The surjectivity of the operator $D_2+i{x}_2^{2k}{D}_1$ from the Gevrey space ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^2\right)$, $s\ge 1$, onto itself and its non-surjectivity from ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^3\right)$ to ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^3\right)$ is proved.

Let α,β,γ,δ ≥ 0 and ϱ:= γβ + αγ + αδ > 0. Let ψ(t) = β + αt, ϕ(t) = γ + δ - γt, t ∈ [0,1]. We study the existence of positive solutions for the m-point boundary value problem ⎧u” + h(t)f(u) = 0, 0 < t < 1, ⎨$\alpha u\left(0\right)-\beta u^{\text{'}}\left(0\right)={\sum }_{i=1}^{m-2}{a}_{i}u\left({\xi }_i\right)$ ⎩$\gamma u\left(1\right)+\delta u^{\text{'}}\left(1\right)={\sum }_{i=1}^{m-2}{b}_{i}u\left({\xi }_i\right)$, where ${\xi }_i\in (0,1)$, $a_i,b_i\in (0,\infty )$ (for i ∈ 1,…,m-2) are given constants satisfying $\varrho -{\sum }_{i=1}^{m-2}{a}_i\varphi \left({\xi }_i\right)>0$, $\varrho -{\sum }_{i=1}^{m-2}{b}_i\psi \left({\xi }_i\right)>0$ and $\Delta :=\left|\begin{array}{cc}-{\sum }_{i=1}^{m-2}{a}_i\psi \left({\xi }_i\right)& \varrho -{\sum }_{i=1}^{m-2}{a}_i\varphi \left({\xi }_i\right)\\ \varrho -{\sum }_{i=1}^{m-2}{b}_i\psi \left({\xi }_i\right)& -{\sum }_{i=1}^{m-2}{b}_i\varphi \left({\xi }_i\right)\end{array}\right|<0$. We show the existence of positive solutions if f is either superlinear or sublinear by a simple application of a fixed point theorem in cones. Our result extends a result established by Erbe and...

The problem of uniqueness of an entire or a meromorphic function when it shares a value or a small function with its derivative became popular among the researchers after the work of Rubel and Yang (1977). Several authors extended the problem to higher order derivatives. Since a linear differential polynomial is a natural extension of a derivative, in the paper we study the uniqueness of a meromorphic function that shares one small function CM with a linear differential polynomial, and...

For a real number $q>1$ and a positive integer $m$, let $Y_m\left(q\right):=\left\{{\sum }_{i=0}^n{ϵ}_i{q}^i:{ϵ}_i\in \left\{0,\pm 1,...,\pm m\right\},n=0,1,...\right\}$. In this paper, we show that $Y_m\left(q\right)$ is dense in $ℝ$ if and only if $q<m+1$ and $q$ is not a Pisot number. This completes several previous results and answers an open question raised by Erdös, Joó and Komornik [8].

Let: $𝐘=\left({𝐘}_i\right)$, where ${𝐘}_i=\left(Y_{i,1},...,Y_{i,d}\right)$, $i=1,2,\cdots$, be a $d$-dimensional, identically distributed, stationary, centered process with uniform marginals and a joint cdf $F$, and $F_n\left(𝐱\right):=\frac{1}{n}{\sum }_{i=1}^n𝕀\left(Y_{i,1}\le x_1,\cdots ,Y_{i,d}\le x_d\right)$ denote the corresponding empirical cdf. In our work, we prove the almost sure central limit theorem for an empirical process $B_n=\sqrt{n}\left(F_n-F\right)$ under some weak dependence conditions due to Doukhan and Louhichi. Some application of the established result to copula processes is also presented.

Let $A$ be a closed set of $M\simeq {\mathbb{R}}^n$, whose conormai cones $x+y_x^{*}\left(A\right)$, $x\in A$, have locally empty intersection. We first show in §1 that $\text{dist}\left(x,A\right)$, $x\in M\setminus A$ is a $C^1$ function. We then represent the n microfunctions of ${\mathcal{C}}_{A|X}$, $X\simeq {\mathbb{C}}^n$, using cohomology groups of ${\mathcal{O}}_X$ of degree 1. By the results of § 1-3, we are able to prove in §4 that the sections of ${\left.{\mathcal{C}}_{A|X}\right|}_{{\dot{\pi }}^{-1}\left(x_0\right)}$, $x_0\in \partial A$, satisfy the principle of the analytic continuation in the complex integral manifolds of ${\left\{H\left({\varphi }_i^C\right)\right\}}_{i=1,\mathrm{\dots },m}$, $\left\{{\varphi }_i\right\}$ being a base for the linear hull of ${\gamma }_{x_0}^{*}\left(A\right)$ in $T_{x_0}^{*}M$; in particular we get ${\left.{\mathrm{\Gamma }}_{A\times {}_{M}T{}^{*}{}_{M}X}\left({\mathcal{C}}_{A|X}\right)\right|}_{\partial A\times {}_M\dot{T}{}^{*}{}_{M}X}=0$. When $A$is a half space with...

This paper concerns the following system of nonlinear third-order boundary value problem: $$u_i^{\text{'}\text{'}\text{'}}\left(t\right)+f_i(t,u_1\left(t\right),\cdots ,u_n\left(t\right),u_1^{\text{'}}\left(t\right),\cdots ,u_n^{\text{'}}\left(t\right))=0,0<t<1,i\in \{1,\cdots ,n\}$$ with the following multi-point and integral boundary conditions: $$\left\{\begin{array}{c}{u}_i\left(0\right)=0u_i^{\text{'}}\left(0\right)=0u_i^{\text{'}}\left(1\right)=\sum _{j=1}^p{\beta }_{j,i}{u}_i^{\text{'}}\left({\eta }_{j,i}\right)+{\int }_0^1{h}_i(u_1\left(s\right),\cdots ,u_n\left(s\right))ds\hfill \end{array}\right.$$ where ${\beta }_{j,i}>0$, $0<{\eta }_{1,i}<\cdots <{\eta }_{p,i}<\frac{1}{2}$, $f_i:[0,1]\times {ℝ}^n\times {ℝ}^n\to ℝ$ and $h_i:[0,1]\times {ℝ}^n\to ℝ$ are continuous functions for all $i\in \{1,\cdots ,n\}$ and $j\in \{1,\cdots ,p\}$. Using Guo-Krasnosel’skii fixed point theorem in cone, we discuss the existence of positive solutions of this problem. We also prove nonexistence of positive solutions and we give some examples to illustrate our results.