The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type $$\Delta x\left(n\right)+\sum _{k=-p}^q{a}_k\left(n\right)x(n+k)=0,n>n_0,$$ where $\Delta x\left(n\right)=x(n+1)-x\left(n\right)$ is the difference operator and $\left\{a_k\left(n\right)\right\}$ are sequences of real numbers for $k=-p,...,q$, and $p>0$, $q\ge 0$. We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced.

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 → ∞.

We study a general class of nonlinear elliptic problems associated with the differential inclusion $\beta \left(u\right)-div\left(a\right(x,Du)+F(u\left)\right)\ni f$ in Ω where $f\in L^{\infty }\left(\Omega \right)$. The vector field a(·,·) is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in function spaces we prove existence of renormalized solutions for general $L^{\infty }$-data.

We establish the existence of a T-p(x)-solution for the p(x)-elliptic problem $-div\left(a\right(x,u,\nabla u\left)\right)+g(x,u)=f-divF$ in Ω, where Ω is a bounded open domain of ${ℝ}^N$, N ≥ 2 and $a:\Omega \times ℝ\times {ℝ}^N\to {ℝ}^N$ is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but with only a weak monotonicity condition. The right hand side f lies in L¹(Ω) and F belongs to ${\prod }_{i=1}^N{L}^{p^{\text{'}}\left(·\right)}\left(\Omega \right)$.

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 work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type $${\left(r\left(t\right){\left(z^{\text{'}}\left(t\right)\right)}^{\gamma }\right)}^{\text{'}}+\sum _{i=1}^m{q}_i\left(t\right)x^{{\alpha }_i}\left({\sigma }_i\left(t\right)\right)=0,t\ge t_0,$$ where $z\left(t\right)=x\left(t\right)+p\left(t\right)x\left(\tau \right(t\left)\right)$. Under the assumption ${\int }^{\infty }{\left(r\left(\eta \right)\right)}^{-1/\gamma }\mathrm{d}\eta =\infty$, we consider two cases when $\gamma >{\alpha }_i$ and $\gamma <{\alpha }_i$. Our main tool is Lebesgue’s dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem.

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.

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...

This paper is concerned with the oscillatory behavior of the damped half-linear oscillator ${\left(a\left(t\right){\phi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}+b\left(t\right){\phi }_p\left(x^{\text{'}}\right)+c\left(t\right){\phi }_p\left(x\right)=0$, where ${\phi }_p\left(x\right)={\left|x\right|}^{p-1}\mathrm{sgn}x$ for $x\in ℝ$ and $p>1$. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial...

We study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis $\left(e_k\right)$ is said to be subsequentially minimal if for every normalized block basis $\left(x_k\right)$ of $\left(e_k\right)$, there is a further block basis $\left(y_k\right)$ of $\left(x_k\right)$ such that $\left(y_k\right)$ is equivalent to a subsequence of $\left(e_k\right)$. Sufficient conditions are given for a partly modified mixed Tsirelson space to be subsequentially minimal, and connections with Bourgain’s ℓ¹-index are established. It is also shown that a large class of...

We consider a random, uniformly elliptic coefficient field $a$ on the lattice ${\mathbb{Z}}^d$. The distribution $\langle ·\rangle$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ and $L^1$, respectively, in probability, i.e., they obtained bounds on $\langle |{\nabla }_{x}G(t,x,y){{|}^2\rangle }^{1/2}$ and $\langle \left|{\nabla }_x{\nabla }_{y}G(t,x,y)\right|\rangle$. In particular, the elliptic Green’s function $G(x,y)$ satisfies optimal annealed bounds. In their recent work,...

This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by $$\begin{array}{cccc}\hfill t2& -{\mathrm{div}\left(b\right(\left|u\right|\left)\right|\nabla u|}^{p-2}{\nabla u)+d(\left|u\right|\left)\right|\nabla u|}^p=f-{\mathrm{div}\left(c\left(x\right)\right|u|}^{\alpha })\hfill & \hfill & \text{in}\Omega ,\hfill \\ & u=0\hfill & \hfill & \text{on}\partial \Omega ,\hfill \end{array}t$$ where $\Omega$ is a bounded open set of ${ℝ}^N$ ($N\ge 2$) with $1<p<N$ and $f\in L^1\left(\Omega \right),$ under some growth conditions on the function $b(·)$ and $d(·),$ where $c(·)$ is assumed to be in $L^{\frac{N}{(p-1)}}\left(\Omega \right).$ We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded.