0$,
in which case, for each $y\in X$, there is a connected closed
subset $C$ of $(A+N)^{-1}(y)$ whose dimension at each point is at
least $m=i(A)$ and the projection $P$ maps $C$ onto of $N(A)$,
where $N$ is given by \eqref{e4.1}, or
\item[(iii)] $N(A)\ne \{0\}$ and if $F(s,-u)=-F(s,u)$ for
$(s,u)\in \mathbb{R}^+\times \mathbb{R}$, $k\|K_0\|<1$, and
if $S_0$ is the solution set of \eqref{e4.2} with $y=0$,
then, for any positive
real number $r$ and $B(0,r)=\{x\in H:\|x\| 0$. Let $F:\mathbb{R}^+\times
\mathbb{R}\to \mathbb{C}$ be a Caratheodory function such that
$$
|F(s,u)|\le c(s)+c_0(s)|u|,\quad s\in \mathbb{R}^+,\;u\in \mathbb{R}
$$
where $c(s)\in L_p(\mathbb{R}^+,\mathbb{R})$ and
$c_0(s)\in L_{\infty}(\mathbb{R}^+,\mathbb{C})$.
Then, either
\begin{itemize}
\item[(i)] the equation $Ax=0$ has a unique zero solution, i.e., $i(A)=0$,
in which case \eqref{e4.2} is approximation solvable for each $f\in X$,
$(A+N)^{-1}(\{f\})$ is compact for each $f\in X$ and the cardinal number
$\mathop{\rm card}(A+N)^{-1}(\{f\})$
is constant, finite and positive on each connected component of
$X\setminus (A+N)(\Sigma)$, or
\item[(ii)] $N(A)\ne \{0\}$, i.e., $i(A)=\mathop{\rm dim}N(A)>0$,
in which case, for each $f\in X$, there is a connected closed
subset $C$ of $(A+N)^{-1}(f)$ whose dimension at each point is at least
$m=i(A)$ and the projection $P$ maps $C$ onto of $N(A)$, where $N$
is given by \eqref{e4.1},
or
\item[(iii)] $N(A)\ne \{0\}$ and if $F(s,-u)=-F(s,u)$ for
$(s,u)\in \mathbb{R}^+\times \mathbb{R}$, and if $S_0$ is the solution set
of \eqref{e4.2} with $y=0$, then, for any positive real number $r$
and $B(0,r)=\{x\in H:\|x\| k$ with $c$ depending only on $\lambda, k(s)$ and $Q$. Hence,
it is uniquely solvable by the contraction principle. Since $A+N$
is $A$-proper with respect to $\Gamma _0$, the assertion follows
from Theorem \ref{thm2.1}(a).
\end{proof}
\subsection*{Acknowledgements}
This work was done while the author was on the
sabbatical leave at the Mathematics Department at Rutgers
University, New Brunswick, NJ.
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