We investigate differences of composition operators acting between weighted spaces of holomorphic functions defined on the open unit ball of a Banach space. We give necessary and sufficient conditions for such operators to be bounded resp. compact.

In this paper, the authors study the oscillatory and asymptotic properties of solutions of nonlinear fourth order neutral dynamic equations of the form \begin{equation} \tag{H} (r(t)(y(t)+p(t)y(\alpha(t)))^{{\Delta }^2})^{{\Delta }^2} + q(t)G(y(\beta(t)))h(t)H(y(\gamma(t)))=0 \end{equation} and \begin{equation} \tag{NH} (r(t)(y(t)+p(t)y(\alpha(t)))^{{\Delta }^2})^{{\Delta }^2} + q(t)G(y(\beta(t)))h(t)H(y(\gamma(t)))=f(t), \end{equation} where $\mathbb {T}$ is a time scale with $\sup \mathbb {T}=\infty$, $t \in [t_0,\infty)_\mathbb{T}$, and $t_0\geqslant 0$. They assume that $\int _{t_0}^\infty \frac{\sigma (t)}{r(t)}\Delta t \lt \infty$ and obtain results for various ranges of values of $p(t)$. Examples illustrating the results are included.

We use variational methods to study the existence of at least one positive solution of the following Schrödinger-Poisson system $$ \left\{ \begin{array}{ll} \Delta u +u +l(x)\phi u = k(x)|u|^{{2^*}2}u +\mu h(x)|u|^{q2}u \quad & \ \hbox{in}\ \mathbb{R}^3,\\ \\ \Delta \phi = l(x)u^2\quad & \ \hbox{in} \ \mathbb{R}^3, \end{array} \right. $$ under some suitable conditions on the nonnegative functions $l, k, h$ and constant $\mu\gt 0$, where $2\leq q\lt 2^*$ (critical Sobolev exponent).

The intent of this paper is to develop a framework for discrete calculus of variations with action densities involving a new class of discretization operators. We introduce first the generalized scale derivatives, study their regularity and state some Leibniz formulas. Then, we deduce the discrete EulerLagrange equations for critical points of sampled actions that we compare to existing versions. Next, we investigate the case of general quadratic lagrangians and provide two examples of such lagrangians. At last, we find nontrivial properties for the discretization of a quadratic nulllagrangian.

Motivated by the theory of bisingular pseudodifferential operators, we introduce a two-dimensional version of the Adler-Manin trace. Our construction is rather general in the sense that it involves a twist afforded by an algebra automorphism. That is, starting from an algebra equipped with an automorphism, two twisted derivations, and a twisted invariant trace, we construct an algebra of formal twisted pseudodifferential symbols and define a noncommutative residue. Also, we provide related examples.

In this paper, we study the asymptotic behavior of neutral advanced difference equations of the form \begin{equation*} \Delta ^{m}\left[ x(n)+cx(n+a)\right] +p(n)x(\sigma (n))=0,\qquad m\in \mathbb{N}, \qquad n\geq 0\text{,} \end{equation*} where $c\in \mathbb{R} $, $ \mathbb{N} \ni a\geq 2$, $(\sigma (n))$ is a sequence of positive integers such that $% (\sigma (n))\geq n+2$\ for all $n\geq 0$, $\left( p(n)\right) _{n\geq 0}$ is a sequence of real numbers, $\Delta $ denotes the forward difference operator $\Delta x(n)=x(n+1)x(n)$, and $\Delta ^{j}$ denotes the $j^{th}$ forward difference operator $\Delta ^{j}\left( x(n)\right) =\Delta \left( \Delta ^{j1}\left( x(n)\right) \right) $ for $j=2,3,...,m$. Examples illustrating the results are also given.

In this paper, we consider the Cauchy problem of two models of the theory of heat conduction with three-phase-lag. Under appropriate assumptions on the material parameters, we show the optimal decay rate of the $L^2$norm of solutions. More precisely, we prove that in each model the $L^2$norm of the solution is decaying with the rate $(1+t)^{1/4}$ for initial data in $L^1(\mathbb{R})$. This decay rate is similar to the one of the heat kernel. Some faster decay rates have been also given for some weighted initial data in $L^1(\mathbb{R})$.

We prove pointwise estimates for the heat kernel of a secondorder elliptic operator with Dirichlet boundary conditions on a bounded Lipschitz domain in $\mathbb{R}^n,\, n\geq 1$. Applications to obtain estimates for intrinsic ultracontractivity of the heat semigroup and expected exit time of a Brownian motion are given.