Oscillation of the second order advanced differential equations

We establish a new technique for deducing oscillation of the second order advanced differential equation \begin{equation*} \left(r(t)u'(t)\right)'+p(t)u(\sigma(t))=0 \end{equation*} with help of a suitable equation of the form $$\left(r(t)u'(t)\right)'+q(t)u(t)=0.$$ The comparison principle obtained fills the gap in the theory of oscillation and essentially improves existing criteria. Our technique is based on new monotonic properties of nonoscillatory solutions, and iterated exponentiation is employed. The results are supported with several illustrative examples

New oscillation results to fourth order delay differential equations with damping

This paper is concerned with the oscillation of the linear fourth order delay differential equation with damping \begin{equation*} \left(r_3(t)\left(r_2(t)\left(r_1(t)y'(t)\right)'\right)'\right)'+p(t)y'(t)+q(t)y(\tau(t))=0 \end{equation*} under the assumption that the auxiliary third order differential equation \begin{equation*} \left(r_3(t)\left(r_2(t)z'(t)\right)'\right)'+\frac{p(t)}{r_1(t)}z(t)=0 \end{equation*} is nonoscillatory. In addition, a couple of examples is provided to illustrate the relevance of the main results

Properties of even order linear functional differential equations with deviating arguments of mixed type

This paper is concerned with oscillatory behavior of linear functional differential equations of the type $y^((n))(t) = p(t)y(τ (t))$ with mixed deviating arguments which means that its both delayed and advanced parts are unbounded subset of $(0,∞)$. Our attention is oriented to the Euler type of equation, i.e. when $p(t) ∼ a//t^n$

Property (B) Of Differential Equations With Deviating Argument

. The aim of this paper is to deduce oscillatory and asymptotic behavior of delay differential equation Lnu(t) \Gamma p(t)u(ø(t)) = 0; from the oscillation of a set of the first order delay equations. In this paper we are concerned with the oscillatory and asymptotic behavior of solutions of the differential equations of the form (1) Lnu(t) \Gamma p(t)u(ø (t)) = 0; where n 3 and Ln denotes the general disconjugate differential operator of the form (2) Ln = 1 r n (t) d dt 1 r n\Gamma1 (t) d dt \Delta \Delta \Delta d dt 1 r 1 (t) d dt \Delta r 0 (t) : It is always assumed that r i (t), 0 i n, p(t) and ø (t) are continuous on [t 0 ; 1), p(t) ? 0, ø (t) ! t, ø (t) !1 as t !1 and (3) Z 1 r i (s) ds = 1 for 1 i n \Gamma 1: The operator Ln satisfying (3) is said to be in canonical form. It is well-known that any differential operator of the form (2) can always be represented in a canonical form in an essentially unique way (see Trench [7]). In the sequel we will assume t..

A Comparison Theorem For Linear Delay Differential Equations

. In this paper property (A) of the linear delay differential equation Lnu(t) + p(t)u(ø(t)) = 0; is to deduce from the oscillation of a set of the first order delay differential equations. Let us consider the delay differential equation (1) Lnu(t) + p(t)u(ø (t)) = 0; where n 2 and (2) Lnu(t) = / 1 r n\Gamma1 (t) / \Delta \Delta \Delta ` 1 r 1 (t) u 0 (t) ' 0 \Delta \Delta \Delta ! 0 !0 : We always assume that r i (t), 1 i n \Gamma 1, ø (t) and p(t) are continuous on [t 0 ; 1), r i (t) ? 0, ø (t) ! t and ø (t) ! 1 as t ! 1. Moreover, in the sequel we assume that (3) Z 1 r i (s) ds = 1 for 1 i n \Gamma 1: For convenience we introduce the following notation: L 0 u(t) = u(t); L i u(t) = 1 r i (t) d dt L i\Gamma1 u(t); 1 i n \Gamma 1: Lnu(t) = d dt Ln\Gamma1 u(t): 1991 Mathematics Subject Classification : Primary 34C10. Key words and phrases: Comparison theorem, property (A). .. Received May 16, 1994. JOZEF D ZURINA The domain D(Ln ) of Ln is defined to be th..

The fourth order strongly noncanonical operators

It is shown that the strongly noncanonical fourth order operato

Oscillatory criteria via linearization of half-linear second order delay differential equations

In the paper, we study oscillation of the half-linear second order delay differential equations of the form [formula]. We introduce new monotonic properties of its nonoscillatory solutions and use them for linearization of considered equation which leads to new oscillatory criteria. The presented results essentially improve existing ones

Oscillation and Property B for third-order differential equations with advanced arguments

We establish sufficient conditions for the third-order nonlinear advanced differential equation $$\Big(a(t)[\big(b(t)y'(t)\big)']^{\gamma}\Big)'-p(t)f(y(\sigma(t)))=0$$ to have property B or to be oscillatory. These conditions are based on monotonic properties and estimates of non-oscillatory solutions, and essentially improve known results for differential equations with deviating arguments and for ordinary differential equations

Oscillation results for even-order quasilinear neutral functional differential equations

In this article, we use the Riccati transformation technique and some inequalities, to establish oscillation theorems for all solutions to even-order quasilinear neutral differential equation $$Big(ig[ig(x(t)+p(t)x(au(t))ig)^{(n-1)}ig]^gammaBig)' +q(t)x^gammaig(sigma(t)ig)=0,quad tgeq t_0.$$ Our main results are illustrated with examples

Oscillation of solutions to third-order half-linear neutral differential equations

In this article, we study the oscillation of solutions to the third-order neutral differential equations $$Big(a(t)ig([x(t)pm p(t)x(delta(t))]''ig)^alphaBig)' + q(t)x^alpha(au(t)) = 0.$$ Sufficient conditions are established so that every solution is either oscillatory or converges to zero. In particular, we extend the results obtain in [1] for $a(t)$ non-decreasing, to the non-increasing case