\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 113, pp. 1--26.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2009/113\hfil Homeomorphisms and Fredholm theory]
{Homeomorphisms and Fredholm theory for perturbations of nonlinear
Fredholm maps of index zero with applications}
\author[P. S. Milojevi\'c\hfil EJDE-2009/113\hfilneg]
{Petronije S. Milojevi\'c}
\address{Petronije S. Milojevi\'c \newline
Department of Mathematical Sciences and CAMS,
New Jersey Institute of Technology, Newark, NJ 07102, USA}
\email{milojevi@adm.njit.edu}
\thanks{Submitted March 6, 2009. Published September 12, 2009.}
\subjclass[2000]{47H15, 35L70, 35L75, 35J40}
\keywords{Nonlinear Fredholm mappings; index zero; homeomorphism;
\hfill\break\indent Fredholm alternative;
potential problems; nonlinear boundary conditions;
\hfill\break\indent quasilinear elliptic equations}
\begin{abstract}
We develop a nonlinear Fredholm alternative theory involving
$k$-ball and $k$-set perturbations of general homeomorphisms and
of homeomorphisms that are nonlinear Fredholm maps of index zero.
Various generalized first Fredholm theorems and finite
solvability of general (odd) Fredholm maps of index zero are also
studied. We apply these results to the unique and finite solvability
of potential and semilinear problems with strongly nonlinear boundary
conditions and to quasilinear elliptic equations. The basic tools
used are the Nussbaum degree and the degree theories for nonlinear
$C^1$-Fredholm maps of index zero and their perturbations.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\section{Introduction}
Tromba \cite{t1} proved that a locally injective and proper Fredholm map
$T$ of index zero is a homeomorphism. The purpose of this paper is
to give various extensions of this result to maps of the form $T+C$,
where $T$ is a homeomorphism and $C$ is a $k$-set contractive
map. We prove several Fredholm alternative results and various
extensions of the first Fredholm theorem for this class of maps
using either Nussbaum's degree or the degree theories
for (non) compact perturbations of Fredholm maps as developed by
Fitzpatrick, Pejsachowisz, Rabier, Salter \cite{f1,p1,r2} and
Benevieri, Calamai, Furi \cite{b3,b4,b5}.
Applications to potential equations with nonlinear boundary conditions
and to Dirichlet problems for quasilinear elliptic equations are given.
Let us describe our main results in more detail. Throughout the paper,
we assume that $X$ and $Y$ are infinite dimensional Banach spaces.
In Section 2, we establish a number of nonlinear Fredholm alternatives
involving $k$-ball and $k$-set perturbations $C:X\to Y$ of general
homeomorphisms $T:X\to Y$ as well as of homeomorphisms that are
nonlinear Fredholm maps of index zero. In particular, we obtain
various homeomorphism results for $T+C$ assuming that it is locally
injective, satisfies
\begin{description}
\item[Condition (+)] $\{x_n\}$ is bounded whenever $\{(T+C)x_n\}$ converges,
\end{description}
and $\alpha (C)<\beta (T)$, \cite{f2}, using the set measure
of noncompactness $\alpha$,
\begin{gather*}
\alpha(T)=\sup\{\alpha (T(A))/\alpha (A): A\subset X
\text{ bounded},\; \alpha(A) > 0\},\\
\beta(T)=\inf\{\alpha (T(A))/\alpha (A): A\subset X
\text{ bounded},\; \alpha(A) > 0\}.
\end{gather*}
$\alpha (T)$ and $\beta (T)$ are related to the properties of compactness
and properness of the map $T$, respectively. In particular, these
results show that such homeomorphisms are stable
under $k$-set contractive perturbations.
We also prove such results when $T+C$ is asymptotically close to a
suitable map that is positively homogeneous outside some ball.
In the last part of Section 2, we establish Fredholm alternatives
for equations of the form $Tx+Cx+Dx=f$
with $T$ either a homeomorphism or a Fredholm map of index zero, assuming
that $\alpha(D)< \beta (T)-\alpha(C)$, $T+C$ is asymptotically close
to a k-positive homogeneous map and $D$ quasibounded.
We show that these equations are either uniquely solvable or are
finitely solvable for almost all right hand sides and that the
cardinality of the solution set is constant on
certain connected components in $Y$. In particular, we obtain such
alternatives for $T+D$ when $T$ is either a $c$-expansive homeomorphism
or an expansive along rays local homeomorphism
and $D$ is quasibounded with $\alpha(D) 0$, then $T$ is proper on bounded closed sets.
\item $\beta (T)-\alpha (C)\le \beta (T+C)\le \beta (T)+\alpha (C).$
\item If $T$ is a homeomorphism and $\beta(T) > 0$, then $\alpha(T^{-1})\beta(T)=1$.
\end{enumerate}
If $T:X\to Y$ is a homeomorphism, then (3) implies
$1=\beta(I)=\beta(T^{-1}oT)\le \alpha(T^{-1})\beta(T)$.
Hence, $\beta(T)>0$.
If $L : X\to Y$ is a bounded linear operator, then
$\beta(L) >$ 0 if and only if $\mathop{\rm Im} L $ is closed and
$\dim \ker L< \infty$ and $\alpha(L)\le \|L\|$.
Moreover, one can prove that $L$ is Fredholm if and only if
$\beta(L) > 0$ and $\beta(L^*) > 0$,
where $L^*$ is the adjoint of $L$.
Let $T:U\to Y$ be, as before, a map from an open subset $U$ of a
Banach space $X$ into a Banach space $Y$, and let $p\in U$ be fixed.
Let $B_r(p)$ be the open ball in $X$ centered at $p$ with radius $r$.
Suppose that $B_r(p)\subset U$ and set
$$
\alpha(T|_{B_r(p)})=\sup\{\alpha T(A)/\alpha (A):
A\subset B_r(p) \text{ bounded},\; \alpha(A) > 0\}.
$$
This is non-decreasing as a function of $r$, and clearly
$\alpha(T|_{B_r(p)})\le \alpha(T)$.
