By using averaging techniques, some oscillation criteria for quasilinear elliptic differential equations of second order ${\sum }_{i,j=1}^N{D}_i[A_{ij}{\left(x\right)\left|Dy\right|}^{p-2}{D}_{j}y]+p\left(x\right)f\left(y\right)=0$ are obtained. These results extend and generalize the criteria for linear differential equations due to Kamenev, Philos and Wong.

Fite and Kamenev type oscillation criteria for the second order nonlinear damped elliptic differential equation ${\sum }_{i,j=1}^N{D}_i\left[a_{ij}\left(x\right)D_{j}y\right]+{\sum }_{i=1}^N{b}_i\left(x\right)D_{i}y+p\left(x\right)f\left(y\right)=0$ are obtained. Our results are extensions of those for ordinary differential equations and improve some known oscillation criteria in the literature. Several examples are given to show the significance of the results.

Oscillation theorems are established for forced second order mixed-nonlinear elliptic differential equations ⎧ ${div\left(A\left(x\right)\right|\left|\nabla y\right||}^{p-1}{\nabla y)+⟨b\left(x\right),|\left|\nabla y\right||}^{p-1}\nabla y⟩+C(x,y)=e\left(x\right)$, ⎨ ⎩ $C(x,y)={c\left(x\right)\left|y\right|}^{p-1}y+{\sum }_{i=1}^m{c}_i\left(x\right){\left|y\right|}^{p_i-1}y$ $$under quite general conditions. These results are extensions of the recent results of Sun and Wong, [J. Math. Anal. Appl. 334 (2007)] and Zheng, Wang and Han [Appl. Math. Lett. 22 (2009)] for forced second order ordinary differential equations with mixed nonlinearities, and include some known oscillation results in the literature

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.

The paper is concerned with oscillation properties of $n$-th order neutral differential equations of the form $${[x\left(t\right)+cx\left(\tau \left(t\right)\right)]}^{\left(n\right)}+q\left(t\right)f\left(x\left(\sigma \right(t\left)\right)\right)=0,t\ge t_0>0,$$ where $c$ is a real number with $\left|c\right|\ne 1$, $q\in C([t_0,\infty ),ℝ)$, $f\in C(ℝ,ℝ)$, $\tau ,\sigma \in C([t_0,\infty ),{ℝ}_{+})$ with $\tau \left(t\right)<t$ and ${lim}_{t\to \infty }\tau \left(t\right)={lim}_{t\to \infty }\sigma \left(t\right)=\infty$. Sufficient conditions are established for the existence of positive solutions and for oscillation of bounded solutions of the above equation. Combination of these conditions provides necessary and sufficient conditions for oscillation of bounded solutions of the equation. Furthermore, the results are generalized to equations...

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.

This paper deals with the oscillation problems on the nonlinear differential equation $\left(a\left(t\right)\right|x^{\text{'}}{|}^{p\left(t\right)-2}{x}^{\text{'}}{)}^{\text{'}}+b\left(t\right){\left|x\right|}^{\lambda -2}x=0$ involving $p\left(t\right)$-Laplacian. Sufficient conditions are given under which all proper solutions are oscillatory. In addition, we give a-priori estimates for nonoscillatory solutions and propose an open problem.

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.

Sufficient oscillation conditions involving $lim\; sup$ and $lim\; inf$ for first-order differential equations with non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Grönwall inequality. Examples, numerically solved in MATLAB, are also given to illustrate the applicability and strength of the obtained conditions over known ones.

We introduce a method to treat a semilinear elliptic equation in ${ℝ}^N$ (see equation (1) below). This method is of a perturbative nature. It permits us to skip the problem of lack of compactness of ${ℝ}^N$ but requires an oscillatory behavior of the potential b.

In this note, we consider some elliptic systems on a smooth domain of $R^n$. By using the maximum principle, we can get a more general and complete results of the identical property of positive solution pair, and thus classify the structure of all positive solutions depending on the nonlinarities easily.

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 $i=1,2,...,n$.

Let $f:X\to X$ be a continuous map with the specification property on a compact metric space $X$. We introduce the notion of the maximal Birkhoff average oscillation, which is the “worst” divergence point for Birkhoff average. By constructing a kind of dynamical Moran subset, we prove that the set of points having maximal Birkhoff average oscillation is residual if it is not empty. As applications, we present the corresponding results for the Birkhoff averages for continuous functions on a repeller...

Sufficient conditions are obtained so that every solution of ${[y\left(t\right)-p\left(t\right)y(t-\tau )]}^{\left(n\right)}+Q\left(t\right)G\left(y(t-\sigma )\right)=f\left(t\right)$ where n ≥ 2, p,f ∈ C([0,∞),ℝ), Q ∈ C([0,∞),[0,∞)), G ∈ C(ℝ,ℝ), τ > 0 and σ ≥ 0, oscillates or tends to zero as $t\to \infty$. Various ranges of p(t) are considered. In order to accommodate sublinear cases, it is assumed that ${\int }_0^{\infty }Q\left(t\right)dt=\infty$. Through examples it is shown that if the condition on Q is weakened, then there are sublinear equations whose solutions tend to ±∞ as t → ∞.