The Schaefer method in the theory of Hammerstein integral equations

The Hammerstein integral equation
\begin{equation} x(t)=\int_\Omega k(t,s)f[s, x(s)]\,dt+g(t) \end{equation}
is studied. It is assumed that the linear integral operator $K$ with symmetric kernel $k(t,s)$ acts and is completely continuous or the Hilbert space $H=L_2$. Furthermore, it is assumed that $E_0$ and $E$ ($E_0\subset E\subset H$) are ideal spaces for which the following conditions are fulfilled: a) the operator $K$ acts on the dual space $E'_0$; b) the eigenfunctions of $K$ lie in $E_0$; c) the linear span of the eigenfunctions of $K$ is dense in $E_0$ in the sense of $o$-covergence; d) the operator $~K$ acts from $E_0$ to $E'_0$ (and is completely continuous); e) the operator $f$ acts from $E_0$ to $E'_0$ and transforms bounded sets into $E_0$-weakly sequentially compact sets (acts from $E_0$ to $E'_0$). It is proved that under these hypotheses in the case of a positive definite $K$ a sufficient condition for the solvability of equation $(1)$ is the inequality
\begin{equation} uf(s,u)\leqslant au^2+\omega(s,u) \end{equation}
where $a\lambda<1$ ($\lambda$ is the largest eigenvalue of $K$) and $\omega (s,u)$ contains terms that grow at infinity more slowly than $u^2$.
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