Hence, the following definition makes sense:
$$
\alpha_p(T)= \lim_{r\to 0}\alpha (T|_{B_r(p)}).
$$
Similarly, we define $\beta_p(T)$. We have
$\alpha _p(T) \le \alpha(T)$ and
$\beta_p(T)\ge \beta (T)$ for any $p$. If $T$ is of class $C^1$, then
$\alpha _p(T) =\alpha(T'(p))$ and $\beta_p(T) = \beta (T'(p))$ for
any $p$ \cite{c1}.
Note that for a Fredholm map $T:X\to Y$, $\beta_p(T)>0$ for all $p\in X$.
Recall that a map $T:X\to Y$ is a $c$-expansive map if
$\|Tx-Ty\|\ge c\|x-y\|$ for all $x,y\in X$ and some $c>0$.
\begin{example} \label{exa2.1} \rm
Let $T:X\to Y$ be continuous and $c$-expansive
for some $c>0$. Then $\alpha (T)\ge \beta(T)\ge c$. If $T$ is also
a homeomorphism,
then $\alpha(T^{-1})=1/\beta(T)\le 1/c$.
\end{example}
\begin{example} \label{exa2.2} \rm
Let $T:X\to Y$ be continuous and for each $p\in X$ there is
an $r>0$ such that $T:B(p,r)\to Y$ has the form $Tx=T(p)+L(x-p)+R(x)$,
where $L:X\to Y$ is a continuous linear map such that
$\|Lx\|\ge c_1\|x\|$ for some $c_1>0$ and all x, and $R$ is
a Lipschitz map with Lipchitz constant $c_20$ and therefore $\beta_p(T)\ge c$,
$\alpha_{T(p)}(T^{-1})=1/\beta_p(T)\le 1/c$ for some $c=c(p)>0$.
\end{example}
For a continuous map $F:X\to Y$, let $\Sigma$ be the set of all
points $x\in X$ where $F$ is not locally invertible and let
$\mathop{\rm card}F^{-1}(\{f\})$ be
the cardinal number of the set $F^{-1}(\{f\})$.
We need the following result.
\begin{theorem}[Ambrosetti] \label{thm2.1}
Let $F\in C(X,Y)$ be a proper map. Then
the cardinal number $\mathop{\rm card}F^{-1}(\{f\})$ is constant,
finite (it may be even 0) on each connected component of the
set $Y\setminus F(\Sigma)$.
\end{theorem}
In \cite{m2}, using Browder's theorem \cite{b6}, we have shown that
if $T:X\to Y$ is closed on bounded
closed subsets of X and is a local homeomorphism, then it is
a homeomorphism if and only if it satisfies condition (+).
Now, we shall look at perturbations of such maps.
\begin{theorem}[Fredholm Alternative] \label{thm2.2}
Let $T:X\to Y$ be a homeomorphism and
$C:X\to Y$ be such that $\alpha (C)<\beta (T)$
($T$ be a $c$-expansive homeomorphism and $C$ be k-$\phi$-contraction
with $k0$, $T+C$ is proper on
bounded closed subsets of $X$.
Hence, if $T+C$ satisfies condition (+), then it is a proper map and
therefore closed.
Assume that (i) holds. We shall show that $T+C$ is an open map.
The equation $Tx+Cx=f$ is equivalent to $y+CT^{-1}y=f$, $y=Tx$.
Then the map $CT^{-1}$ is $k/c$-$\phi$-contractive with $k/c<1$ if $T$
is expensive.
If $\alpha (C)<\beta (T)$, then the map $CT^{-1}$ is
$\alpha(CT^{-1})$-contractive
since
$$
\alpha (CT^{-1})\le \alpha (C)\alpha (T^{-1})=\alpha(C)/\beta(T)<1.
$$
Moreover, $I+CT^{-1}$ is (locally) injective since such is $T+C$. Hence,
$I+CT^{-1}$ is an open map (\cite{n2}, see also \cite{d1}).
Thus, $T+C$ is an open map since $T+C=(I+CT^{-1})T$ and therefore
it is a local homeomorphism.
Hence, it is a homeomorphism by Browder's theorem \cite{b6} if (a) holds.
Let (i)(b) hold. Then
$T+C$ is surjective since $(T+C)(X)$ is open and closed, and it is
therefore a homeomorphism.
(ii) Suppose that $T+C$ is not injective (locally injective, respectively).
We have seen above that $CT^{-1}$ is k-contractive, $k<1$, and
$I+tCT^{-1}$ satisfies condition (+) since $C$ is bounded.
Hence, using the homotopy $H(t,x)=x+tCT^{-1}x$ and the degree
theory for condensing maps \cite{n2}, we
get that $I+CT^{-1}$ is surjective. Therefore, $T+C$ is surjective.
Since $T+C$ is proper on $X$,
the cardinal number $\mathop{\rm card}(T+C)^{-1}(f)$ is positive,
constant and finite on each connected component of the set
$Y\setminus (T+C)(\Sigma)$ by Theorem \ref{thm2.1}.
\end{proof}
\begin{remark} \label{rmk2.1} \rm
Under the conditions of Theorem \ref{thm2.2}, we have that condition (+) is
equivalent to $R(T+C)(X)$ is closed when $T+C$ injective.
\end{remark}
\begin{remark} \label{rmk2.2} \rm
If $X$ is an infinite dimensional Banach space and
$T:X\to X$ is a homeomorphism, then $T$ satisfies condition (+)
but it need not be coercive in the sense that
$\|Tx\|\to \infty$ as $\|x\|\to \infty$. It was proved in
\cite[Corollary 8]{c3} that for every proper subset $U\subset X$,
there is a homeomorphism $T:X\to X$ such that it maps $X\setminus U$
into $U$. Then, selecting $U$ to be a ball
in $X$, such a homeomorphism is not coercive.
\end{remark}
\begin{corollary} \label{coro2.1}
Let $T:X\to Y$ be a homeomorphism
and $C:X\to Y$ be continuous and uniformly bounded; i.e.,
$\|Cx\|\le M$ for all x and some $M>0$, and $\alpha(C)<\beta (T)$.
Then either
\begin{itemize}
\item[(i)] $T+C$ is injective, in which case $T+C$ is a homeomorphism, or
\item[(ii)] $T+C$ is not injective, in which case $T+C$ is surjective,
$(T+C)^{-1}(f)$ is compact for
each $f\in Y$, and the cardinal number
$\mathop{\rm card}(T+C)^{-1}(f)$ is positive, constant and finite
on each connected component of the set $Y\setminus (T+C)(\Sigma)$.
