\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 51, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/51\hfil A global description of solutions]
{A global description of solutions to nonlinear
perturbations of the Wiener-Hopf integral equations}
\author[P. S. Milojevi\'c\hfil EJDE-2006/51\hfilneg]
{Petronije S. Milojevi\'c }

\address{Petronije S. Milojevi\'c \newline
 Department of Mathematical Sciences and CAMS,
 New Jersey Institute of Technology, Newark, NJ, USA}
\email{pemilo@m.njit.edu}

\date{}
\thanks{Submitted February 22, 2006. Published April 18, 2006.}
\subjclass[2000]{47H15, 35L70, 35L75, 35J40}
\keywords{Number of solutions;
 covering dimension; Wiener-Hopf equations; \hfill\break\indent
 nonlinear; (pseudo) A-proper maps; surjectivity}

\begin{abstract}
 We establish the solvability, the number of solutions and
 the covering dimension of the solution set of nonlinear
 Wiener-Hopf equations.
 The induced linear mapping is assumed to be of nonnegative index,
 while the nonlinearities are such that projection like methods
 are applicable. Solvability of nonlinear integral equations on
 the real line has been also discussed.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

Consider a nonlinear perturbation of the Wiener-Hopf equation \begin{equation}
\lambda x(s)-\int_0^{\infty} k(s-t)x(t)dt+ (Nx)(s)=y(s)
\label{e1.1} \end{equation} where $k:\mathbb{R}\to \mathbb{C}$ is in
$L_1(\mathbb{R},\mathbb{C})$ and $y(s)\in
L_1(\mathbb{R}^+,\mathbb{C})$ and $N$ is a suitable nonlinear
mapping. The corresponding linear Wiener-Hopf
 equations is
\begin{equation}
\lambda x(s)-\int_0^{\infty} k(s-t)x(t)dt=y(s).  \label{e1.2}
\end{equation}
Let $i(\lambda)$ be the index of the homogeneous equation corresponding
to \eqref{e1.2} with $y(s)=0$.
Then, if $i(\lambda)>0$, the homogeneous equation has an
$i(\lambda)$-dimensional space of solutions, and the nonhomogeneous
linear equation \eqref{e1.2}
has infinitely many solutions for each y in a suitable space.
If $i(\lambda)=0$, then \eqref{e1.2} has a unique solution for each y. These results
have been proven in the seminal paper by Krein [9]. Detailed
study of \eqref{e1.2} can be found in Corduneanu \cite{c1}. Many problems in mathematical
physics lead to \eqref{e1.2}. In particular, it appears in studying questions of
transfer of radiant energy. We also note that the study of \eqref{e1.2} with the integral
over $\mathbb{R}$ is much simpler and is based on using integral
Fourier transform (see Section 5).

In this paper, we shall extend these results to the nonlinear
perturbed Wiener-Hopf equation \eqref{e1.1}. As mentioned above,
if $x(s)$  is a solution of \eqref{e1.2}, then one also has the
$i(\lambda)$-dimensional plane of solutions of \eqref{e1.2}
consisting of $\{ x(s)+ x_0(s): x_0(s) \hbox{ in the null space of
} A \}$, where $A$ is the linear mapping defined by \eqref{e1.2}. 
The results for \eqref{e1.1} we will prove consist in
describing the manner in which this $i(\lambda)$-dimensional
aspect of the solutions of
 \eqref{e1.2} persists in the global description of the set of
 solutions of the nonlinear  \eqref{e1.1}.


Since $N$ is nonlinear, one cannot expect the solutions of
\eqref{e1.1}  to have any linear structure. We prove various
dimension results for the solution set of \eqref{e1.1} with
$i(\lambda)>0$, where by dimension we will mean the natural
extension of the linear concept of dimension, namely, the Lebesgue
covering dimension. Moreover, if $i(\lambda)=0$, we prove that
\eqref{e1.1} has a constant number of solutions on certain
connected components for almost all $f$.


Our method consists in reformulating \eqref{e1.1} as an operator
equation of the form
\begin{equation}
Ax+N(u,x)=f, \;\;(u,x)\in D\subset
\mathbb{R}^i\times X, \label{e1.3}
\end{equation}
where $X$ is a Banach space,
$D$ is an open subset in $\mathbb{R}^i\times X$, and $A+N :D\to X$
is an A-proper mapping. Then, under suitable assumptions on A and
N, we prove some abstract results for \eqref{e1.3} using some
dimension results of \cite{f1,i1,m2,m3,m4}, which are needed in the
study of \eqref{e1.1}.


In Section 2, we give some definitions and state our multiplicity
and dimension results for \eqref{e1.1}. Section 3 is devoted to
proving some abstract results for \eqref{e1.3} needed in Section
2. We prove the results stated in Section 2 in Section 4.
Moreover, we look at special classes of nonlinearities  that are
of integral Hammerstein or Nemitskii type. Possible extensions of
these results to integral equations on the real line are indicated
briefly in Section 5.



\section{Multiplicity and dimension results for semi-abstract equation 
\eqref{e1.1}}


Let us first recall some facts about the Wiener-Hopf equation
\eqref{e1.2}, where $k:\mathbb{R}\to \mathbb{C}$ is in
$L_1(\mathbb{R},\mathbb{C})$ and $y(s)\in
L_1(\mathbb{R}^+,\mathbb{C})$ ( see \cite{c1,k2}). 
Let $Y=L_1(\mathbb{R},\mathbb{C})$
and $x(s)=y(s)=0$ for $s<0$. Then \eqref{e1.2} becomes
\begin{equation}
\lambda x(s)-\int_{-\infty}^{\infty} k(s-t)x(t)dt=z(s) \label{e2.1}
\end{equation}
 where $z(s)=y(s)$ for $s\ge 0$ and
$z(s)=-\int_0^{\infty}k(s-t)x(t)dt$ for $s<0$. Let $\hat k(\xi)$
be the Fourier transform of $k$, i.e.,
$$
 \hat k(\xi)=\int_{-\infty}^{+\infty}k(t)e^{i\xi t}dt.
 $$
Applying the Fourier transform to \eqref{e2.1}, we get
$\lambda \hat x(\xi)- \hat k(\xi) \hat x(\xi)=\hat z(\xi)$
 on $\mathbb{R}$.
Let $X=L_p(\mathbb{R}^+,\mathbb{C})$ and $K:X\to X$ be given by
\begin{equation}
Kx(s)=\int_0^{\infty}k(s-t)x(t)dt,\;\;s\in \mathbb{R}^+. \label{e2.2}
\end{equation}
It turns out that
$\lambda I-K:X\to X$ is a Fredholm mapping if and only if
$\lambda -\hat k(\xi)\ne 0$
in $\mathbb{R}\cup \{-\infty,+\infty\}$ and the index
 $i(\lambda)=$index$(\lambda I-K)= -w(\Gamma_{\lambda},0)$ for
$\Gamma_{\lambda}=\{\lambda -\hat k(\xi) :
-\infty \le \xi \le +\infty \}$. $\Gamma _{\lambda }$
is a closed curve and
$w(\Gamma _{\lambda},0)$ is the winding number.
If $i(\lambda)\ge 0$, then
$\mathop{\rm dim}N(\lambda I-K)=i(\lambda)$ and the range
 $\mathbb{R}(\lambda I-K)=X$. If $i(\lambda)<0$, then
$\mathop{\rm dim}N(\lambda I-K)=\{\emptyset\}$ and
$\mathop{\rm codim}\mathbb{R}(\lambda I-K)=-i(\lambda)$. Suppose that
$\lambda -\hat k(\xi)\ne   0$.  Then $\lambda I-K$ is of index zero if,
for example, $k(t-s)\; (0\le t,s<\infty)$ is a symmetric kernel, that
is, if $k(t)$ (-$\infty <t<\infty$)
is an even function. In this case $\hat k(\xi)$ is also an even function.
Another interesting case is when $k(t-s)$ is a hermitian kernel,
i.e., $k(-t)=\overline{k(t)}$. Then $\hat k(\xi)$
is real, and $\lambda -\hat k(\xi)\ne   0$ if and only if it is positive.
Hence $\lambda I-K$ is of index zero in this case if
$\lambda -\hat k(\xi)> 0$. Throughout the paper $A$ will stand for the
operator $\lambda I-K$ for some $\lambda \in \mathbb{C}$.


