Oscillation Of Half-linear Differential Equations Research Articles

Oscillation of half-linear differential equations with mixed type of argument

This paper is devoted to the study of the oscillatory behavior of half-linear functional differential equations with deviating argument of the form ( E ) ( r ( t ) ( y ′ ( t ) ) α ) ′ = p ( t ) y α ( τ ( t ) ) . . We introduce new technique based on monotonic properties of nonoscillatory solutions to offer new oscillatory criteria for ( E ) . We will show that presented results essentially improve existing ones even for linear differential equations.

New Criteria for Oscillation of Half-Linear Differential Equations with p-Laplacian-like Operators

In this paper, new oscillatory properties for fourth-order delay differential equations with p-Laplacian-like operators are established, using the Riccati transformation and comparison method. Moreover, our results are an extension and complement to previous results in the literature. We provide some examples to examine the applicability of our results.

Integral condition for oscillation of half-linear differential equations with damping

The purpose of this paper is to provide an oscillation theorem that can be applied to half-linear differential equations with time-varying coefficients. A parametric curve by the coefficients is focused in order to obtain our theorem. This parametric curve is a generalization of the curve given by the characteristic equation of the second-order linear differential equation with constant coefficients. The obtained theorem is proved by transforming the half-linear differential equation to a standard polar coordinates system and using phase plane analysis carefully.

Generalized Prüfer angle and oscillation of half-linear differential equations

In this paper, we introduce a new modification of the half-linear Prüfer angle. Applying this modification, we investigate the conditional oscillation of the half-linear second order differential equation (∗)tα−1r(t)Φ(x′)′+tα−1−ps(t)Φ(x)=0,Φ(x)=|x|p−1sgnx,where p>1, α≠p, and r,s are continuous functions such that r(t)>0 for large t. We present conditions on the functions r,s which guarantee that Eq. (∗) behaves like the Euler type equation [tα−1Φ(x′)]′+λtα−1−pΦ(x)=0, which is conditionally oscillatory with the oscillation constant λ0=|p−α|p∕pp.

Conditional Oscillation of Half-Linear Differential Equations with Coefficients Having Mean Values

We prove that the existence of the mean values of coefficients is sufficient for second-order half-linear Euler-type differential equations to be conditionally oscillatory. We explicitly find an oscillation constant even for the considered equations whose coefficients can change sign. Our results cover known results concerning periodic and almost periodic positive coefficients and extend them to larger classes of equations. We give examples and corollaries which illustrate cases that our results solve. We also mention an application of the presented results in the theory of partial differential equations.

Oscillation of half-linear differential equations with asymptotically almost periodic coefficients

We investigate second-order half-linear differential equations with asymptotically almost periodic coefficients. For these equations, we explicitly find an oscillation constant. If the coefficients are replaced by constants, our main result (concerning the conditional oscillation) reduces to the classical one. We also mention examples and concluding remarks. MSC:34C10, 34C15.

Oscillation of Half-Linear Differential Equations with Delay

We study the half-linear delay differential equation , , We establish a new a priori bound for the nonoscillatory solution of this equation and utilize this bound to derive new oscillation criteria for this equation in terms of oscillation criteria for an ordinary half-linear differential equation. The presented results extend and improve previous results of other authors. An extension to neutral equations is also provided.

The oscillation of half-linear differential equations with an oscillatory coefficient

We derive lower bounds for the distance between consecutive zeros of a solution of the second order half-linear differential equation (¦y′ (t) ¦ α − 1y′ (t)) ′ + q (t) ¦y (t) ¦ α − 1 y (t) = 0 , (∗) when q (t): (t 0, ∞) → R is locally integrable for some t 0 > 0. Then we apply these results to the following equations: (p(t) ¦y′ (t) ¦ α − 1y′ (t)) ′ + q (t) ¦y (t) ¦ α − 1 y (t) = 0 , and ∑ i=1 N |Dy| n-2D iy +h(|x|)|y| n-2y=0 forx∈Ω t 0 , where 1. (i) p ϵ C ([t 0, ∞), (0, ∞)) and ∝ ∞ t 0 p (t) −1 α dt = ∞ 2. (ii) D i = t6 t6x i , D = (D 1, …, D N), Ω t 0 = {x ϵ R N: ¦x¦≥ t 0} is an exterior domain h ϵ C ([t 0, ∞), R) 3. (iii) α > 0 is a constant; n > 1 and N > 2 are integers.