Abstract

In this paper it is demonstrated that the inequality $$\biggl(\int_G|u|^p\,dx\biggr)^{1/p}\leq A_p\biggl(\int_D|\nabla u|^p\,dx \biggr)^{1/p},\quad u|_{\partial D}=0,1\leq p\leq\infty$$ holds, where $G\subset D\subset R^2$, $D$ is a convex domain and constant $A_p$ is expressed in terms of areas of $G$ and~$D$.

Keywords: Poincaré-Friedrich's inequality, $L^p$-space.

MSC: 26D10, 35P15

Abstract

The problem of necessary conditions for the existence of periodical solutions of the system $X'=\f(t)P(X)+Q(t)$ is considered, where $\f(t)$, $Q_i(t)$ ($i=1,\dots,n$) are continuous functions belonging to certain classes.

Keywords: Periodical solution, differential system.

MSC: 34C25

Abstract

In this paper, we study an initial value problem for a one-dimensional system of thermoelasticity. Using an a priori bound and a density argument, we prove the existence and uniqueness of a generalized solution.

Keywords: Coupled system, thermoelasticiy, a priori bound.

MSC: 35K22, 58D25, 73B30

Abstract

In this article we consider stability of nonlinear equations which have the following form: $$Ax+F(x)=b, \tag1$$ where $F$ is any function, $A$ is a linear operator, $b$ is given and $x$ is an unknown vector. We give (under some assumptions about function $F$ and operator $A$) a generalization of inequality: $$\frac{\|X_{1}-X_{2}\|}{\|X_{1}\|}\leq \|A\|\|A^{-1}\|\frac{\|b_{1}-b_{2}\|}{\|b_{1}\|} \tag2$$ (equation (2) estimates the relative error of the solution when the linear equation $Ax=b_{1}$ becomes the equation $Ax=b_{2}$) and a generalization of inequality: $$\frac{\|X_{1}-X_{2}\|}{\|X_{1}\|}\leq \|A_{1}^{-1}\|\|A_{1}\|\left(\frac{\|b_{1}-b_{2}\|}{\|b_{1}\|}+ \|A_{1}\|\|A_{2}^{-1}\|\frac{\|b_{2}\|}{\|b_{1}\|}\cdot \frac{\|A_{1}-A_{2}\|}{\|A_{1}\|}\right) \tag3$$ (equation (3) estimates the relative error of the solution when the linear equation $A_1x=b_{1}$ becomes the equation $A_2x=b_{2}$).

Keywords: Numerical stability, nonlinear equations.

MSC: 65J15

Abstract

In this paper we give a mapping theorems on $\aleph$-spaces by means of strong compact-covering mappings and $\sigma$-mappings. As an application, we get characterizations of quotient (pseudo-open) $\sigma$-images of metric spaces.

Keywords: $\aleph$-spaces; strong compact-covering mappings; $\sigma$-mappings; $k$-spaces; Fréchet spaces.

MSC: 54E99, 54C10

Abstract

In this paper we give some properties of Banach algebras of bounded operators $B(X)$, when $X$ is a BK space. We then study the solvability of the equation $Ax=b$ for $b\in\{s_{\alpha },s_{\alpha}^{{{}^{\circ}}},s_{\alpha }^{( c)},l_{p}( \alpha)\}$ with $\alpha\in U^{+}$ and $1\leq p<\infty$. We then deal with the equation $T_{a}x=b$, where $b\in\chi(\Delta ^{k})$ for $k\geq 1$ integer, $\chi\in\{s_{\alpha },s_{\alpha }^{{{}^{\circ}}},s_{\alpha}^{(c)},l_{p}(\alpha)\}$, $1\leq p<\infty$ and $T_{a}$ is a Toeplitz triangle matrix. Finally we apply the previous results to infinite tridiagonal matrices and explicitly calculate the inverse of an infinite tridiagonal matrix. These results generalize those given in [4,~9].

Keywords: Infinite linear system, sequence space, BK space, Banach algebra, bounded operator.

MSC: 40C05, 46A45