Next, let us recall the idea of covering dimension.  If $D$ is a
topological space, and k is a positive integer, then $D$ is said
to have covering dimension equal to $k$ provided that $k$ is the
smallest integer with the property that whenever $U$ is a family
of open subsets of $D$ whose union covers $D$, there exists a
refinement, $U'$, of $U$, whose union also covers $D$, and no
subfamily of $U'$ consisting of more than $k+1$ members has
nonempty intersection. If $D$ fails to have this refinement
property for each positive integer, then $D$ is said to have
infinite dimension. When $a\in D$, we say that $D$ has a dimension
$k$ at $a$ if each neighborhood, in $D$, of $a$ has dimension at
least $k$. In the absence of a manifold structure on $D$, the
concept of dimension is the natural way in which to describe its
size.

For a mapping $T:X\to Y$, let $\Sigma$ be  the set of all points
$x\in X$ where $T$ is not locally invertible, and let
card$T^{-1}(\{f\})$ be the cardinal number of the set
$T^{-1}(\{f\})$. Let $X=L_p(\mathbb{R}^+,\mathbb{C})$, $1\le
p<\infty$ and $P:X\to N(A)$ be the projection onto $N(A)$.
Throughout the paper we assume that a nonlinear mapping $N$ is
quasibounded with the quasinorm
$$
|N|=\limsup_{\|x\|\to \infty}\|Nx\|/\|x\|<\infty
$$


\begin{theorem}[Nonlinear Fredholm Alternative] \label{thm2.1}
 Let $A=\lambda I-K:X\to X$ be a Fredholm mapping of index
$i(A)\ge 0$ induced by \eqref{e1.2} and $N:X\to X$ be a k-ball
contractive mapping with $k$ and $|N|$ sufficiently small.
Then, either
\begin{itemize}

\item[(i)] the equation $Ax=0$ has a unique zero solution, i.e,
$i(A)=0$, in which case \eqref{e1.1} is approximation solvable
for each $y\in X$ and $(A+N)^{-1}(\{y\})$ is compact for each
$y\in X$ and the cardinal number $\mathop{\rm card}(A+N)^{-1}(\{y\})$
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 $y\in X$, 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 $N(A)$.
\end{itemize}
\end{theorem}

The uniqueness result in Theorem \ref{thm2.1} with $i(A)=0$ and $N$
a k-contraction has been proven
by Corduneanu \cite{c1} using the contraction mapping principle.
Next, we shall look at monotone like perturbations.


\begin{theorem} \label{thm2.2}
Let $A:H=L_2(\mathbb{R}^+,\mathbb{R})\to H$
be a Fredholm mapping of index $i(A) = 0$ induced by \eqref{e1.2}
and $N:H\to H$ be such that either
\begin{itemize}

\item[(a)] $A$ is monotone and $N$ is bounded continuous and of
type $(S_+)$ with $|N|$ sufficiently small, or


\item[(b)] $A$ is c-strongly monotone for some $c>0$ and $N=N_1+N_2$
with $N_1$ monotone and $N_2$ is $k$-contractive for some $k<c$
with $|N_2|$ sufficiently small.

\end{itemize}
Then, the equation $Ax=0$ has a unique zero solution and \eqref{e1.1}
is approximation solvable for each $y\in H$ and
$(A+N)^{-1}(\{y\})$ is compact for each $y\in H$ and the cardinal
number card$(A+N)^{-1}(\{y\})$ is constant, finite and positive
on each connected component of $H\setminus (A+N)(\Sigma)$.
\end{theorem}

Under just the monotonicity assumption on $N$, we can only prove
the solvability of \eqref{e1.1}.


\begin{theorem} \label{thm2.3}
Let A be a monotone Fredholm mapping of index
$i(A) = 0$ induced by \eqref{e1.2} and $N:H\to H$ be continuous,
bounded and monotone such that $|N|$ is sufficiently small.
Then \eqref{e1.1} has a solution for each $y\in H$.
\end{theorem}


For odd nonlinearities, we have the following result.


\begin{theorem} \label{thm2.4}
Let A be a Fredholm mapping of index $i(A)>0$ induced by \eqref{e1.2}
and $N:X\to X$ be a continuous odd
$k$-ball contractive mapping with $k$ sufficiently small. Let $S_0$
be the solution set of \eqref{e1.1} with $y=0$. Then, for any
positive real number $r$ and $B(0,r)=\{x\in X:\|x\|<r\}$, the
dimension of $S_0\cap \partial B(0,r)$ is at least $i(A)-1$, when
$i(A)>1$, and $S_0\cap \partial B(0,r)\}$ contains at least two
points when $i(A)=1$.
\end{theorem}


\begin{remark} \label{rmk2.1} \rm
 If $i(A)=0$, then \eqref{e1.1} is equivalent
to the Hammerstein operator equation  $x+A^{-1}Nx= A^{-1}y$.
Then the recent results of the author \cite{m6} apply to
\eqref{e1.1} with $F$ of pseudo monotone type and $A$ selfadjoint
or P-(quasi) positive.
\end{remark}

\section{Nonlinear Fredholm alternative and the number of solutions}

We begin by proving some generalizations of the first Fredholm
theorem and the Fredholm alternative to general (pseudo) A-proper
mappings. In the case of A-proper mappings, we also establish the
number of solutions of these equations and the dimension of the
solution set. These results are needed for the proofs of the
theorems in Section 2. There is a large literature on nonlinear
Fredholm theory which has dealt only with the existence results
\cite{b1}. The index and the covering dimension results for
various operator equations involving $A$-proper and epi-maps can
be found in  \cite{f1,i1,m2,m4} and the literature in there.



Let us recall some definitions first. Let $\{X_n\}$ and $\{Y_n\}$
be finite dimensional subspaces of Banach spaces $X$ and  $Y$
respectively such that $\dim X_n-\dim Y_n=i\ge 0$ for each $n$ and
dist$ (x,X_n)\to 0$ as $n\to\infty$ for each $x\in X$. Let
$P_n:X\to Y_n$ and $Q_n:Y\to Y_n$ be linear projections onto $X_n$
and $Y_n$ respectively such that $P_nx\to x$ for each $x\in X$ and
$\delta=$max $\|Q_n\|<\infty$. Then $\Gamma_i=\{X_n,P_n;Y_n,Q_n\}$
is a projection scheme for $(X,Y)$. Such schemes are needed for
studying multiparameter problems and nonlinear perturbations of
Fredholm mappings of index $i\ge 0$.


