If $(\Omega ,\Sigma ,\mu )$ is a finite measure space and $X$ a Banach space, in this note we show that $L_{w^{*}}^1(\mu ,X^{*})$, the Banach space of all classes of weak* equivalent $X^{*}$-valued weak* measurable functions $f$ defined on $\Omega$ such that $\parallel f\left(\omega \right)\parallel \le g\left(\omega \right)$ a.e. for some $g\in L_1\left(\mu \right)$ equipped with its usual norm, contains a copy of $c_0$ if and only if $X^{*}$ contains a copy of $c_0$.

The paper deals with the periodic boundary value problem (1) $L_{4}x\left(t\right)+a\left(t\right)x\left(t\right)\in F(t,x\left(t\right))$, $t\in J=[a,b]$, (2) $L_{i}x\left(a\right)=L_{i}x\left(b\right)$, $i=0,1,2,3$, where $L_{0}x\left(t\right)=a_{0}x\left(t\right)$, $L_{i}x\left(t\right)=a_i\left(t\right)L_{i-1}x\left(t\right)$, $i=1,2,3,4$, $a_0\left(t\right)=a_4\left(t\right)=1$, $a_i\left(t\right)$, $i=1,2,3$ and $a\left(t\right)$ are continuous on $J$, $a\left(t\right)\ge 0$, $a_i\left(t\right)>0$, $i=1,2$, $a_1\left(t\right)=a_3\left(t\right)·F(t,x):J\times R\to${nonempty convex compact subsets of $R$}, $R=(-\infty ,\infty )$. The existence of such periodic solution is proven via Ky Fan’s fixed point theorem.

Let $X$ be a reflexive Banach space and $A$ be a closed operator in an $L_1$-space of $X$-valued functions. Then we characterize the range $R\left(A\right)$ of $A$ as follows. Let $0\ne {\lambda }_n\in \rho \left(A\right)$ for all $1\le n<\infty$, where $\rho \left(A\right)$ denotes the resolvent set of $A$, and assume that ${lim}_{n\to \infty }{\lambda }_n=0$ and ${sup}_{n\ge 1}\parallel {\lambda }_n{({\lambda }_n-A)}^{-1}\parallel <\infty$. Furthermore, assume that there exists ${\lambda }_{\infty }\in \rho \left(A\right)$ such that $\parallel {\lambda }_{\infty }{({\lambda }_{\infty }-A)}^{-1}\parallel \le 1$. Then $f\in R\left(A\right)$ is equivalent to ${sup}_{n\ge 1}{\parallel {({\lambda }_n-A)}^{-1}f\parallel }_1<\infty$. This generalizes Shaw’s result for scalar-valued functions.

Integral criteria are established for $dim{V}_i\left(p\right)=0$ and $dim{V}_i\left(p\right)=1,i\in \{0,1\}$, where $V_i\left(p\right)$ is the space of solutions $u$ of the equation $$u^{\text{'}\text{'}}+p\left(t\right)u=0$$ satisfying the condition $${\int }^{+\infty }\frac{u^2\left(s\right)}{s^i}ds<+\infty .$$