Order Half-linear Differential Equations Research Articles

Nonoscillation of higher order half-linear differential equations

We establish nonoscillation criteria for even order half-linear differential equations. The principal tool we use is the Wirtinger type inequality combined with various perturbation techniques. Our results extend nonoscillation criteria known for linear higher order differential equations.

The Distribution of Zeroes and Critical Points of Solutions of a Second Order Half-Linear Differential Equation

This paper reuses an idea first devised by Kwong to obtain upper bounds for the distance between a zero and an adjacent critical point of a solution of the second order half-linear differential equation(p(x)Φ(y'))'+q(x)Φ(y)=0, withp(x),q(x)>0,Φ(t)=|t|r-2t, andrreal such thatr>1. It also compares it with other methods developed by the authors.

Gaps between zeros of second-order half-linear differential equations

In this paper, for second order half-linear differential equations, we will establish some new inequalities of Lyapunov’s type. These inequalities give results related to the spacing between consecutive zeros of a solution and the spacing between a zero of a solution and/or a zero of its derivative. The results also yield conditions for disfocality, disconjugacy and lower bounds for an eigenvalue of a boundary value problem. The main results will be proved by making use of some generalizations of Opial and Wirtinger type inequalities. Some examples are considered to illustrate the main results.

Oscillation criterion for half-linear differential equations with periodic coefficients

In this paper, we present an oscillation criterion for second order half-linear differential equations with periodic coefficients. The method is based on the nonexistence of a proper solution of the related modified Riccati equation. Our result can be regarded as an oscillatory counterpart to the nonoscillation criterion by Sugie and Matsumura (2008). These two theorems provide a complete half-linear extension of the oscillation criterion of Kwong and Wong (2003) dealing with the Hill’s equation.

Comparison theorems for even order half-linear differential equations

A generalization of Picone’s formula to the case of half-linear differential operators of the even order is given and comparison results concerning the associated differential equations are established with the help of this formula.

Asymptotic stability of two dimensional systems of linear difference equations and of second order half-linear differential equations with step function coefficients

Asymptotic stability of two dimensional systems of linear difference equations and of second order half-linear differential equations with step function coefficients

Oscillation and nonoscillation of perturbed half-linear Euler differential equations

Using general results for (non)oscillation of the second order half-linear differential equation we establish new oscillation and nonoscillation criteria for this equation when it is viewed as a perturbation of half-linear Euler differential equation.

New oscillation criteria for forced second order half-linear differential equations with damping

Using variational principle and Riccati technique, new oscillation criteria for forced second order half-linear differential equation with damped term are established, which improve a recent paper by Li and Cheng [W.T. Li, S.S. Cheng, An oscillation criterion for nonhomogenuous half-linear differential equations, Appl. Math. Lett. 15 (2002) 259–263] and generalized some of the results in the literature.

Nonoscillation and oscillation of second order half-linear differential equations

We study the oscillation problems for the second order half-linear differential equation [ p ( t ) Φ ( x ′ ) ] ′ + q ( t ) Φ ( x ) = 0 , where Φ ( u ) = | u | r − 1 u with r > 0 , 1 / p and q are locally integrable on R + ; p > 0 , q ⩾ 0 a.e. on R + , and ∫ 0 ∞ p − 1 / r ( t ) d t = ∞ . We establish new criteria for this equation to be nonoscillatory and oscillatory, respectively. When p ≡ 1 , our results are complete extensions of work by Huang [C. Huang, Oscillation and nonoscillation for second order linear differential equations, J. Math. Anal. Appl. 210 (1997) 712–723] and by Wong [J.S.W. Wong, Remarks on a paper of C. Huang, J. Math. Anal. Appl. 291 (2004) 180–188] on linear equations to the half-linear case for all r > 0 . These results provide corrections to the wrongly established results in [J. Jiang, Oscillation and nonoscillation for second order quasilinear differential equations, Math. Sci. Res. Hot-Line 4 (6) (2000) 39–47] on nonoscillation when 0 < r < 1 and on oscillation when r > 1 . The approach in this paper can also be used to fully extend Elbert's criteria on linear equations to half-linear equations which will cover and improve a partial extension by Yang [X. Yang, Oscillation/nonoscillation criteria for quasilinear differential equations, J. Math. Anal. Appl. 298 (2004) 363–373].

Oscillation and nonoscillation criteria for half-linear second order differential equations

Oscillatory properties of the second order half-linear differential equation are investigated. The studied equation is viewed as a perturbation of a nonoscillatory equation of the same form. Conditions on the perturbation term are given which guarantee that the original equation is oscillatory (nonoscillatory).

Some Oscillation Theorems for Second Order Differential Equations

In this paper we establish some oscillation or nonoscillation criteria for the second order half-linear differential equation $$(r(t)\Phi (u'(t)))' + c(t)\Phi (u(t)) = 0,$$ where(i) r,c ∈ C([t 0, ∞), ℝ := (− ∞, ∞)) and r(t) > 0 on [t 0, ∞) for some t 0 ⩾ 0;(ii) Φ(u) = |u|p−2 u for some fixed number p > 1.We also generalize some results of Hille-Wintner, Leighton and Willet.

Interval criteria for oscillation of second order half-linear differential equations with damping

Interval oscillation criteria are established for second order half-linear differential equations with damping that are different from most known ones in the sense that they are based only on a sequence of subintervals of $ [t_0, \infty)$, rather than on the whole half-line. The results extend the integral averaging technique and include earlier results.

Oscillation of Second Order Half-Linear Differential Equations with Damping

Abstract This paper is concerned with a class of second order half-linear damped differential equations. Using the generalized Riccati transformation and the averaging technique, new oscillation criteria are obtained which are either extensions of or complementary to a number of the existing results.

Nonoscillation theory for second order half-linear differential equations in the framework of regular variation

Criteria are established for nonoscillation of all solutions of the second order half-linear differential equation A % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $(\mid y^\prime \mid^{\alpha-1}y^\prime)^\prime + q(t)\mid y \mid^{\alpha -1}y = 0, t \geq 0,$ where α > 0 is a constant and q: [0, ∞) → ℝ is continuous. The criteria are designed to exhibit the role played by the integral of q(t) in guaranteeing the existence of nonoscillatory solutions of (A) in specific classes of regularly varying functions in the sense of Karamata.

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.

Oscillation criteria for second order half-linear differential equations with functional arguments

Oscillation criteria for second order half-linear differential equations with functional arguments

On a class of fourth order half-linear differential equations

On a class of fourth order half-linear differential equations