A mapping $T:D \subset X \to Y$ is said to be  {\em
approximation-proper} ({\em $A$-proper} for short) with respect to
$ \Gamma_i$ if (i) $Q _{ n}T:D \cap X _{ n} \to Y _{ n}$ is
continuous for each $n$ and (ii) whenever
$ \{ x _{ n _{ k}} \in D \cap X _{ n _{ k}} \}$ is bounded and
$\|Q _{ n _{ k}}Tx _{ n _{k}}-Q _{ n _{ k}}f\| \to 0$ for some
$f \in Y$, then a subsequence $x _{ n _{ k(i)}} \to x$ and $Tx=f$.
 $T$ is said to be {\em pseudo A-proper} with respect to $ \Gamma_i $
 if in (ii) above we do not require that a subsequence of
$ \{ x _{ n _{ k}} \}$ converges to $x$ for which $Tx=f$.
If $f$ is given in advance, we say that $T$ is (pseudo)
$A$-proper at $f$.

We state now a number of examples of $A$-proper and pseudo $A$-proper mappings.
To look at $\phi$-condensing mappings, we recall
that the {\em set measure of noncompactness} of a bounded set $D \subset X$
is defined as $ \gamma (D)= \inf \{ d>0:D$ has a finite covering by
sets of diameter less than $d \}$.
The {\em ball-measure of noncompactness} of $D$ is defined as
$ \chi (D)= \inf \{ r>0|D \subset \cup _{ i=1} ^{ n}B(x _{ i},r),
\; x \in X, \; n \in N \}$.
Let $ \phi $ denote either the set or the ball-measure of noncompactness.
Then a mapping $N:D \subset X \to  X$ is said to be {\em $k-\phi$ contractive}
({\em $ \phi $-condensing}) if $ \phi (N(Q))\le k\phi (Q)$
(respectively $ \phi (N(Q))< \phi (Q)$) whenever $Q \subset D$ (with
$ \phi (Q) \ne 0$). A $k$- ball contractive vector field with $k<1$ is
$A$-proper \cite{n1}.

Recall that a function $N:X\to X^*$ is said to
be of type $(S_+)$ if $x_n\rightharpoonup x$ and
$\limsup (Nx_n,x_n-x)\le 0$ imply
that $x_n\to x$. A ball-condensing perturbation of a $c$-strongly
monotone mapping, i.e., $(Tx-Ty,x-y)\ge c\|x-y\|^2$ for all $x,y\in X$,
 is an $A$-proper mapping.


We need the following classes of $A$-proper mappings.


\begin{example}(\cite{m2}) \label{ex3.1} \rm
Let $A:X\to Y$ be a linear Fredholm mapping
of index $i(A)\ge 0$, $X=N(A)\oplus \tilde X$, $Y=Y_0\oplus R(A)$,
$\|Ax\|\ge c\|x\|$
for some $c>0$ and all $x\in \tilde X$,
and $N:X\to Y$ be a bounded and continuous k-ball contraction with $k<c$.
Then $A,A+N:X\to Y$ are $A$-proper with respect to
$\Gamma _i=\{X_0\oplus X_n,Y_0\oplus Y_n,\tilde Q_n\}$
where $\cup_{n\ge 1}X_n$ is dense in $\tilde X$,
 $Y_n=A(X_n)$, $\tilde Q_n(y_0+y_1)=y_0+Q_ny_1$
and $Q_n:\tilde Y\to Y_n$ are projections onto $Y_n$ for each n. Moreover,
if $Ax_n+Nx_n=f$ with $\{x_n=x_{0n}+x_{1n}\}\in X=N(A)\oplus \tilde X$ and $\{x_{0n}\}$ both
bounded, then $\{x_n\}$ has a convergent subsequence.
\end{example}



\begin{example} \label{ex3.2} \rm
Let $A:X\to X^*$ be monotone and $N:X\to X^*$ be of type $(S_+)$. Then
$A+N:X\to X^*$ is $A$-proper with respect to
$\Gamma _0=\{X_n,R(P_n^*),P_n^*\}$, where $P_n:X\to X_n$
are projections onto $X_n$.
\end{example}


\begin{example}[\cite{m1,t1}] \label{ex3.3} \rm
Let $A:X\to X^*$ be continuous and c-strongly monotone and $N:X\to X^*$
be continuous and k-ball contractive. Then $A+N:X\to X^*$ is $A$-proper with respect to
$\Gamma _0=\{X_n,R(P_n^*),P_n^*\}$, where $P_n:X\to X_n$
are projections onto $X_n$.
\end{example}

We say that a mapping $T:X\to Y$ satisfies condition (+) if
whenever $Tx_n\to f$ in $Y$ then $\{x_n\}$ is bounded in $X$.
$T$ is locally injective at $x_0\in X$ if there is a neighborhood
$U(x_0)$ of $x_0$ such that $T$ is injective on $U(x_0)$. $T$ is locally
injective on $X$ if it is locally injective at each point $x_0\in X$.
A continuous mapping $T:X\to Y$ is said to be locally invertible
 at $x_0\in X$ if there are a neighborhood $U(x_0)$ and a neighborhood
$U(T(x_0))$ of $T(x_0)$ such that $T$ is a homeomorphism of $U(x_0)$
onto $U(T(x_0))$. It is locally
invertible on $X$ if it is locally invertible at each point $x_0\in X$.


We need the following basic theorem on the number of solutions of
nonlinear equations for A-proper mappings \cite{m5}.


\begin{theorem} \label{thm3.1}
Let $T:X\to Y$ be a continuous $A$-proper mapping w.r.t. $\Gamma_0$ that satisfies
condition (+). Then
\begin{itemize}

\item[(a)] The set $T^{-1}(\{f\})$ is compact (possibly empty)
for each $f\in Y$.

\item[(b)] The range $R(T)$ of  $T$ is closed and connected.

\item[(c)] $\Sigma$ and $T(\Sigma)$ are closed subsets of $X$ and $Y$,
respectively, and $T(X\setminus \Sigma)$ is open in $Y$.

