Let $X$ and $Y$ be vector spaces over the same field $K$. Following the terminology of Richard Arens [Pacific J. Math. 11 (1961), 9–23], a relation $F$ of $X$ into $Y$ is called linear if $\lambda F\left(x\right)\subset F\left(\lambda x\right)$ and $F\left(x\right)+F\left(y\right)\subset F(x+y)$ for all $\lambda \in K\setminus \left\{0\right\}$ and $x,y\in X$. After improving and supplementing some former results on linear relations, we show that a relation $\Phi$ of a linearly independent subset $E$ of $X$ into $Y$ can be extended to a linear relation $F$ of $X$ into $Y$ if and only if there exists a linear subspace $Z$ of $Y$ such that $\Phi \left(e\right)\in Y|Z$ for all $e\in E$. Moreover, if...

We consider the equation $$-{\left(r\left(x\right)y^{\text{'}}\left(x\right)\right)}^{\text{'}}+q\left(x\right)y\left(x\right)=f\left(x\right),x\in ℝ(*)$$ where $f\in L_p\left(ℝ\right)$, $p\in (1,\infty )$ and $$\begin{array}{c}r>0,q\ge 0,\frac{1}{r}\in L_1^{\mathrm{loc}}\left(ℝ\right),q\in L_1^{\mathrm{loc}}\left(ℝ\right),\\ \underset{\left|d\right|\to \infty }{lim}{\int }_{x-d}^x\frac{\mathrm{d}t}{r\left(t\right)}·{\int }_{x-d}^{x}q\left(t\right)\mathrm{d}t=\infty .\end{array}$$ In an earlier paper, we obtained a criterion for correct solvability of ($*$) in $L_p\left(ℝ\right),$ $p\in (1,\infty ).$ In this criterion, we use values of some auxiliary implicit functions in the coefficients $r$ and $q$ of equation ($*$). Unfortunately, it is usually impossible to compute values of these functions. In the present paper we obtain sharp by order, two-sided estimates (an estimate of a function $f\left(x\right)$ for $x\in (a,b)$ through a function $g\left(x\right)$ is sharp by order if $c^{-1}\left|g\left(x\right)\right|\le \left|f\left(x\right)\right|\le c\left|g\left(x\right)\right|,$ ...

Let $G$ be a locally compact group. We continue our work [A. Ghaffari: $\Gamma$-amenability of locally compact groups, Acta Math. Sinica, English Series, 26 (2010), 2313–2324] in the study of $\Gamma$-amenability of a locally compact group $G$ defined with respect to a closed subgroup $\Gamma$ of $G\times G$. In this paper, among other things, we introduce and study a closed subspace $A_{\Gamma }^p\left(G\right)$ of $L^{\infty }\left(\Gamma \right)$ and then characterize the $\Gamma$-amenability of $G$ using $A_{\Gamma }^p\left(G\right)$. Various necessary and sufficient conditions are found for a locally compact...

$\alpha$-normality and $\beta$-normality are properties generalizing normality of topological spaces. They consist in separating dense subsets of closed disjoint sets. We construct an example of a Tychonoff $\beta$-normal non-normal space and an example of a Hausdorff $\alpha$-normal non-regular space.