On Nonoscillation of Mixed Advanced-Delay Differential Equations with Positive and Negative Coefficients

For a mixed (advanced--delay) differential equation with variable delays and coefficients $$\dot{x}(t) \pm a(t)x(g(t)) \mp b(t)x(h(t)) = 0, t\geq t_0$$ where $$a(t)\geq 0, b(t)\geq 0, g(t)\leq t, h(t)\geq t$$ explicit nonoscillation conditions are obtained.Comment: 17 pages; 2 figures; to appear in Computers & Mathematics with Application

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

HILLE-KNESER-TYPE CRITERIA FOR SECOND-ORDER DYNAMIC EQUATIONS ON TIME SCALES

We consider the pair of second-order dynamic equations, (r(t)(xΔ)γ)Δ + p(t)xγ(t) = 0 and (r(t)(xΔ)γ)Δ + p(t)xγσ (t) = 0, on a time scale T, where γ \u3e 0 is a quotient of odd positive integers. We establish some necessary and sufficient conditions for nonoscillation of Hille-Kneser type. Our results in the special case when T = R involve the well known Hille-Kneser-type criteria of second-order linear differential equations established by Hille. For the case of the second-order half-linear differential equation, our results extend and improve some earlier results of Li and Yeh and are related to some work of Dosly and Rehak and some results of Rehak for half-linear equations on time scales. Several examples are considered to illustrate the main results

Functional Inequalities: New Perspectives and New Applications

This book is not meant to be another compendium of select inequalities, nor does it claim to contain the latest or the slickest ways of proving them. This project is rather an attempt at describing how most functional inequalities are not merely the byproduct of ingenious guess work by a few wizards among us, but are often manifestations of certain natural mathematical structures and physical phenomena. Our main goal here is to show how this point of view leads to "systematic" approaches for not just proving the most basic functional inequalities, but also for understanding and improving them, and for devising new ones - sometimes at will, and often on demand.Comment: 17 pages; contact Nassif Ghoussoub (nassif @ math.ubc.ca) for a pre-publication pdf cop

Oscillation theorems for nonlinear differential equations

We establish new oscillation theorems for the nonlinear differential equation $$[a(t)\psi(x(t))|x'(t)|^{\alpha-1}x'(t)]'+q(t)f(x(t))=0, \alpha>0$$ where $a,q:[t0,\infty)\rightarrow R, \psi,f:R\rightarrow R$ are continuous, $a(t)>0$ and $\psi(x)>0$, $xf(x)>0$ for $x\not=0$. These criteria involve the use of averaging functions

Oscillation criteria for nonlinear second-order differential equations with damping

Some new oscillation criteria are given for general nonlinear second order ordinary differential equations with damping of the form x′′ + p(t)x′ + q(t) f (x) = 0, where f is with or without monotonicity. Our results generalize and extend some earlier results of Deng.Наведено деякі нові осцнляційні критерії для загальних нелінійних звичайних диференціальних рівнянь другого порядку із затуханням вигляду x" + p(t)x' + q(t)f(x) = 0, де функція f або монотонна, або немонотонна. Наведені результати узагальнюють та розширюють деякі результати, отримані раніше Денгом

Oscillation Results for Emden–Fowler Type Differential Equations

AbstractThe third order nonlinear differential equationx‴+a(t)x′+b(t)f(x)=0, (∗)is considered. We present oscillation and nonoscillation criteria which extend and improve previous results existing in the literature, in particular some results recently stated by M. Greguš and M. Greguš, Jr., (J. Math. Anal. Appl.181, 1994, 575–585). In addition, contributions to the classification of solutions are given. The techniques used are based on a transformation which reduces (∗) to a suitable disconjugate form. To this aim auxiliary results on the asymptotic behavior of solutions of a second order linear differential equation associated to (∗) are stated. They are presented in an independent form because they may be applied also to simplify and improve other qualitative problems concerning differential equations with quasiderivatives

Oscillation Criteria for Nonlinear Differential Equations of Second Order with Damping Term

2000 Mathematics Subject Classification: 34C10, 34C15.Some new criteria for the oscillation of all solutions of second order differential equations of the form (d/dt)(r(t)ψ(x)|dx/dt|α−2(dx/dt))+ p(t)φ(|x|α−2x,r(t) ψ(x)|dx/dt|α−2(dx/dt))+q(t)|x|α−2 x=0, and the more general equation (d/dt)(r(t)ψ(x)|dx/dt|α−2(dx/dt))+p(t)φ(g(x),r(t) ψ(x)|dx/dt|α−2 (dx/dt))+q(t)g(x)=0, are established. our results generalize and extend some known oscillation criterain in the literature

On Nonoscillatory Solutions of Two-Dimensional Nonlinear Dynamical Systems

During the past years, there has been an increasing interest in studying oscillation and nonoscillation criteria for dynamical systems on time scales that harmonize the oscillation and nonoscillation theory for the continuous and discrete cases in order to combine them in one comprehensive theory and eliminate obscurity from both. We not only classify nonoscillatory solutions of two-dimensional systems of first-order dynamic equations on time scales but also guarantee the existence of such solutions using the Knaster, Schauder-Tychonoff and Schauder’s fixed point theorems. The approach is based on the sign of components of nonoscillatory solutions. A short introduction to the time scale calculus is given as well. Examples are significant in order to see if nonoscillatory solutions exist or not. Therefore, we give several examples in order to highlight our main results for the set of real numbers R, the set of integers Z and qN0 = {1, q, q2, q3, …}, q >1, which are the most well-known time scales