\item[(d)] $\mathop{\rm card}T^{-1}(\{f\})$ is constant and finite
(it may be 0) on each connected
component of the open set $Y\setminus T(\Sigma)$.
\end{itemize}
\end{theorem}

Next, we shall prove a nonlinear Fredholm alternative for nonlinear
perturbations of linear Fredholm mappings
\begin{equation}
 Ax+Nx=f  \label{e3.1}
\end{equation}
where $A:X\to Y$ is a linear Fredholm mapping of index $i(A)\ge 0$
with $R(A)=Y$, and $N$ is a nonlinear quasibounded mapping with
$|N|$  sufficiently small. If $X_0$ is the null space of $A$, then
$X=X_0\oplus \tilde X$ and
$i(A)=$$\mathop{\rm dim}X_0$. Let $P:X\to X_0$ be a
linear projection onto $X_0$. Let $\Gamma_0 =\{X_n,Y_n, Q_n\}$ be
an approximation scheme for $(\tilde X,Y)$ with sup$\|Q_n\|$ finite and define a new
approximation scheme $\Gamma_i =\{X_0\oplus X_n, Y_n, Q_n\}$ for
$(X,Y)$. Then $\mathop{\rm dim}X_0\oplus X_n-$ $\mathop{\rm dim}Y_n=$ $\mathop{\rm dim}X_0=i(A)$ for each
$n$. Let $\Sigma=\{x\in X:A+N\hbox{ is not invertible  at }x\}$


\begin{theorem}[Nonlinear Fredholm Alternative] \label{thm3.2}
Let $A:X\to Y$ be a linear Fredholm mapping of index $i(A)\ge 0$
with $R(A)=Y$, and $N:X\to Y$ be a continuous nonlinear mapping.

\noindent (a) If  $A, A+N:X\to Y$ are $A$-proper
with respect to $\Gamma_i =\{X_0\oplus X_n, Y_n,Q_n\}$ and $|N|<c$
with $c$ sufficiently small,
then either
\begin{itemize}

\item[(i)] $N(A)=\{0\}$, in which case \eqref{e3.1} is approximation
solvable for each $f\in Y$ and $(A+N)^{-1}(\{f\})$ is compact for
 each $f\in Y$ and the cardinal number card$(A+N)^{-1}(\{f\})$
is constant, finite and positive on each connected component of
the set $Y\setminus (A+N)(\Sigma)$,
or

\item[(ii)] $N(A)\ne \{0\}$ and then
for each $f\in Y$ $(=N(A^*)^{\perp})$ there is
a connected closed
subset $C$ of $(A+N)^{-1}(\{f\})$ whose dimension at each point is at least
$i(A)$ and the projection $P$ maps $C$ onto $X_0$.

\end{itemize}

\noindent (b) If $N(A)=\{0\}$ and $A+N$ is pseudo $A$-proper with
respect to $\Gamma_0$ and $\delta |N|<c$, where $\delta=\hbox{max}\|Q_n\|$,
 then $(A+N)(X)=Y$.
\end{theorem}


\begin{proof} (a) First, assume that $N(A)=\{0\}$. We claim that
the homotopy $H(t,x)=Ax+tNx$ satisfies
condition (+), i.e., if $H(t_n,x_n)\to f$ then $\{x_n\}$ is bounded in $X$.
Since $A$ is a continuous bijection, it follows that for some $c>0$
$$
\|Ax\|\ge c\|x\|,\;\;x\in X.
$$
Let $\epsilon >0$ be such that $|N|+\epsilon <c$ and
$R=R(\epsilon)>0$ such that
$$
\|Nx\|\le (|N|+\epsilon)\|x\|\quad \hbox{for all }\|x\|\ge R.
$$
Then, for $x\in X\setminus B(0,R)$, we get that
$$
\|Ax+tNx\|\ge (c-|N|-\epsilon)\|x\|
$$
and therefore, $\|H(t,x)\|=\|Ax+tNx\|\to \infty$ as $\|x\|\to \infty$
independently of $t$.
Hence, condition $(+)$ holds. Arguing by contradiction, we see that
for each $f\in Y$ there is an $r>R$ and $\gamma >0$ such that
$$
\|H(t,x)-tf\|\ge \gamma \;\hbox{for all }t\in [0,1],\;x\in \partial B(0,r).
$$
Since $H_t$ is an $A$-proper homotopy, this implies that there is an
$n_0\ge 1$ such that
$$
Q_nH(t,x)\ne tQ_nf \;\hbox{for all}\;t\in [0,1],\;x\in
\partial B(0,r)\cap X_n,\; n\ge n_0.
$$
By the Brouwer degree properties and the $A$-properness of $H_1$,
there is an $x_n\in B(0,r)\cap X_n$ such
that $Q_nAx_n+Q_nNx_n=Q_nf$ and a subsequence
$x_{n_k}\to x$ with $Ax-Nx=f$. Hence, part (i) follows from
Theorem \ref{thm3.1}.


Next, let $N(A)\ne \{0\}$. For a given $f\in Y$, let $Bx=Nx-f$.
We need to show that $A+N:X_0\oplus \tilde X\to Y$ is complemented by
the projection $P$ of $X$ onto $X_0$. To that end, it suffices to show 
(see \cite{f1})
that $\mathop{\rm deg}(Q_n(A+B)|X_n,X_n,0)\ne 0$
for all large n. Define the homotopy $H_n:[0,1]\times X_n\to Y_n$
by $H_n(t,x_1)= Q_nAx_1+tQ_nBx_1$. Consider the restriction of
 $A+B$ to $\tilde X$. Since $A$ restricted
to $\tilde X$ is a bijection from $\tilde X$ onto $Y$, as in the first
case we get that there are $n_0\ge 1$ and
$r\ge R$ such that  $H_n(t,x_1)\ne 0$ for $x_1\in X_n$ with
$n\ge n_0$ and $t\in [0,1]$.
Thus, for each
$n\ge n_0$, $\mathop{\rm deg}(Q_n(A+B)|X_n,X_n,0)
=\mathop{\rm deg}(Q_nA|X_n,X_n,0)\ne 0$.

Next, we need to show that the projection $P:X=X_0\oplus \tilde X\to X_0$
is proper on $ (A+B)^{-1}(f)$. To see this, it suffices to show that
if $\{x_n\}\subset X_0\times \tilde X$
is such that $Ax_n+Bx_n=f$ and $\{Px_n\}$ is bounded,
 then $\{x_n\}$ is bounded since
the $A$-proper mapping $A+B$ with respect to $\Gamma_i$ is proper when 
restricted to bounded sets.
We have that $x_n=x_{0n}+x_{1n}$ with
$x_{0n}\in X_0$ and $x_{1n}\in \tilde X$
and $c\|x_{1n}\|\le \|Ax_{1n}\|\le (|N|+\epsilon)(\|x_{0n}\|+\|x_{1n}\|)+\|f\|$
for some $\epsilon >0$ with $|N|+\epsilon <c$ if $\|x_n\|\ge R$.
This implies that $\{x_{1n}\}$ is bounded by the boundedness of $\{x_{0n}\}$.
Since $\{x_{0n}\}=\{Px_n\}$ is bounded, it follows that $\{x_n\}$ is
also bounded. Hence, the conclusions of
(a)-(ii) follow from \cite[Theorem 1.2]{f1}.