\end{itemize}
\end{corollary}
\begin{proof}
We note first that the equation $Tx+Cx=f$ is equivalent to
$y+CT^{-1}y=f$, with $y=Tx$. Using the uniform boundedness of $C$,
it is easy to see that for each $f\in Y$,
$\|H(t,y)=y+tCT^{-1}y-tf\|\to \infty $ as
$\|y\|\to \infty$ uniformly in $t\in [0,1]$. Hence, the Nussbaum degree
$\deg (I-CT^{-1},B(0,r),f)\ne 0$ for some $r>0$ large. Thus,
the equation $Tx+Cx=f$ is solvable for each $f\in Y$.
Hence, (i) follows from Theorem \ref{thm2.2}(i)(b).
We have remarked before that $T+C$ is proper on bounded closed subsets.
Next, we shall show that $T+C$ is a proper map. Let $K\subset Y$ be
compact and $x_n\in (T+C)^{-1}(K)$.
Then $y_n=Tx_n+Cx_n\in K$ and we may assume that $y_n\to y\in K$.
Since $C$ is uniformly bounded,
$\{Tx_n\}$ is bounded. Set $z_n=Tx_n$ and note that
$y_n=(I+CT^{-1})z_n\in K$. Hence,
$z_n\in (I+CT^{-1})^{-1}(y_n)\subset (I+CT^{-1})^{-1}(K)$ and
$\{z_n\}$ bounded. Since $I+CT^{-1}$ is proper
on bounded closed subsets, we may assume that
$z_n\to z\in B(0,r)\cap (I+CT^{-1})^{-1}(K)$ for some $r>0$.
Hence, $x_n=T^{-1}z_n\to T^{-1}z=x\in (T+C)^{-1}(K)$ since
$T+C$ is proper on bounded closed subsets.
Thus, $T+C$ is a proper map and (ii) follows by Theorem \ref{thm2.1}.
\end{proof}
When $T$ is Fredholm of index zero, then the injectivity of $T+C$ can
be replaced by the local injectivity.
\begin{theorem}[Fredholm Alternative] \label{thm2.3}
Let $T:X\to Y$ be a Fredholm map
of index zero and $C:X\to Y$ be such that $\alpha (C)<\beta (T)$.
Then either
\begin{itemize}
\item[(i)] $T+C$ is locally injective, in which
case it is an open map and it is a homeomorphism if and only if
one of the following conditions holds
\begin{itemize}
\item[(a)] $T+C$ is closed ( in particular, proper, or
satisfies condition {\rm (+)}),
\item[(b)] T+C is injective and $R(T+C)$ is closed, or
\end{itemize}
\item[(ii)] $T+C$ is not locally injective, in which case,
assuming additionally that $T$ is locally injective and $T+tC$ satisfies
condition {\rm (+)}, the equation $Tx+Cx=f$ is solvable for each
$f\in Y$ with $(T+C)^{-1}(f)$ compact
and the cardinal number $\mathop{\rm card}(T+C)^{-1}(f)$ is positive,
constant and finite
on each connected component of the set $Y\setminus (T+C)(\Sigma)$.
\end{itemize}
\end{theorem}
\begin{proof}
As observed before, $T+C$ is proper on bounded closed subsets of $X$.
If it satisfies condition (+), then it is proper. Let (i) hold.
Then $T+C$ is an open map by a theorem of Calamai \cite{c1}. Hence,
$T+C$ is a local homeomorphism. If (a) holds, then $T+C$ is a
homeomorphism if and only if it is a closed map by Browder's
theorem \cite{b6}.
If (b) holds, then T+C is surjective since $R(T+C)$ is open and closed.
Hence, it is a homeomorphism.
Let (ii) hold. Then $T$ is a homeomorphism
by part (i)-a) (or by Tromba's theorem \cite{t1}) since it is proper on
bounded closed subsets and satisfies condition (+).
Hence, the conclusions follow as in Theorem \ref{thm2.2}(ii).
\end{proof}
Condition $\alpha (C)<\beta (T)$ in (i) can be replaced by
$\alpha_p(C)<\beta_p(T)$ for each
$p\in X$ since T+C is also open in this case \cite{c1}. This condition
always holds if $T$ is a Fredholm map
of index zero and C is compact. In view of this remark, we have
the following extension of Tromba's homeomorphism result for
proper locally injective Fredholm maps of index zero \cite{t1}.
\begin{corollary} \label{coro2.2}
Let $T:X\to Y$ be a Fredholm map of index zero,
$C:X\to Y$ be compact and $T+C$ be locally injective and closed.
Then $T+C$ is a homeomorphism.
\end{corollary}
\begin{corollary} \label{coro2.3}
Let $T:X\to Y$ be a locally injective closed Fredholm map of index zero,
$C:X\to Y$ be continuous and uniformly bounded and $\alpha(C)<\beta (T)$.
Then either
\begin{itemize}
\item[(i)] $T+C$ is locally injective, in which case $T+C$ is a
homeomorphism, or
\item[(ii)] $T+C$ is not locally injective, in which case $T+C$ is
surjective, $(T+C)^{-1}(f)$ is compact for each $f\in Y$, and the
cardinal number $\mathop{\rm card}(T+C)^{-1}(f)$ is positive,
constant and finite on each connected component
of the set $Y\setminus (T+C)(\Sigma)$.
\end{itemize}
\end{corollary}
\begin{proof}
$T$ is a homeomorphism by Corollary \ref{coro2.2}. We have shown in the proof
of Corollary \ref{coro2.1} that $T+C$ is proper
and surjective. If $T+C$ is locally injective, then it is a
homeomorphism by
Theorem \ref{thm2.3}(i)(a). Part (ii) follows from Theorem \ref{thm2.1}.
