### An existence theorem for Hammerstein integral equations.

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This note is an announcement of results contained in the papers [4], [5], [6] concerning isomorphic properties of Banach spaces in projective tensor products (for this definition and some property we refer to [1]). At the end, some new result is obtained too.

We show that the equality $W(E,{F}^{*})=K(E,{F}^{*})$ is a necessary condition for the validity of certain results about isomorphic properties in the projective tensor product $E{\otimes}_{\pi}F$ of two Banach spaces under some approximation property type assumptions.

In the paper [5] L. Drewnowski and the author proved that if $X$ is a Banach space containing a copy of ${c}_{0}$ then ${L}_{1}(\mu ,X)$ is complemented in $cabv(\mu ,X)$ and conjectured that the same result is true if $X$ is any Banach space without the Radon-Nikodym property. Recently, F. Freniche and L. Rodriguez-Piazza ([7]) disproved this conjecture, by showing that if $\mu $ is a finite measure and $X$ is a Banach lattice not containing copies of ${c}_{0}$, then ${L}_{1}(\mu ,X)$ is complemented in $cabv(\mu ,X)$. Here, we show that the complementability of ${L}_{1}(\mu ,X)$ in $cabv(\mu ,X)$ together...

We show that as soon as ${c}_{0}$ embeds complementably into the space of all weakly compact operators from $X$ to $Y$, then it must live either in ${X}^{*}$ or in $Y$.

We consider a perturbed Cauchy problem like the following $$\text{(PCP)}\left\{\begin{array}{c}{x}^{\text{'}}=A(t,x)+B(t,x)\phantom{\rule{4pt}{0ex}}x\left(0\right)={x}_{0}\hfill \end{array}\right.$$ and we present two results showing that (PCP) has a solution. In some cases, our theorems are more general than the previous ones obtained by other authors (see [4], [8], [9], [11], [13], [17], [18]).

We show results about the existence and the nonexistence of a projection from the space $L({L}^{1}\left(\lambda \right),X)$ of all linear and bounded operators from ${L}^{1}\left(\lambda \right)$ into $X$ onto the subspace $R({L}^{1}\left(\lambda \right),X)$ of all representable operators.

Using some known lifting theorems we present three-space property type and permanence results; some of them seem to be new, whereas other are improvements of known facts.

In the first part of the paper we prove some new result improving all those already known about the equivalence of the nonexistence of a projection (of any norm) onto the space of compact operators and the containment of ${c}_{0}$ in the same space of compact operators. Then we show several results implying that the space of compact operators is uncomplemented by norm one projections in larger spaces of operators. The paper ends with a list of questions naturally rising from old results and the results...

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