(b) Since $A$ is an $A$-proper injection, it is easy to see that there is
an $n_0\ge 1$ and $c>0$ such that for each $n\ge n_0$
$$
\|Q_nAx\|\ge c\|x_n\|\quad\hbox{for all }x\in X_n.
$$
Let $\epsilon >0$ be such that $|N|+\epsilon <c/\delta$ and
$R=R(\epsilon)>0$ such that
$$
\|Nx\|\le (|N|+\epsilon)\|x\|\quad \hbox{for all }\|x\|\ge R.
$$
Define the homotopy $Q_nH(t,x)=Q_nAx+(1-t)Q_nNx+(1-t)Q_nf$ on
$[0,1]\times X_n$ for $n\ge n_0$.
It follows easily from the above remarks that $Q_nH(t,x)\ne 0$
on $\partial B(0,R)\cap X_n$
for each $n\ge n_0$. Thus, the there is an $x_n\in B(0,R)\cap X_n$
such that $Q_nAx_n+Q_nNx_n=Q_nf$ for each $n\ge n_0$.
Hence, $Ax+Nx=f$ for some $x\in X$ by the
pseudo $A$-properness of $A+N$.
\end{proof}


\begin{remark} \label{rmk3.1} \rm
Theorem \ref{thm3.2} is valid also for Fredholm mappings of index zero under
the additional assumption that the range $R(N)\subset R(A)$ \cite{m3}.
Just the solvability of $x-Kx+Nx=y$ with $K$ and $N$ compact under
this condition was first established
by Kachurovskii \cite{k1}.
\end{remark}


Next, we shall look at  odd mappings. Recall
that  a nonempty and symmetric subset $A$ of $X\setminus \{0\}$ has
genus $p$, $\gamma(A)=p$, if there is an odd continuous mapping
$\phi:A\to \mathbb{R}^p\setminus \{0\}$ and
$p$ is the smallest integer having this property;
$\gamma (\emptyset)=0$. We also have
that $\mathop{\rm dim}A\ge \gamma(A)-1$. The following dimension
result from \cite{m2} for odd mappings will be needed in the sequal.



\begin{theorem} \label{thm3.3}
Let $T:X\to Y$ be an odd $A$-proper mapping
at zero with respect to $\Gamma _i$ with $i\ge 1$. For each $r>0$, let
$S_r=\{x\in \partial B(0,r):Tx=0\}$. Then the genus $\gamma
(S_r)\ge i$ and the dimension $\mathop{\rm dim}(S_r)\ge \gamma (S_r)-1$ if
$i>1$, and $S_r$ contains at least two points when $i=1$.
\end{theorem}



\section{Proofs of Theorems \ref{thm2.1}-\ref{thm2.4} and special cases}

In this section, we provide proofs to the results in Section 2 and
look at some special classes of $N$. To that end we need to have
suitable approximation schemes. It is known (R. Beals) that for
$1\le p<\infty $ and a finite dimensional Banach space $M$,
$L_p(\mathbb{R},M)$ is a $\pi_1$-space and thus has a monotone
Schauder basis, i.e. there is an increasing sequence of finite
dimensional subspaces $\{X_n\}$ of $L_p(\mathbb{R},M)$ and linear
projections of $L_p$ onto $X_n$ with $\|P_n\|=1$ and $\cup_{n\ge
1} X_n$ dense in $L_p$. If $S$ is any closed interval or
$(0,\infty)$, then $L_p(S,M)$ is also a $\pi_1$-space as a
subspace of $L_p(\mathbb{R},M)$, since each $f\in L_p(S,M)$ is in
$L_p(\mathbb{R},M)$ if we set $f=0$ outside of $S$.


Equation \eqref{e1.1} is equivalent to the operator equation
$Ax+Nx=y$ in $X$.


\begin{proof}[Proof of Theorem \ref{thm2.1}]
 By our assumptions $A$ and $A+N$ are $A$-proper  mappings with respect to
$\Gamma_i =\{X_0\oplus X_n,A(X_n),P_n\}$ by Example \ref{ex3.1},  where
$P_n:X\to A(X_n)$ are the projections onto $A(X_n)$ with sup$\|P_n\|$ finite. 
Thus, the
conclusions of the theorem follow from Theorem \ref{thm3.2}. since
$R(A)=X$.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm2.2}]
 By our assumptions $A+N:H\to H$ is $A$-proper with respect to
$\Gamma _0=\{A(H_n),P_n\}$ by Example \ref{ex3.2}, where $\cup_{h\ge 1}H_n$
is dense in $H$. Moreover,
$A$ is also $A$-proper with respect to $\Gamma_0$ by Example \ref{ex3.1}. 
If (b) holds, then $A+N_1$ c-strongly
monotone and $N_2$ is k-ball contractive. Hence, $A$ and $A+N$ are $A$-proper 
with respect to $\Gamma_0$
by Example \ref{ex3.3}. Since $R(A)=H$ in either case, the conclusions of
the theorem follow from Theorem \ref{thm3.2}(a).
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm2.3}]
 Note that the mapping $A$ is $A$-proper with respect to 
 $\Gamma_0=\{A(H_n),P_n\}$
by Example \ref{ex3.1}, and $A+N$ is pseudo $A$-proper with respect to
$\Gamma_0$, where $\cup_{n\ge 1}H_n$ is dense in $H$.
Moreover, $|N|$ is sufficiently small. Hence, the conclusion of
the theorem follows
from Theorem \ref{thm3.2}(b).
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm2.4}]
The mapping $A+N:X\to X$ is odd and $A$-proper with respect to
$\Gamma_i$ by Example \ref{ex3.1}. Hence, the conclusions follow from
Theorem \ref{thm3.3}.
\end{proof}

Next, we shall look at various classes of $k$-ball contractive mappings
$N$. Let $X=L_p(\mathbb{R}^+,\mathbb{C})$ with $1<p<\infty$.
First, we assume that the operator $N$ is formally given by the formula
\begin{equation}
(Nx)(s)= \int_0^{\infty}k_0(s,t)F(t,x(t))dt,\quad
s\in \mathbb{R}^+, \label{e4.1}
\end{equation}
with $k_0(s,t)$ and $F$ satisfying appropriate conditions. Let
\[
K_0x(s)=\int_0^{\infty}k_0(s,t)x(t)dt,\quad s\in \mathbb{R}^+.
\]
Then $N=K_0F$ and \eqref{e1.1} becomes
\begin{equation}
 \lambda x(s)-\int_0^{\infty} k(s-t)x(t)dt
+ \int_0^{\infty}k_0(s,t)F(t,x(t))dt=y(s),\quad s\in \mathbb{R}^+.
  \label{e4.2}
\end{equation}
In the operator form, \eqref{e4.2} is $Ax+K_0Fx=y$. For a given
$k_0(s)$, define
$$
\tilde K_0x(s)=\int_0^{\infty}k_0(s-t)x(t)dt,\quad s\in \mathbb{R}^+.
$$


\begin{corollary}[Nonlinear Fredholm Alternative] \label{coro4.1}
Let $A=\lambda I-K:X\to X$ be a Fredholm mapping of index $i(A)\ge 0$
induced by \eqref{e1.2} and $k_0(s,t)$ be a measurable
complex-valued function of $(s,t)$ for $0\le s,t <\infty$ and
$$
|\int_0^{\infty}k_0(s,t)x(t)dt|\le |\int_0^{\infty}k_0(s-t)x(t)dt|,
\quad s\in \mathbb{R}^+
$$
for $x\in X^*=L_q$ and some
$k_0(s)\in L_1(\mathbb{R},\mathbb{C})\cap L_{\infty}(\mathbb{R},\mathbb{C})$ 
with
$(\tilde K_0x,x)\ge 0$ for all $x\in X^*$.
Assume that for $p\ge 2$,
$F:\mathbb{R}^+\times \mathbb{R}\to \mathbb{C}$ is a Caratheodory function
such that $F(s,0)\in X^*$ and
$$
|F(s,u)-F(s,v)|\le k|u-v|^{p-1}\quad \hbox{for all }
s\in \mathbb{R}^+,\;u,v\in \mathbb{R}
$$
and some $k$ such that $k\|K_0\|$ is sufficiently small.
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 uniquely approximation
solvable for each $y\in X$ with respect to $\Gamma_0$ for $X$,
or

