This paper discusses the existence and multiplicity of solutions for a class of $p\left(x\right)$-Kirchhoff type problems with Dirichlet boundary data of the following form $$\left\}\begin{array}{cc}-\left(a+b{\int }_{\Omega }\frac{1}{p\left(x\right)}{\left|\nabla u\right|}^{p\left(x\right)}dx\right){\mathrm{div}\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u)=f(x,u),\hfill & in\Omega \hfill \\ u=0\hfill & on\partial \Omega ,\hfill \end{array}\right.$$ where $\Omega$ is a smooth open subset of ${ℝ}^N$ and $p\in C\left(\overline{\Omega }\right)$ with $N<p^{-}={inf}_{x\in \Omega }p\left(x\right)\le p^{+}={sup}_{x\in \Omega }p\left(x\right)<+\infty$, $a$, $b$ are positive constants and $f:\overline{\Omega }\times ℝ\to ℝ$ is a continuous function. The proof is based on critical point theory and variable exponent Sobolev space theory.

It is known that the removal of an edge from a graph G cannot decrease a domination number γ(G) and can increase it by at most one. Thus we can write that γ(G) ≤ γ(G-e) ≤ γ(G)+1 when an arbitrary edge e is removed. Here we present similar inequalities for the weakly connected domination number ${\gamma }_w$ and the connected domination number ${\gamma }_c$, i.e., we show that ${\gamma }_w\left(G\right)\le {\gamma }_w(G-e)\le {\gamma }_w\left(G\right)+1$ and ${\gamma }_c\left(G\right)\le {\gamma }_c(G-e)\le {\gamma }_c\left(G\right)+2$ if G and G-e are connected. Additionally we show that ${\gamma }_w\left(G\right)\le {\gamma }_w(G-Eₚ)\le {\gamma }_w\left(G\right)+p-1$ and ${\gamma }_c\left(G\right)\le {\gamma }_c(G-Eₚ)\le {\gamma }_c\left(G\right)+2p-2$ if G and G - Eₚ are connected and Eₚ = E(Hₚ) where Hₚ of order...

Duffus wrote in his 1978 Ph.D. thesis, “It is not obvious that $P$ is connected and $P^P\cong Q^Q$ imply that $Q$ is connected”, where $P$ and $Q$ are finite nonempty posets. We show that, indeed, under these hypotheses $Q$ is connected and $P\cong Q$.

This paper contains a systematic analysis of a natural measure theoretic notion of connectedness for sets of finite perimeter in ${ℝ}^N$, introduced by H. Federer in the more general framework of the theory of currents. We provide a new and simpler proof of the existence and uniqueness of the decomposition into the so-called $M$-connected components. Moreover, we study carefully the structure of the essential boundary of these components and give in particular a reconstruction formula of a set...

Let $P\left(z\right)={\sum }_{\nu =0}^n{a}_{\nu }{z}^{\nu }$ be a polynomial of degree at most $n$ which does not vanish in the disk $\left|z\right|<1$, then for $1\le p<\infty$ and $R>1$, Boas and Rahman proved $${∥P\left(Rz\right)∥}_p\le ({∥R^n+z∥}_p/{∥1+z∥}_p){∥P∥}_p.$$ In this paper, we improve the above inequality for $0\le p<\infty$ by involving some of the coefficients of the polynomial $P\left(z\right)$. Analogous result for the class of polynomials $P\left(z\right)$ having no zero in $\left|z\right|>1$ is also given.

Let $\mu$ be a probability measure on $[0,1]$ which is invariant and ergodic for $T_a\left(x\right)=ax\mathrm{𝚖𝚘𝚍}1$, and $0<\mathrm{𝚍𝚒𝚖}\mu <1$. Let $f$ be a local diffeomorphism on some open set. We show that if $E\subseteq ℝ$ and $\left(f\mu \right){\left|{}_E\sim \mu \right|}_E$, then $f^{\text{'}}\left(x\right)\in \left\{\pm a^r:r\in \mathbb{Q}\right\}$ at $\mu$-a.e. point $x\in f^{-1}E$. In particular, if $g$ is a piecewise-analytic map preserving $\mu$ then there is an open $g$-invariant set $U$ containing supp $\mu$ such that $g\left|{}_U\right.$ is piecewise-linear with slopes which are rational powers of $a$. In a similar vein, for $\mu$ as above, if $b$ is another integer and $a,b$ are not powers of a common integer, and if $\nu$ is...

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 $K\subset {ℝ}^m$ ($m\ge 2$) be a compact set; assume that each ball centered on the boundary $B$ of $K$ meets $K$ in a set of positive Lebesgue measure. Let $C_0^{\left(1\right)}$ be the class of all continuously differentiable real-valued functions with compact support in ${ℝ}^m$ and denote by ${\sigma }_m$ the area of the unit sphere in ${ℝ}^m$. With each $\varphi \in C_0^{\left(1\right)}$ we associate the function $$W_K\varphi \left(z\right)=\frac{1}{{\sigma }_m}\underset{{ℝ}^m\setminus K}{\to }\int \mathrm{g}rad\varphi \left(x\right)·\frac{z-x}{{|z-x|}^m}x$$ of the variable $z\in K$ (which is continuous in $K$ and harmonic in $K\setminus B$). $W_K\varphi$ depends only on the restriction ${\varphi |}_B$ of $\varphi$ to the boundary $B$ of $K$. This gives rise to a linear operator $W_K$...

We consider functions $u\in W_0^{m,1}\left(\Omega \right)$, where $\Omega \subset {ℝ}^N$ is a smooth bounded domain, and $m\ge 2$ is an integer. For all $j\ge 0,1\le k\le m-1$, such that $1\le j+k\le m$, we prove that $\frac{{\partial }^{i}u\left(x\right)}{d{\left(x\right)}^{m-j-k}}\in W_0^{k,1}\left(\Omega \right)$ with $∥{\partial }^k{\left(\frac{{\partial }^{i}u\left(x\right)}{d{\left(x\right)}^{m-j-k}}\right)}_{L^1\left(\Omega \right)}\le C∥u∥{}_{W^{m,1}\left(\Omega \right)}$, where $d$ is a smooth positive function which coincides with dist$(x,\partial \Omega )$ near $\partial \Omega$, and ${\partial }^l$ denotes any partial differential operator of order $l$.

In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X,d,\mu )$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B(x,r)$ converge to $f\left(x\right)$ when $r$ converges to $0$. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....