The author studies the commutators generated by a suitable function a(x) on ℝⁿ and the oscillatory singular integral with rough kernel Ω(x)|x|ⁿ and polynomial phase, where Ω is homogeneous of degree zero on ℝⁿ, and a(x) is a BMO function or a Lipschitz function. Some mapping properties of these higher order commutators on $L^p\left(ℝⁿ\right)$, which are essential improvements of some well known results, are given.

We consider the second order self-adjoint differential equation (1) (r(t)y’(t))’ + p(t)y(t) = 0 on an interval I, where r, p are continuous functions and r is positive on I. The aim of this paper is to show one possibility to write equation (1) in the same form but with positive coefficients, say r₁, p₁ and to derive a sufficient condition for equation (1) to be oscillatory in the case p is nonnegative and ${\int }^{\infty }[1/r\left(t\right)]dt$ converges.

We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term $$x^{\text{'}\text{'}\text{'}}\left(t\right)+q\left(t\right)x^{\text{'}}\left(t\right)+r\left(t\right){\left|x\right|}^{\lambda }\left(t\right)\mathrm{sgn}x\left(t\right)=0,t\ge 0.$$ We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case $\lambda \le 1$ and if the corresponding second order differential equation $h^{\text{'}\text{'}}+q\left(t\right)h=0$ is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.

We prove the $L^p$ and $H^1$ boundedness of oscillatory singular integral operators defined by Tf = p.v.Ω∗f, where $\Omega \left(x\right)=e^{i\Phi \left(x\right)}K\left(x\right)$, K(x) is a Calderón-Zygmund kernel, and Φ satisfies certain growth conditions.

The paper investigates the singular initial problem[4pt] ${\left(p\left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}+q\left(t\right)f\left(u\left(t\right)\right)=0,u\left(0\right)=u_0,u^{\text{'}}\left(0\right)=0$[4pt] on the half-line $[0,\infty )$. Here $u_0\in [L_0,L]$, where $L_0$, $0$ and $L$ are zeros of $f$, which is locally Lipschitz continuous on $ℝ$. Function $p$ is continuous on $[0,\infty )$, has a positive continuous derivative on $(0,\infty )$ and $p\left(0\right)=0$. Function $q$ is continuous on $[0,\infty )$ and positive on $(0,\infty )$. For specific values $u_0$ we prove the existence and uniqueness of damped solutions of this problem. With additional conditions for $f$, $p$ and $q$ it is shown that the problem has for each specified...

We study the oscillatory behavior of the second-order quasi-linear retarded difference equation $$\Delta \left(p\left(n\right){\left(\Delta y\left(n\right)\right)}^{\alpha }\right)+\eta \left(n\right)y^{\beta }(n-k)=0$$ under the condition ${\sum }_{n=n_0}^{\infty }{p}^{-\frac{1}{\alpha }}\left(n\right)<\infty$ (i.e., the noncanonical form). Unlike most existing results, the oscillatory behavior of this equation is attained by transforming it into an equation in the canonical form. Examples are provided to show the importance of our main results.

Consider the third order nonlinear dynamic equation $x^{\Delta \Delta \Delta }\left(t\right)+p\left(t\right)f\left(x\right)=g\left(t\right)$, (*) on a time scale which is unbounded above. The function f ∈ C(,) is assumed to satisfy xf(x) > 0 for x ≠ 0 and be nondecreasing. We study the oscillatory behaviour of solutions of (*). As an application, we find that the nonlinear difference equation $\Delta ³x\left(n\right)+n^{\alpha }{\left|x\right|}^{\gamma }sgn\left(n\right)=(-1)ⁿn^c$, where α ≥ -1, γ > 0, c > 3, is oscillatory.

In this paper we have considered completely the equation $$y^{\text{'}\text{'}\text{'}}+a\left(t\right)y^{\text{'}\text{'}}+b\left(t\right)y^{\text{'}}+c\left(t\right)y=0,(*)$$ where $a\in C^2([\sigma ,\infty ),R)$, $b\in C^1([\sigma ,\infty ),R)$, $c\in C\left(\right[\sigma ,\infty ),R)$ and $\sigma \in R$ such that $a\left(t\right)\le 0$, $b\left(t\right)\le 0$ and $c\left(t\right)\le 0$. It has been shown that the set of all oscillatory solutions of (*) forms a two-dimensional subspace of the solution space of (*) provided that (*) has an oscillatory solution. This answers a question raised by S. Ahmad and A.  C. Lazer earlier.

Sufficient conditions are presented for all bounded solutions of the linear system of delay differential equations $${(-1)}^{m+1}\frac{d^m{y}_i\left(t\right)}{d{t}^m}+\sum _{j=1}^n{q}_{ij}{y}_j(t-h_{jj})=0,m\ge 1,i=1,2,...,n,$$ to be oscillatory, where $q_{ij}\epsilon (-\infty ,\infty )$, $h_{jj}\in (0,\infty )$, $i,j=1,2,...,n$. Also, we study the oscillatory behavior of all bounded solutions of the linear system of neutral differential equations $${(-1)}^{m+1}\frac{d^m}{d{t}^m}(y_i\left(t\right)+c{y}_i(t-g))+\sum _{j=1}^n{q}_{ij}{y}_j(t-h)=0,$$ where $c$, $g$ and $h$ are real constants and $i=1,2,...,n$.

Recently there has been an increasing interest in studying $p\left(t\right)$-Laplacian equations, an example of which is given in the following form $$\left(\right|u^{\text{'}}{\left(t\right)|}^{p\left(t\right)-2}{u}^{\text{'}}{\left(t\right))}^{\text{'}}+c\left(t\right){\left|u\left(t\right)\right|}^{q\left(t\right)-2}u\left(t\right)=0,t>0.$$ In particular, the first study of sufficient conditions for oscillatory solution of $p\left(t\right)$-Laplacian equations was made by Zhang (2007), but to our knowledge, there has not been a paper which gives the oscillatory conditions by utilizing Riccati inequality. Therefore, we establish sufficient conditions for oscillatory solution of nonlinear differential equations...