\item[(ii)] $N(A)\ne \{0\}$, i.e., $i(A)=$$\mathop{\rm dim}N(A)>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 $B(0,r)=\{x\in H:\|x\|<r\}$, the dimension of
$S_0\cap \partial B(0,r)$ is at least $i(A)-1$, when $i(A)>1$ and
$S_0\cap \partial B(0,r)$ contains at least two points when
$i(A)=1$.
\end{itemize}
\end{corollary}


\begin{proof} We claim that $K_0:X^*\to X$, where $K_0$ is given above.
Indeed, by Holder's inequality, for each $x\in X^*$,
\begin{align*}
\big|\int_0^{\infty}k_0(s-t)x(t)dt\big|
&\le \int_0^{\infty}|k_0(s-t)|^{1/p}(|k_0(s-t)|^{1/q}|x(t)|)dt\\
&\le \Big(\int_0^{\infty}|k_0(s-t)|dt\Big)^{1/p}
\Big(\int_0^{\infty}|k_0(s-t)| |x(t)|^qdt\Big)^{1/q},
\end{align*}
and therefore
$$
|\int_0^{\infty}k_0(s-t)x(t)dt|^p
 \le \|k_0\|_{L_1}(\int_0^{\infty}|k_0(s-t)| |x(t)|^qdt)^{p/q}.
$$
The integral on the right hand side is in $L_{\infty}\cap L_1$ and
therefore it is in $L_{p/q}(\mathbb{R}^+,\mathbb{C})$ since $p/q=p-1\ge 1$.
Hence, $\tilde K_0:X^*\to X$ and is
continuous since the linear mapping $\tilde K_0$ is monotone.
Moreover,
$\|K_0x\|_{L_p}\le \|\tilde K_0x\|_{L_p}\le \|\tilde K_0\|\;\|x\|_{L_q}$.
Hence, $K_0:X^*\to X$ is continuous.

Next, (a) implies that $(Fx)(s)=F(s,x(s))$ maps $X$ into $X^*$ and therefore,
the nonlinear mapping given by $N=K_0F$, i.e., by \eqref{e4.1},
maps $X$ into itself. Moreover, for each $x,y\in L_p$
$$
\|Fx-Fy\|^q = \int_0^{\infty}|F(t,x(t))-F(t,y(t))|^qdt\le k^q\|x-y\|^p.
$$
Hence, $F$ is a $k$-contraction and $\|Nx-Ny\|\le k\|K_0\|\;\|x-y\|$
 with $k_1=k\|K_0\|<1$.
Thus, $N$ is a $k_1$-ball contraction.
If $i(A)=0$, then the equation $Ax+Nx=y$ is equivalent to
$x+A^{-1}Nx=A^{-1}y$, where $A^{-1}N$ is
a $k_2=k_1/c$-contraction with $k_2<1$, since $\|x-Kx\|\ge c\|x\|$ and $c>k_1$. 
Hence, it is
uniquely solvable by the contraction principle. This proves (i),
while (ii) follows from Theorem \ref{thm2.1} and (iii) follows from
Theorem \ref{thm2.4}.
\end{proof}


\begin{corollary}[Nonlinear Fredholm Alternative] \label{coro4.2}
 Let $A=\lambda I-K:X\to X$ be a Fredholm mapping of index $i(A)\ge 0$
induced by \eqref{e1.2} and $k_0(s,t)$ be a measurable
complex valued function of $(s,t)$ for $0\le s,t <\infty$ and either
\begin{equation}
\sup_{s\in \mathbb{R}^+}\int_0^{\infty}|k_0(s,t)|dt < \infty  \label{e4.3}
\end{equation}
or
\begin{equation}
|k_0(s,t)|\le |k_0(s-t)|,\;\;s,t\in \mathbb{R}^+. \label{e4.4}
\end{equation}
for some $k_0\in L_1(\mathbb{R}^+,\mathbb{C})$. Assume that
$F:\mathbb{R}^+\times \mathbb{R}\to \mathbb{C}$ is a Caratheodory
function such that $F(s,0)\in L_p(\mathbb{R}^+,\mathbb{R})$ and
$$
|F(s,u)-F(s,v)|\le k|u-v|\quad \hbox{for all } s\in \mathbb{R}^+,
\;u,v\in \mathbb{R}
$$
and some $k$ sufficiently small.
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 uniquely approximation solvable for each $y\in X$ with respect
to $\Gamma_0$ for $X$,
or

\item[(ii)] $N(A)\ne \{0\}$, i.e., $i(A)=$$\mathop{\rm dim}N(A)>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 $(s,u)\in
\mathbb{R}^+\times \mathbb{R}$, and if $S_0$ is the solution set
of \eqref{e4.2} with $y=0$, then, for any positive real number $r$
and $B(0,r)=\{x\in H:\|x\|<r\}$, the dimension of $S_0\cap
\partial B(0,r)$ is at least $i(A)-1$, when $i(A)>1$ and $S_0\cap
\partial B(0,r)$ contains at least two points when $i(A)=1$.
\end{itemize}
\end{corollary}


\begin{proof}
 Let $N=K_0F$ be given by \eqref{e4.1}. If \eqref{e4.3} holds,
then $K_0$ maps $L_p(\mathbb{R}^+,\mathbb{C})$ into itself,
$1\le p\le \infty$,
and is a $\|K_0\|$-contraction by the Riesz convexity theorem
(see \cite[Chapter 11]{j1}). If \eqref{e4.4} holds, then
$K_0 :L_p(\mathbb{R}^+,\mathbb{C})\to L_p(\mathbb{R}^+,\mathbb{C})$
is continuous by Young's inequality and
$\|K_0\|\le \|k_0\|_1$ for each $1\le p\le \infty$.
If $(Fx)(s)=F(s,x(s))$, then $F:L_p\to L_p$ is a $k$-contraction.
Thus, $N=K_0F$ is a $k_1=k\|K_0\|$-contraction in $L_p$, and is
therefore $k_1$-ball contractive.
The conclusions now follow from Theorems \ref{thm2.1} and \ref{thm3.3}
 since (i) follows as in Corollary \ref{coro4.1}.
\end{proof}

\begin{corollary}[Nonlinear Fredholm Alternative] \label{coro4.3}
 Let $A=\lambda I-K:X\to X$ be a Fredholm mapping of index $i(A)\ge 0$
induced by \eqref{e1.2} and $k_0(s,t)$ be a measurable
complex valued function of $(s,t)$ for $0\le s,t <\infty$ and
$$
\int_0^{\infty}[\int_0^{\infty}|k_0(s,t)|^pds]^{q/p}dt\le c^q,
$$
for $1<p<\infty$, $q=p/(p-1)$, $c>0$. 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 $B(0,r)=\{x\in H:\|x\|<r\}$, the dimension of $S_0\cap
\partial B(0,r)$ is at least $i(A)-1$, when $i(A)>1$ and $S_0\cap
\partial B(0,r)$ contains at least two points when $i(A)=1$.
\end{itemize}
\end{corollary}


\begin{proof} The mapping $K_0$ defined above
is a completely continuous linear operator in $L_p$
with $\|K_0\|\le c$  \cite{c1}. Moreover, $(Fx)(s)=F(s,x(s))$ is a continuous
bounded mapping from $L_p(\mathbb{R}^+,\mathbb{C})$ into itself.
Hence, $N=K_0F:L_p(\mathbb{R}^+,\mathbb{C})\to L_p(\mathbb{R}^+,\mathbb{C})$
is a compact mapping and the conclusions follow from
Theorems \ref{thm2.1}, \ref{thm3.1} and \ref{thm3.3}.
\end{proof}