\end{proof}
The following lemmas, needed later on, give a number of particular
conditions on $T$ and $C$ that imply condition (+). Recall
that a map $C$ is quasibounded if, for some $k>0$,
$$
|C|=\limsup_{\|x\|\to \infty}\|Cx\|/\|x\|^k<\infty.
$$
\begin{lemma} \label{lem2.1}
Suppose that $T,C:X\to Y$ and either one of the following
conditions holds
\begin{itemize}
\item[(i)] $\|Cx\|\le a\|Tx\|+b$ for some constants $a\in [0,1)$ and $b>0$ and all $\|x\|$ large
and $\|Tx\|\to \infty$ as $\|x\|\to \infty$.
\item[(ii)] There exist constants $c,c_0, k>0$ and $R>R_0$ such that
$$
\|Tx\|\ge c\|x\|^k-c_0\quad \text{for all }\|x\|\ge R.
$$
and $C$ is quasibounded with the quasinorm $|C|0$ such that if $Ax=0$, then
$\|x\|\le M$
\item[(ii)] $A$ is injective
\item[(iii)] $A$ is locally injective and \eqref{eq2.1} holds for
all $\lambda >0$.
\end{itemize}
Then there exist constants $c>0$ and $R>R_0$ such that
\begin{equation}
\|Ax\|\ge c\|x\|^k\quad \text{for all }\|x\|\ge R\label{eq2.2}
\end{equation}
and, in addition, $A^{-1}$ is bounded when (ii) holds. Moreover, if $A$ is
positively k-homogeneous, then $Ax=0$ has only the trivial solution if
and only if \eqref{eq2.2} holds.
(b) If $T:X\to Y$ is asymptotically close to A with $|T-A|$ sufficiently
small, then $T$ also satisfies \eqref{eq2.2} with $c$ replaced
by $c-|T-A|$.
\end{lemma}
Let us connect this with eigenvalue problems. Let $T, C$ be
asymptotically close to
$k$-positive homogeneous maps $T_0$ and $C_0$, respectively.
We say that $\mu$ is not an eigenvalue
of $T_0$ relative to $C_0$ if $T_0x=\mu C_0x$ implies that $x=0$.
Then one can use Lemma \ref{lem2.2} to show that $T-\mu C$ satisfies
condition \eqref{eq2.2} provided that $\mu$ is not an eigenvalue
of $T_0$ relative to $C_0$.
\begin{lemma} \label{lem2.3}
(i) Let $A:X=X^{**}\to X^*$ be k-positive homogeneous such
that $(Ax,x)\ge m\|x\|^k$ for all
$x\in X$ and some $m>0$, $k\ge 2$, $G:X\to X^*$ be k-positive homogeneous,
$g(x)=(Gx,x)$ be weakly continuous and
$f(x)=(Ax+Gx,x)$ be weakly lower semicontinuous and positive definite,
i.e., $f(x)>0$ for $x\ne 0$. Then
$$
(Ax+Gx,x)\ge c\|x\|^k\quad \text{for all }x\in X\text{ and some }c>0.
$$
(ii) If $T,C:X\to X^*$ are asymptotically close to $A$ and $G$,
respectively with $|T-A|$ and $|C-G|$ sufficiently small, then there
is a $c_1>0$ and an $R>0$ such that
\begin{equation}
\|Tx+Cx\|\ge c_1\|x\|^{k-1}\quad \text{for all }\|x\|\ge R.
\label{eq2.3}
\end{equation}
\end{lemma}
\begin{proof}
Let $c=\inf_{\|x\|=1}f(x)$. If $c=0$, then there is a sequence
$\{x_n\}$ such that $f(x_n)\to c=0$. We may assume that
$x_n\rightharpoonup x_0$.
Since $f$ is weakly lower semicontinuous, we get
$$
\lim_{n\to \infty}f(x_n)\ge f(x_0).
$$
Hence, $f(x_0)\le 0$ and therefore $x_0=0$ by the positive
definiteness of $f$.
On the other hand,
$$
\lim_{n\to \infty}f(x_n)\ge m
\lim_{n\to \infty}\|x\|^k-\lim_{n\to \infty}|(Gx_n,x_n)|\ge m>0
$$
since $g(x)$ is weakly continuous and $\|x_n\|=1$. This is a
contradiction, and therefore
$c>0$. By the k-positive homogeneity of f we get that
$f(x/\|x\|)\ge c$ for each
$x\ne 0$. Thus, $f(x)\ge c\|x\|^k$.
(ii) Let $\epsilon >0$ be such that $2\epsilon +|T-A|+|C-G|0$ in part (i). Then there is an $R>0$ such that
$\|Tx-Ax\|\le (\epsilon +|T-A|)\|x\|^{k-1}$ and
$\|Cx-Gx\|\le (\epsilon +|C-G|)\|x\|^{k-1}$ for all
$\|x\|\ge R$. This implies
\eqref{eq2.3} with $c_1=c-2\epsilon-|T-A|-|C-G|$.
\end{proof}
For $G:X\to X^*$, define $g(x)=(Gx,x)$. If $G$ is weakly continuous,
$g$ need not be weakly
lower semicontinuous. It is easy to show that
(1) g is weakly lower semicontinuous if G is weakly continuous and
monotone, i.e., $(Gx-Gy,x-y)\ge 0$,
or G is completely continuous, i.e., $Gx_n\to Gx$ if
$x_n\rightharpoonup x$.
(2) If $A:X\to X^*$ is such that $h(x)=(Ax,x)$ is weakly lower
semicontinuous and $G$ is completely continuous, then
$f(x)=(Ax+Gx,x)$ is weakly lower semicontinuous.
\begin{lemma}[\cite{m2}] \label{lem2.4}
Let $T:X\to Y$ be a homeomorphism and $C:X\to Y$.
\begin{itemize}
\item[(i)] If $I+CT^{-1}:Y\to Y$ is proper on bounded closed subsets of $Y$ and
if either $T$ or C is bounded, then $T+C$ is proper on bounded closed subsets of $X$.
\item[(ii)] If $T^{-1}$ is bounded and $I+CT^{-1}:Y\to Y$ satisfies
condition {\rm (+)}, then
$T+C$ satisfies condition {\rm (+)}. Conversely, if either $T$ or C is bounded, and
$T+tC$, $t\in[0,1]$, satisfies condition {\rm (+)}, or
$C$ has a linear growth and $T^{-1}$ is quasibounded with a sufficiently small quasinorm,
then $I+tCT^{-1}:Y\to Y$ satisfies condition {\rm (+)}, $t\in [0,1]$.
