Let $D$ be a nonempty compact subset of a Banach space $X$ and denote by $S\left(X\right)$ the family of all nonempty bounded closed convex subsets of $X$. We endow $S\left(X\right)$ with the Hausdorff metric and show that there exists a set $ℱ\subset S\left(X\right)$ such that its complement $S\left(X\right)\setminus ℱ$ is $\sigma$-porous and such that for each $A\in ℱ$ and each $\tilde{x}\in D$, the set of solutions of the best approximation problem $\parallel \tilde{x}-z\parallel \to min$, $z\in A$, is nonempty and compact, and each minimizing sequence has a convergent subsequence.

We consider the Robin eigenvalue problem $\Delta u+\lambda u=0$ in $\Omega$, $\partial u/\partial \nu +\alpha u=0$ on $\partial \Omega$ where $\Omega \subset {ℝ}^n$, $n\ge 2$ is a bounded domain and $\alpha$ is a real parameter. We investigate the behavior of the eigenvalues ${\lambda }_k\left(\alpha \right)$ of this problem as functions of the parameter $\alpha$. We analyze the monotonicity and convexity properties of the eigenvalues and give a variational proof of the formula for the derivative ${\lambda }_1^{\text{'}}\left(\alpha \right)$. Assuming that the boundary $\partial \Omega$ is of class $C^2$ we obtain estimates to the difference ${\lambda }_k^D-{\lambda }_k\left(\alpha \right)$ between the $k$-th eigenvalue of the Laplace operator with...

In this paper we deal with the Cauchy problem for differential inclusions governed by $m$-accretive operators in general Banach spaces. We are interested in finding the sufficient conditions for the existence of integral solutions of the problem $x^{\text{'}}\left(t\right)\in -Ax\left(t\right)+f(t,x\left(t\right))$, $x\left(0\right)=x_0$, where $A$ is an $m$-accretive operator, and $f$ is a continuous, but non-compact perturbation, satisfying some additional conditions.

Given a space $X$, its $G_{\delta }$-subsets form a basis of a new space $X_{\omega }$, called the $G_{\delta }$-modification of $X$. We study how the assumption that the $G_{\delta }$-modification $X_{\omega }$ is homogeneous influences properties of $X$. If $X$ is first countable, then $X_{\omega }$ is discrete and, hence, homogeneous. Thus, $X_{\omega }$ is much more often homogeneous than $X$ itself. We prove that if $X$ is a compact Hausdorff space of countable tightness such that the $G_{\delta }$-modification of $X$ is homogeneous, then the weight $w\left(X\right)$ of $X$ does not exceed $2^{\omega }$ (Theorem 1)....