Theorem \ref{thm2.1} also applies when $N$ is the Nemitskii operator,
i.e., $(Nx)(s)=F(s,x(s))$ for some function $F$. Then \eqref{e1.1}
becomes
\begin{equation}
\lambda x(s)-\int_0^{\infty} k(s-t)x(t)dt+ F(t,x(t))=y(s),
\quad s\in \mathbb{R}^+ \label{e4.5}
\end{equation}
We have the following result.

\begin{corollary}[Nonlinear Fredholm Alternative] \label{coro4.4} 
Let $A=\lambda I-K$, $A:X=L_p(\mathbb{R}^+,\mathbb{R})\to X$, 
be a Fredholm mapping of index $i(A)\ge 0$ induced by \eqref{e1.2}
and $F:\mathbb{R}^+\times \mathbb{R}\to \mathbb{R}$ be a Caratheodory
function such that $F(s,0)\in X$ and
$$
|F(s,u)-F(u,v)|\le k|u-v|\quad \hbox{ for all }
s\in \mathbb{R}^+,\;\;u,v\in \mathbb{R}
$$
for some $k$ sufficiently small.
Then, either
\begin{itemize}

\item[(i)] the equation $Ax=0$ has a unique zero solution, i.e.,
 $i(A)=0$, in which case \eqref{e4.5}
is uniquely approximation solvable for each $y\in X$ with respect
to $\Gamma_0$ for $X$,
or

\item[(ii)] $N(A)\ne \{0\}$, i.e., $i(A)=\mathop{\rm dim}N(A)>0$,
in which case, for each $f\in H$, 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 $Nx=F(s,x(s))$,
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 $B(0,r)=\{x\in H:\|x\|<r\}$, the dimension
of $S_0\cap \partial B(0,r)$ is at least $i(A)-1$, when $i(A)>1$
and $S_0\cap \partial B(0,r)$ contains at least two points when $i(A)=1$.
\end{itemize}
\end{corollary}

\begin{proof}
 Condition (a) implies that $(Nx)(s)=F(s,x(s))$ is a k-contraction
from $X$ into itself. Hence, $N$ is a $k$-ball
contraction. Part (i) follows as in Corollary \ref{coro4.1},
while (ii)-(iii)
follow from Theorem \ref{thm3.2}-\ref{thm3.3} since $R(A)=H$.
\end{proof}

Next, we shall look at some special cases of Theorems
\ref{thm2.2}-\ref{thm2.4}
with nonlinearities of the form $(Nx)(s)=F(s,x(s))$.


\begin{corollary} \label{coro4.5}
Let $A:H=L_2(\mathbb{R}^+,\mathbb{R})\to H$
be a Fredholm mapping of index $i(A) = 0$ induced by \eqref{e1.2}
and $F:\mathbb{R}^+\times \mathbb{R}\to \mathbb{R}$ be a Caratheodory
function such that
\begin{itemize}

\item[(a)] $|F(s,u)|\le c(s)+c_0(s)|u|$, $s\in \mathbb{R}^+$,
$u\in \mathbb{R}$,
where $c(s)\in L_2(\mathbb{R}^+)$ and
$c_0(s)\in L_{\infty}(\mathbb{R}^+)$ with
$\|c_0\|_2$ sufficiently small, and either


\item[(b)] $A$ is monotone, $F(s,u)$ is strictly monotone increasing
in $u$ for each fixed $s$ and
$$
F(s,u)u\ge c_2|u|^2-c_1(s),\quad \hbox{for all }
s\in \mathbb{R}^+, u\in \mathbb{R}
$$
for some constant $c_2>0$ and a function $c_1(s)\in L_1(\mathbb{R}^+)$,
or

\item[(c)] $A$ is c-strongly monotone for some $c>0$ and $F=F_1+F_2$
 with $F_1(s,u)$ monotone increasing in u for each fixed s and
$$
|F_2(s,u)-F_2(s,v)|\le k|u-v|\quad \hbox{for all }
 s\in \mathbb{R}^+,\;\;u,v\in \mathbb{R}
$$
and some $k<c$.
\end{itemize}
Then, the equation $Ax=0$ has a unique zero solution and \eqref{e4.5}
is approximation solvable for each $y\in H$,
$(A+N)^{-1}(\{y\})$ is compact for each $y\in H$ and the cardinal
number card$(A+N)^{-1}(\{y\})$ is constant, finite and positive
on each connected component of $H\setminus (A+N)(\Sigma)$,
where $(Nx)(s)=F(s,x(s))$.
\end{corollary}

\begin{proof} Again, $N:H\to H$ is bounded and continuous. Condition
(b) implies that $N:H\to H$ is of type $(S_+)$ (cf., e.g., \cite{b1}).
Since $A$ is monotone, $A+N$ is
$A$-proper with respect to $\Gamma _0=\{A(H_n),P_n\}$ by
Example \ref{ex3.2},
where $\cup_{h\ge 1}H_n$ is dense in $H$. Moreover,
$A$ is also $A$-proper with respect to $\Gamma_0$ by Example \ref{ex3.1}.
If (c) holds, then $A+N_1$ is c-strongly
monotone and $N_2$ is k-ball contractive. Hence, $A$ and $A+N$ are $A$-proper 
with respect to $\Gamma_0$.
by Example \ref{ex3.3}. Since $R(A)=H$ in either case, the conclusions of 
the theorem follow from Theorems \ref{thm3.1}-\ref{thm3.2}.
\end{proof}


\begin{corollary} \label{coro4.6}
Let $A$ be a Fredholm mapping of index $i(A) = 0$ induced by \eqref{e1.2}
and $F:\mathbb{R}^+\times \mathbb{R}\to \mathbb{R}$ be a
 Caratheodory function such that
\begin{itemize}
\item[(a)] $|F(s,u)|\le c(s)+c_0|u|$, $s\in \mathbb{R}^+$,
$u\in \mathbb{R}$,
where $c(s)\in L_2(\mathbb{R}^+,\mathbb{R})$, $c_0>0$ is sufficiently
small and

\item[(b)] $A$ is monotone and $F(s,u)$ is monotone increasing in
$u$ for each fixed $s$.
\end{itemize}
Then \eqref{e4.5} has a solution for each $y\in H$.
\end{corollary}


\begin{proof}
The mapping $A+N:H\to H$ is bounded, continuous and monotone since
such are $A$ and $N$. Hence, $A$ is $A$-proper with respect to
$\Gamma_0=\{A(H_n),P_n\}$ by Example \ref{ex3.1},
and $A+N$ is pseudo $A$-proper with respect to $\Gamma_0$, where
$\cup_{n\ge 1}H_n$ is dense in $H$.
Moreover, $|N|$ is sufficiently small. Hence, the conclusion of
the theorem follows from Theorem \ref{thm3.2}(b).
\end{proof}