\end{itemize}
\end{lemma}
\begin{lemma} \label{lem2.5}
Let $T:X\to Y$ be a $C^1$ local homeomorphism and $C:X\to Y$ be
$c$-expansive. Then $T+C$ is locally expansive and therefore it is
locally injective on X.
\end{lemma}
\begin{proof}
Let $p\in X$ be fixed. By Example \ref{exa2.3}, there is an $r=r(p)>0$ such that
$T:B(p,r)\to Y$ is $c(p)$-expansive for some $c(p)\in (0,c)$.
Hence, $T+C$ is $c-c(p)$-expansive on B and is therefore locally
injective on X.
\end{proof}
Next, we shall prove some nonlinear extensions of the Fredholm
alternative to set contractive like perturbations of
homeomorphisms as well as of Fredholm maps of index zero that are
asymptotically close to positive k-homogeneous maps.
\begin{theorem}[Fredholm alternative] \label{thm2.4}
Let $T:X\to Y$ be a homeomorphism and $C,D:X\to Y$ be continuous
maps such that $\alpha(D)<\beta(T)-\alpha(C)$ with $|D|$
sufficiently small (T be a $c$-expansive homeomorphism and
$C$ be a k-$\phi$-contraction with
$k0$
such that $ A(\lambda x)=\lambda^k Ax$
for all $\|x\|\ge R_0$, all $\lambda \ge 1$ with $A^{-1}(0)$
bounded and $|T+C-A|$
sufficiently small. Then either
\begin{itemize}
\item[(i)] $T+C+D$ is injective, in which case $T+C+D$ is a
homeomorphism, or
\item[(ii)] $T+C+D$ is not injective, in which case
the solution set $(T+C+D)^{-1}(\{f\})$
is nonempty and compact for each $f\in Y$ and the cardinal number
$\mathop{\rm card}(T+C+D)^{-1}(\{f\})$
is constant, finite and
positive on each connected component of the set
$Y\setminus (T+C+D)(\Sigma)$.
\end{itemize}
\end{theorem}
\begin{proof}
Since $T$ is a homeomorphism, it is proper and $\beta (T)>0$. Moreover,
$T+C$ satisfies condition (+) by Lemma \ref{lem2.2}(i)(a).
Since $\alpha (C)<\beta (T)$, $T+C$ is proper on bounded closed subsets
of X and is therefore proper by condition (+). It follows that
$T+C$ is a homeomorphism by Theorem \ref{thm2.2}(i)(a).
Next, we claim that $H_t=T+C+tD$ satisfies condition (+).
Let $y_n=(T+C+t_nD)x_n\to y$ as $n\to \infty$ with $t_n\in [0,1]$,
and suppose that $\|x_n\|\to \infty$. Then, by Lemma \ref{lem2.2}(b) with
$c_1=c-|T+C|$,
$$
c_1\|x_n\|^k-c_0\le \|(T+C)x_n\|\le \|y_n\|+(|D|+\epsilon)\|x_n\|^k
$$
for all $n$ large and any $\epsilon >0$ fixed.
Dividing by $\|x_n\|^k$ and letting $n\to \infty $, we get that
$c\le |D|$. This contradicts our assumption that $|D|$ is
sufficiently small and
therefore condition (+) holds for $H_t$.
Let (i) hold. Since $\alpha (D)< \beta (T)-\alpha (C)\le \beta(T+C)$
and $H_1=T+C+D$ satisfies condition (+), $T+C+D$ is a homeomorphism
by Theorem \ref{thm2.2}(i)(a).
Next, let (ii) hold.
Since $\alpha (D)< \beta(T+C)$, $T+C$ is a homeomorphism and $H_t$
satisfies condition (+), the equation $Tx+Cx+Dx=f$ is solvable
for each $f$ by Theorem \ref{thm2.2}(ii). Moreover, $T+C+D$ is proper on
closed bounded sets since
$\beta (T+C+D)\ge \beta (T)-\alpha (C+D)
\ge \beta (T)-\alpha (C)-\alpha (D)>0$.
Hence, the map $T+C+D$ is proper on X by condition (+), and the
other conclusions follow from Theorem \ref{thm2.1}.
\end{proof}
If $C=0$ in Theorem \ref{thm2.4}, then the injectivity of $T+D$ can be weaken to
local injectivity when $T$ is $c$-expansive.
\begin{corollary} \label{coro2.4}
Let $T:X\to Y$ be a $c$-expansive homeomorphism and
$D:X\to Y$ be a continuous map such that $\alpha(D)0\quad\text{for all }x\in H,
$$
or
$$
\sup_{\|h\|=1}Re(T'(x)h,h)<0\quad \text{for all } x\in H,
$$
\item[(d)] $X$ is reflexive and $T:X\to X^*$ is a $C^1$ potential map.
\end{itemize}
Then either
\begin{itemize}
\item[(i)] $T+D$ is locally injective, in which case $T+D$ is a
homeomorphism, or
\item[(ii)] $T+D$ is not locally injective, in which case
the solution set $(T+D)^{-1}(\{f\})$
is nonempty and compact for each $f\in Y$ and the cardinal number
$\mathop{\rm card}(T+D)^{-1}(\{f\})$ is
constant, finite and positive on each connected component of
the set $Y\setminus (T+D)(\Sigma)$.
\end{itemize}
\end{corollary}
\begin{proof}
$T$ is a homeomorphism by Chang-Shujie \cite{c2} in parts (a), (d),
and by Her\-nandez-Nashed \cite{h1} in parts (b), (c).
Since $\|Tx\|\ge c\|x\|-\|T(0)\|$, Corollary \ref{coro2.4} applies.
\end{proof}
Recall that a map $T:X\to Y$ is expansive along rays if for each
$y\in Y$, there is a $c(y)>0$ such that $\|Tx-Ty\|\ge c(y)\|x-y\|$
for all $x,y\in T^{-1}([0,y])$, where $[0,y]=\{ty:0\le t\le 1\}$.
\begin{theorem} \label{thm2.5}
An expansive along rays local homeomorphism $T:X\to Y$ is a
homeomorphism. A locally expansive local homeomorphism is a
homeomorphism if $m=\inf_xc(x)>0$. In general, a locally expansive
local homeomorphism need not be a homeomorphism.
