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.

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.

Let $X$ be an Archimedean Riesz space and $𝒫\left(X\right)$ its Boolean algebra of all band projections, and put ${𝒫}_e=\{Pe:P\in 𝒫\left(X\right)\}$ and ${ℬ}_e=\{x\in X:x\wedge (e-x)=0\}$, $e\in X^{+}$. $X$ is said to have Weak Freudenthal Property ($WFP$) provided that for every $e\in X^{+}$ the lattice $lin{𝒫}_e$ is order dense in the principal band $e^{dd}$. This notion is compared with strong and weak forms of Freudenthal spectral theorem in Archimedean Riesz spaces, studied by Veksler and Lavrič, respectively. $WFP$ is equivalent to $X^{+}$-denseness of ${𝒫}_e$ in ${ℬ}_e$ for every $e\in X^{+}$, and every Riesz space with sufficiently many...