $$\Delta(a_{n}(\Delta^{2}(x_{n} + p_{n}x_{n-\delta}))^{\alpha}) + q_{n} {x^{\alpha}_{n-\tau}} = 0,\quad n\geq 0, \tag{E}$$ where $\alpha > 0, 0\leq p_{n}\leq p < \infty.$ By using Riccati transformation we estabilish some new sufficient conditions which ensure that every solution of equation (E) is either oscillatory or converges to zero. These results improve some known results in the literature. Examples are provided to illustrate the main results