\end{theorem}
\begin{proof}
Note first that if $T$ is a $c$-expansive local homeomorphism,
then it is an open map. Since $R(T)$ is also closed, $T$ is surjective.
Since it is injective, it is a homeomorphism.
Next, assume that $T$ is just expansive along rays.
We may assume that
$T(0)=0$. Since $T$ is a local homeomorphism, there is a ball $B$
about $0$ in Y and a continuous local inverse
$g:B\to X$ of $T$ with $g(0)=0$. Next, we shall continue the local
inverse $g$ along each ray from 0 as far out as possible.
To that end, let D be the set of all points $y\in Y$ such that
there is a continuous inverse g of $T$ defined on the ray
$[0,y]=\{ty:0\le t\le 1\}$ and $g(0)=0$. It is known that $D$ is an
open subset of $Y$, the value of $g(y)$ depends
uniquely on $T$ and $y$, the map $T^{-1}$ defined by $T^{-1}(y)=g(y)$
is an inverse of $T$ on $D$ and $T^{-1}$ is continuous
(see John \cite{j1} for details). Next, it has been shown
in Hernandez-Nashed \cite{h1} that $D=Y$. Hence, $T$ is a homeomorphism.
If $T$ is a locally expensive
local homeomorphism with $m>0$, then $T$ is a homeomorphism by
John's theorem \cite{j1} since the scalar
derivative of $T$ is $D_x^-T=\liminf_{y\to x}\|Tx-Ty\|/\|x-y\|=c(x)$
and $m=\inf_xD^-_xT>0$.
A $C^1$ local homeomorphism is locally expansive by Example \ref{exa2.3},
but it need not be a homeomorphism.
\end{proof}
\begin{corollary} \label{coro2.6}
Let $T:X\to Y$ be an expansive along rays local homeomorphism
and $D:X\to Y$ be a continuous maps such that $\alpha(D)<\beta (T)$
and $\|Dx\|\le M$ for all $x\in X$ and some $M>0$.
Then either
\begin{itemize}
\item[(i)] $T+D$ is injective, in which case $T+D$ is a homeomorphism,
or
\item[(ii)] $T+D$ is not injective, in which case the solution set
$(T+D)^{-1}(\{f\})$
is nonempty and compact for each $f\in Y$ and the cardinal number
$\mathop{\rm card}(T+D)^{-1}(\{f\})$ is constant, finite and
positive on each connected component of the set
$Y\setminus (T+D)(\Sigma)$.
\end{itemize}
\end{corollary}
The proof of the above corollary follows from Theorem \ref{thm2.5} and
Corollary \ref{coro2.1}.
Next, we shall give another extension of the Fredholm Alternative to
perturbations of nonlinear Fredholm maps of index zero.
\begin{theorem}[Fredhom Alternative] \label{thm2.6}
Let $T:X\to Y$ be a Fredholm map of index zero and
$C,D:X\to Y$ be continuous maps such that
$\alpha(D)< \beta (T)-\alpha(C)$
with $|D|$ sufficiently small, where
$$
|D|=\limsup_{\|x\|\to \infty}\|Dx\|/\|x\|^k< \infty.
$$
Assume that either $\|Tx+Cx\|\ge c\|x\|^k- c_0$ for all
$\|x\|\ge R$ for some R, c and $c_0$, or $T+C$ is asymptotically close
to a continuous, closed (in particular, proper) on bounded
and closed subsets of X positive k-homogeneous map $A$,
outside some ball in $X$, i.e., there are $k\ge 1$ and $R_0>0$ such that
$ A(\lambda x)=\lambda^k Ax$ for all $\|x\|\ge R_0$, all
$\lambda\ge 1$ and $(A)^{-1}(0)$ bounded with $|T+C-A|$
sufficiently small. Then either
\begin{itemize}
\item[(i)] $T+C+D$ is locally injective, in which case $T+C+D$
is a homeomorphism, or
\item[(ii)] $T+C+D$ is not locally injective, in which case, assuming
additionally that $T+C$ is locally injective, the solution set
$(T+C+D)^{-1}(\{f\})$ is nonempty and compact for each
$f\in Y$ and the cardinal number
$\mathop{\rm card}(T+C+D)^{-1}(\{f\})$ is constant, finite and
positive on each connected component of the set
$Y\setminus (T+C+D)(\Sigma)$.
\end{itemize}
\end{theorem}
\begin{proof}
Let (i) hold. Since $\alpha (D)< \beta (T)-\alpha (C)\le \beta(T+C)$,
we get $\beta(T+C+D)>0$ and therefore $T+C+D$ is proper on bounded
closed subsets of X. We have that $T+C+D$ satisfies condition (+)
as in Theorem \ref{thm2.4}. Hence, $T+C+D$ is proper on X and so it is
closed on $X$.
Since $\alpha(C+D)<\beta(T)$ and $T$ is Fredholm of index zero,
$T+C+D$ is an open map by Calamai's theorem \cite{c1}.
Hence, $T+C+D$ is surjective since the range $(T+C+D)(X)$ is both
open and closed. Since $T+C+D$ is a local homeomorphism, it
is a homeomorphism by the Banach-Mazur theorem.
Let (ii) hold. Since $\alpha (C)<\beta (T)$ and $T+C$ is locally
injective and satisfies condition (+) by Lemma \ref{lem2.2}(i)(a),
$T+C$ is a homeomorphism by Theorem \ref{thm2.3}(i).
Since $\alpha (D)< \beta (T)-\alpha (C)\le \beta(T+C)$,
the conclusions follow as in Theorem \ref{thm2.4}(ii).
