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The Schaefer method in the theory of Hammerstein integral equations
P. P. Zabreiko
Abstract:
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
$u^2$.
Bibliography: 10 titles.
UDC:
517.948.33
MSC: Primary
45G05; Secondary
45A05,
46E30,
47G05,
47H15 Received: 12.03.1970
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