  This paper considers the existence and multiplicity of fixed points for the
integral operator \begin{equation*} {\mathcal{T}}u(t)=\lambda
\,\int_{0}^{T}k(t,s)\,f(s,u(s),u^{\prime }(s),\dots ,u^{(m)}(s))\,\dif s,\quad
t\in \lbrack 0,T]\equiv I, \end{equation*} where $\lambda >0$ is a positive
parameter, $k:I\times I\rightarrow \mathbb{R% }$ is a kernel function such that
$k\in W^{m,1}\left( I\times I\right) $, $m$ is a positive integer with $m\geq
1$, and $f:I\times \mathbb{R}^{m+1}\rightarrow \lbrack 0,+\infty \lbrack $ is a
$L^{1}$-Carath\'{e}odory function.
  The existence of solutions for these Hammerstein equations is obtained by
fixed point index theory on new type of cones. Therefore some assumptions must
hold only for, at least, one of the derivatives of the kernel or, even, for the
kernel, on a subset of the domain. Assuming some asymptotic conditions on the
nonlinearity $f$, we get sufficient conditions for multiplicity of solutions.
  Two examples will illustrate the potentialities of the main results, namely
the fact that the kernel function and/or some derivatives may only be positive
on some subintervals, which can degenerate to a point. Moreover, an application
of our method to general Lidstone problems improves the existent results on the
literature in this field.