\end{proof}
\begin{corollary} \label{coro2.7}
Let $T,C:X\to Y$ and $T+C$ be Fredholm maps of
index zero such that $\alpha (C)<\beta (T)$ and $|C|$ sufficiently
small, where
$$
|C|=\limsup_{\|x\|\to \infty}\|Cx\|/\|x\|^k< \infty.
$$
Assume that either $\|Tx\|\ge c\|x\|^k- c_0$ for all $\|x\|\ge R$ for
some $R$, $c$ and $c_0$, or $T$ is asymptotically close to a
continuous, closed (in particular, proper)
on bounded and closed subsets of $X$ positive k-homogeneous map $A$,
outside some ball in $X$; i.e., there are $k\ge 1$ and $R_0>0$ such that
$$
A(\lambda x)=\lambda^k Ax
$$
for all $\|x\|\ge R_0$, all $\lambda\ge 1$ and $(A)^{-1}(0)$ bounded
with $|T+C-A|$ sufficiently small.
Then either
\begin{itemize}
\item[(i)] T+C is locally injective, in which case T+C is a homeomorphism, or
\item[(ii)] T+C is not locally injective, in which case, assuming
additionally that $T$ is locally injective,
the solution set $(T+C)^{-1}(\{f\})$ is nonempty and compact for
each $f\in Y$ and the cardinal number
$\mathop{\rm card}(T+C)^{-1}(\{f\})$ is constant, finite and
positive on each connected component of the open and dense set
of regular values $R_{T+C}=Y\setminus (T+C)(S)$ of $Y$, where
$S$ is the set of singular points of $T+C$.
\end{itemize}
\end{corollary}
\begin{proof}
Part (i) and the surjectivity of $T+C$ follow from Theorem \ref{thm2.4}
(with $C$ replaced by $D$). As before, we have that
$T+C$ is proper on bounded and closed set and satisfies condition (+).
Hence, it is proper and the other conclusions follow from the general
theorem on nonlinear Fredholm maps of index zero (see \cite{z1}).
\end{proof}
\section{Finite solvability of equations with perturbations
of odd Fredholm maps of index zero}
In this section, we shall study perturbations of Fredholm maps of
index zero assuming that the
maps are odd. We shall first look at compact perturbations and use
the Fitzpatrick-Pejsachowisz-Rabier-Salter degree.
\begin{theorem}[Generalized First Fredholm Theorem] \label{thm3.1}
Let $T:X\to Y$ be a Fredholm map of index zero that is proper on
bounded and closed subsets of $X$ and
$C,D:X\to Y$ be compact maps with $|D|$ sufficiently small, where
$$
|D|=\limsup_{\|x\|\to \infty}\|Dx\|/\|x\|^k< \infty.
$$
Assume that $T+C$ is odd, asymptotically close to a continuous,
closed (in particular, proper)
on bounded and closed subsets of X positive k-homogeneous map $A$,
outside some ball in $X$, i.e., there exists $R_0>0$ such that
$$
A(\lambda x)=\lambda^k Ax
$$
for all $||x||\ge R_0$, for all
$\lambda\ge 1$ and some $k\ge 1$, and $\|x\|\le M<\infty$
if $Ax=0$ and $|T+C-A|$ sufficiently small.
Then the equation $Tx+Cx+Dx=f$ is solvable for each $f\in Y$ with
$(T+C+D)^{-1}(\{f\})$ compact and the cardinal number
$\mathop{\rm card}(T+C+D)^{-1}(\{f\})$
is constant, finite and positive on each connected component
of the set $Y\setminus (T+C+D)(\Sigma)$.
\end{theorem}
\begin{proof} Step 1. Let $p$ be a base point of $T$.
By Lemma \ref{lem2.2}, there is an $R>0$ such that
condition \eqref{eq2.2} holds for $T+C$. Define the homotopy
$H(t,x)=Tx+Cx+tDx-tf$ for $t\in [0,1]$.
Since $T+C$ satisfies \eqref{eq2.2}, it is easy to show that
$H(t,x)$ satisfies condition (+) and
therefore $H(t,x)\ne 0$ for $(t,x)\in [0,1]\times \partial B(0,R_1)$
for some $R_1\ge R$.
Next, we note that $H(t,x)$ is a compact perturbation $tD$ of the
odd map $T+C$ with $T$ Fredholm of
index zero. Then, by the homotopy theorem \cite[Corollary 7.2]{r2}
and the Borsuk theorem for such maps of Rabier-Salter \cite{r2},
we get that the Fitzpatrick-Pejsachowisz-Rabier-Salter degree
\begin{align*}
\deg_{T,p}(H_1,B(R_1,0),0)
&=\deg_{T,p}(T+C+D-f,B(R_1,0),0)\\
&=\nu\deg_{T,p}(T+C,B(R_1,0),0)\ne 0
\end{align*}
where $\nu$ is $1$ or $-1$.
Step 2. $T$ has no base point. Then pick a point $q\in X$ and
let $A:X\to Y$ be a continuous linear map with finite dimensional
range such that $T'(q)+A$ is invertible. Then
$T+A$ is Fredholm of index zero, proper on $\bar{B}(0,r)$ and $q$
is a base point of $T+A$. Moreover, $T+C+D=(T+A)+ (C-A)+D$.
We have reduced the problem to the case
when there is a base point for $T+A$ and the maps $T+A$, $C-A$
and $D$ satisfy the same
conditions of the theorem as the maps $T$, $C$ and $D$.
Then, as in Step 1,
\begin{align*}
\deg_{T+A,q}(H_1,B(R_1,0),0)
&= \deg_{T+A,q}((T+A)+(C-A)+D-f,B(R_1,0),0)\\
&= \nu\deg_{T+A,q}(T+C,B(R_1,0),0)\ne 0
\end{align*}
where $\nu$ is $1$ or $-1$.
Hence, by the existence theorem of this degree, we have that
$Tx+Cx+Dx=f$ is solvable in either case. Next, since $T+C+D$
is continuous and satisfies condition (+), it is proper since
it is proper on bounded closed sets as a compact perturbation
of such a map. Hence, the second part of the theorem follows
from Theorem \ref{thm2.1}.
\end{proof}
\begin{remark} \label{rmk3.1} \rm
Earlier generalizations of the first Fredholm theorem to
condensing vector fields, maps of type $(S_+)$, monotone like maps and
(pseudo) A-proper maps assumed the
homogeneity of $T$ with $Tx=0$ only if $x=0$ (see \cite{h2,m1,n1}
and the references therein).