Let us look at some examples of monotone $A$. The monotonicity of $A$
implies that $\lambda \|x\|^2\ge (Kx,x)$ in $L_2$.
Hence, let us look at some positive definite
$K$, i.e., $(Kx,x)\ge 0$ on $L_2$. This means that
\begin{equation}
\int_0^{\infty}\int_0^{\infty}k(s-t)x(t)\overline{x(s)}\,ds\,dt\ge 0
 \label{e4.6}
\end{equation} for all $x\in L_2(\mathbb{R}^+,\mathbb{C})$. The positive
definitness of $K$ implies that its kernel $k(s-t)$ is hermitian
symmetric, i.e., $k(-t)=\overline{k(t)}$. If $k$ is real, that
means that it is an even function. Hence, in either case $K$ is
selfadjoint and $\lambda I-K$ is of index zero if $\lambda -\hat
k(\xi)\ne 0$. If $k\in L_2(\mathbb{R}^+)$ is continuous, then
condition \eqref{e4.6} is equivalent to (see \cite{b2})
\begin{equation}
\Sigma _{i,j=1}^nk(x_i-x_j)\bar{\xi _i}\xi _j\ge 0  \label{e4.7}
\end{equation}
for all positive integers n,  $(x_1,\dots ,x_n)\in \mathbb{R}^n$ and
$(\xi _1,\dots ,\xi _n)\in \mathbb{C}^n$. A complex valued function $k(t)$
satisfying \eqref{e4.7} is called a positive definite function.
The functions $e^{-c|t|}$  for $c>0$, $e^{-t^2}$, $(1+t^2)^{-1}$
are positive definite by a theorem of Mathias since their Fourier
transforms are positive and integrable. Moreover, any real, even,
continuous function $k(t)$ which is convex on $(0,\infty)$, i.e.,
$k((t_1+t_2)/2)\le (k(t_1)+k(t_2))/2$, and such that
$\lim_{t\to \infty}k(t)=0$ is also positive definite (P\'olya). If, for
example, $k(t)=e^{-c|t|}$ with $c>0$, then $\hat
k(\xi)=2c/(\xi^2+c^2)$ and the corresponding linear map $\lambda
I-K$ given by \eqref{e1.2} is Fredholm of index zero in $L_2$ for
each $\lambda \notin [0,2/c]$ since $\lambda -\hat k(\xi)\ne 0$.
Note that the spectrum of $K$ is $[0,2/c]$ and so $K$ is not
compact.

\section{Nonlinear perturbations of integral equations on the real line}


The study of \eqref{e1.2} with the integral over $\mathbb{R}$, i.e., of
\begin{equation}
 \lambda x(s)-\int_{-\infty}^{\infty} k(s-t)x(t)dt=y(s)  \label{e5.1}
\end{equation}
where $k:\mathbb{R}\to \mathbb{C}$ is in
$L_1(\mathbb{R},\mathbb{C})$ and $y(s)\in
L_1(\mathbb{R},\mathbb{C})$, is much simpler and is based on using
integral Fourier transform. Namely, if $k(t)\in
L_1(-\infty,\infty)$ and $\lambda -\hat k(\xi)\ne 0$ in
$(-\infty,+\infty)$, then \eqref{e4.1} has a unique solution in
$L_1(-\infty,\infty)$ for each $y\in L_1(-\infty,\infty)$
(Wiener). Hence, one can study nonlinear perturbations of such linear
equations
\begin{equation}
\lambda x(s)-\int_{-\infty}^{\infty} k(s-t)x(t)dt+ (Nx)(s)=y(s)
 \label{e5.2}
\end{equation}
with a suitable nonlinear mapping $N$, as above, or using the
theory of Hammerstein equations. Let $K$ be a linear map defined
by the integral in \eqref{e5.1} and $A=\lambda I-K$. Then
\eqref{e5.2} is equivalent to $ Ax+Nx=y$ and the results of
Sections 2 and 4 (with $i(A)=0$) are valid for \eqref{e5.2}
under the corresponding assumption on $k$ and $N$. The problem is
much more difficult if we assume that the kernel is general, i.e.,
if we look at
\begin{equation}
 \lambda x(s)-\int_{-\infty}^{\infty} k(s,t)x(t)dt+ (Nx)(s)=y(s).
 \label{e5.3}
\end{equation} Its operator form is $(\lambda I-K)x+Nx=y$ in
$X=L_p(\mathbb{R},\mathbb{C})$. Under some general conditions on
$K$, it has been shown in \cite{a1} that $A=\lambda I-K$  is
continuously invertible in $X$ if $N(A)=\{0\}$. Hence, again the
results of Sections 2 and 4 (with $i(A)=0$) are valid for
\eqref{e5.3} with the corresponding assumptions imposed on
$k(s,t)$ and $N$. For the sake of illustration, we just state
explicitely the following result when $N$ is the Nemitskii
mapping. Denote by $BC(\mathbb{R})$ the space of complex valued
continuous and bounded functions on $\mathbb{R}$.


\begin{theorem} \label{thm5.1}
Let $k\in L_1(\mathbb{R},\mathbb{C})$, $X=L_p(\mathbb{R},\mathbb{C})$
for $1<p<\infty$, $Q\subset \mathbb{C}$ be compact and convex and
$\lambda \ne 0$. Let, for every $z\in L_Q$, the equation
\begin{equation}
\lambda x(s)-\int_{-\infty}^{\infty}k(s-t)z(t)x(t)dt=0,
\quad s\in \mathbb{R}  \label{e5.4}
\end{equation}
have only the trivial solution in $BC(\mathbb{R})$
and $F:\mathbb{R}^+\times \mathbb{R}\to \mathbb{R}$ be a
Caratheodory function such that $F(s,0)\in X$ and
$$
|F(s,u)-F(u,v)|\le k|u-v|\;\hbox{ for all}\; s\in \mathbb{R}^+,
\quad u,v\in \mathbb{R}
$$
for some $k$ sufficiently small. Then, for each $z\in L_Q$, the equation
\begin{equation}
\lambda x(s)-\int_{-\infty}^{\infty}k(s-t)z(t)x(t)dt + F(s,x(s))=y(s),
\quad s\in \mathbb{R}  \label{e5.5}
\end{equation}
is uniquely approximation solvable for each $y\in X$ with respect
to $\Gamma_0$ for $X$.
\end{theorem}


\begin{proof} Let $A:X\to X$ be the linear map defined by \eqref{e5.4}.
Then $A$ is continuously invertible on $X$ \cite{a1}. If
$(Nx)(s)=F(s,x(s))$, then $N:X\to X$ and \eqref{e5.5} is
equivalent to $x+A^{-1}Nx=A^{-1}y$, where $A^{-1}N$ is a
$k_1=k/c$-contraction with $k_1<1$, since $\|x-Kx\|\ge c\|x\|$ and
$c>k$ 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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\end{document}