Oscillation criteria for nonlinear delay differential equations of second order

We prove oscillation theorems for the nonlinear delay differential equation $\left( \left\vert y^{\prime }(t)\right\vert ^{\alpha -2}y^{\prime}(t)\right) ^{\prime }+q(t)\left\vert y(\tau (t))\right\vert ^{\beta-2}y(\tau (t))=0, t\geq t_{\ast }>0,$ where $\beta >1,$ $\alpha >1,$ $q(t)\geq 0$ and locally integrable on $[t_{\ast },\infty ),$ $\tau (t)$ is a continuous function satisfiying $0<\tau (t)\leq t$ and lim$_{t\rightarrow \infty }\tau (t)=\infty .$ The results obtained essentially improve the known results in the literature and can be applied to linear and half-linear delay type differential equations

Oscillation criteria for second order superlinear neutral delay differential equations

New oscillation criteria for the second order nonlinear neutral delay differential equation $[y(t)+p(t)y(t-\tau )]^{^{\prime \prime}}+q(t)\,f(y(g(t)))=0$, $t\geq t_{0}$ are given. The relevance of our theorems becomes clear due to a carefully selected example

Oscillation criteria for second-order neutral delay differential equations

New sufficient conditions for oscillation of second-order neutral half-linear delay differential equations are given. Our results essentially improve, complement and simplify a number of related ones in the literature, especially those from a recent paper [R. P. Agarwal, Ch. Zhang, T. Li, Appl. Math. Comput. 274(2016), 178–181.]. An example illustrates the value of the results obtained

Oscillation Criteria for Second‐Order Neutral Damped Differential Equations with Delay Argument

The chapter is devoted to study the oscillation of all solutions to second‐order nonlinear neutral damped differential equations with delay argument. New oscillation criteria are obtained by employing a refinement of the generalized Riccati transformations and integral averaging techniques

Oscillation Criteria for Forced Second Order Mixed Type Quasilinear Delay Differential Equations

This article presents new oscillation criteria for the second-order delay differential equation $$(p(t) (x'(t))^{alpha})' + q(t) x^{alpha}(t - au) + sum_{i = 1}^{n} q_{i}(t) x^{alpha_{i}}(t - au) = e(t)$$ where $au geq 0$, $p(t) in C^1[0, infty)$, $q(t),q_{i}(t), e(t) in C[0, infty)$, $p(t) > 0$, $alpha_1 >dots > alpha_{m} > alpha > alpha_{m+1} > dots > alpha_{n} > 0 (n > mgeq 1)$, $alpha_1, dots , alpha_{n}$ and $alpha$ are ratio of odd positive integers. Without assuming that $q(t), q_{i}(t)$ and $e(t)$ are nonnegative, the results in [6,8] have been extended and a mistake in the proof of the results in [3] is corrected

Nonoscillation of Second-Order Dynamic Equations with Several Delays

Existence of nonoscillatory solutions for the second-order dynamic equation (A0xΔ)Δ(t)+∑i∈[1,n]ℕAi(t)x(αi(t))=0 for t∈[t0,∞)T is investigated in this paper. The results involve nonoscillation criteria in terms of relevant dynamic and generalized characteristic inequalities, comparison theorems, and explicit nonoscillation and oscillation conditions. This allows to obtain most known nonoscillation results for second-order delay differential equations in the case A0(t)≡1 for t∈[t0,∞)R and for second-order nondelay difference equations (αi(t)=t+1 for t∈[t0,∞)N). Moreover, the general results imply new nonoscillation tests for delay differential equations with arbitrary A0 and for second-order delay difference equations. Known nonoscillation results for quantum scales can also be deduced

Oscillation Criteria for Fourth Order Nonlinear Positive Delay Differential Equations with a Middle Term

In this article, we establish some new criteria for the oscillation of fourth order nonlinear delay differential equations of the form (Equation presented) provided that the second order equation (Equation presented) is nonoscillatiory or oscillatory. This equation with g(t) = t is considered in [8] and some oscillation criteria for this equation via certain energy functions are established. Here, we continue the study on the oscillatory behavior of this equation via some inequalities

New oscillation criteria for second-order delay differential equations with mixed nonlinearities

We establish new oscillation criteria for second-order delay differential equations with mixed nonlinearities of the form p t x t n i 1 p i t x t−τ i n i 1 q i t |x t−τ i | αi sgn x t−τ i e t , t ≥ 0, where p t , p i t , q i t , and e t are continuous functions defined on 0, ∞ , and p t &gt; 0, p t ≥ 0, and No restriction is imposed on the potentials p i t , q i t , and e t to be nonnegative. These oscillation criteria extend and improve the results given in the recent papers. An interesting example illustrating the sharpness of our results is also provided