\end{remark}
Next, we provide some generalizations of the Borsuk-Ulam principle
for odd compact perturbations of the identity. The first result
generalizes Theorem \ref{thm3.1} when $D=0$.
\begin{theorem} \label{thm3.2}
Let $T:X\to Y$ be a Fredholm map of index zero that is
proper on closed bounded subsets of $X$ and $C:X\to Y$ be compact
such that $T+C$ is odd outside some ball $B(0,R)$.
Suppose that $T+C$ satisfies condition {\rm (+)}. Then $Tx+Cx=f$ is solvable,
$(T+C)^{-1}(f)$ is compact for each $f\in Y$ and
the cardinal number $\mathop{\rm card}(T+C)^{-1}(f)$ is positive
and constant on each connected component of $Y\setminus (T+C)(\Sigma)$.
\end{theorem}
\begin{proof}
Condition (+) implies that for each $f\in Y$ there is an $r=r_f>R$
and $\gamma >0$ such that
$$
\|Tx+Cx-tf\|\ge \gamma \quad \text{for all }t\in [0,1],\|x\|=r.
$$
The homotopy $H(t,x)=Tx+Cx-tf$ is admissible for the Rabier-Salter
degree and $H(t,x)\ne 0$ on
$[0,1]\times \partial B(0,r)$. Hence, by the homotopy
\cite[Corollary 7.2]{r2}, if
$p\in X$ is a base point of $T$, then
$$
\deg_{T,p}(T+C-f,B(0,r),0)=\nu \deg_{T,p}(T+C,B(0,r),0)\ne 0
$$
since $\nu $ is plus or minus one and the second degree is odd by the
generalized Borsuk theorem in \cite{r2}. If $T$ has no base point,
then proceed as in Step 2 of the proof of Theorem \ref{thm3.1}.
Hence, the equation $Tx+Cx=f$ is solvable in either case.
The second part follows from Theorem \ref{thm2.1} since $T+C$ is proper
on bounded closed subsets and satisfies condition (+).
\end{proof}
Next, we shall prove a more general version of Theorem \ref{thm3.2}.
\begin{theorem} \label{thm3.3}
Let $T:X\to Y$ be a Fredholm map of index zero that is
proper on closed bounded subsets of $X$ and $C_1,C_2:X\to Y$
be compact such that $T+C_1$ is odd outside some ball $B(0,R)$.
Suppose that $H(t,x)=Tx+C_1x+tC_2x-tf$ satisfies condition {\rm (+)}.
Then $Tx+C_1x+C_2x=f$ is solvable for each $f\in Y$ with
$(T+C_1+C_2)^{-1}(f)$ compact and the cardinal number
$\mathop{\rm card}(T+C_1+C_2)^{-1}(f)$ is positive and constant
on each connected component of $Y\setminus (T+C_1+C_2)(\Sigma)$.
\end{theorem}
\begin{proof}
Condition (+) implies that for each $f\in Y$ there is an $r=r_f>R$
with $0\notin H([0,1]\times \partial B(0,r)$.
If $p$ is a base points of $T$, then by \cite[Theorem 7.1]{r2},
$$
\deg_{T,p}(H(1,.),B(0,r),0)=\nu \deg_{T,p}(T_1+C_1,B(0,r),0)\ne 0
$$
where $\nu\in \{-1,1\}$.
Next, if $T$ has no a base point, then pick $q\in X$ and
let $A$ be a continuous linear map from $X$ to $Y$ with finite
dimensional ranges such that $T'(q)+A$ is invertible. Then we can
rewrite $H$ as $H(t,x)=(T+A)x+(C_1-A)x+tC_2x-tf$, where
$T+A$, $C_1-A$ and $C_2$ satisfy all the conditions of the theorem
and $T+A$ has a base point. As in the first case, we get that
\begin{align*}
\deg_{T+A,q}(H(1,.),B(0,r),0)
&=\deg_{T+A,q}((T+A)+(C_1-A)+C_2-f,B(0,r),0)\\
&=\nu \deg_{T+A,q}((T+A)+(C_1-A),B(0,r),0)\ne 0
\end{align*}
where $\nu\in \{-1,1\}$.
Hence, the equation $Tx+C_1x+C_2x=f$ is solvable in either case.
The other conclusions follow from Theorem \ref{thm2.1}.
\end{proof}
Next, we shall study $k$-set contractive perturbations of
Fredholm maps of index zero. Denote by $\deg_{BCF}$ the degree
of Benevieri-Calamai-Furi.
When $T+C$ is not locally injective,
we have the following extension of Theorem \ref{thm2.6}.
\begin{theorem}[Generalized First Fredholm Theorem] \label{thm3.4}
Let $T:X\to Y$ be a Fredholm map of index zero and
$C,D:X\to Y$ be continuous maps such that
$\alpha(D)< \beta (T)-\alpha(C)$
with $|D|$ sufficiently small, where
$$
|D|=\limsup_{\|x\|\to \infty}\|Dx\|/\|x\|^k< \infty.
$$
Assume that $T+C$ is asymptotically close to a continuous,
closed (in particular, proper)
on bounded and closed subsets of X positive k-homogeneous map $A$,
outside some ball in $X$ for all $\lambda\ge 1$ and some $k\ge 1$,
$\|x\|\le M<\infty$ if $Ax=0$, $|T+C-A|$ sufficiently small, and
$\deg_{BCF}(T+C,B(0,r),0)\ne 0$ for all large $r$.
Then the equation $Tx+Cx+Dx=f$ is solvable for each $f\in Y$ with
$(T+C+D)^{-1}(\{f\})$
compact and the cardinal number $\mathop{\rm card}(T+C+D)^{-1}(\{f\})$
is constant, finite and positive on each connected component of
the set $Y\setminus (T+C+D)(\Sigma)$.
\end{theorem}
\begin{proof}
Since $\beta (T+C)\ge \beta (T)-\alpha (C)>0$, we have that $T+C$ is
proper on closed bounded sets and satisfies condition \eqref{eq2.2}
for some $R>0$ by Lemma \ref{lem2.2} with $c=c-|T-C|$.
Moreover, $\alpha (D+C)\le \alpha (D)+\alpha (C)<\beta (T)$.
Let $f\in Y$ be fixed and $\epsilon >0$ and $R_1>R$ such that
$|D|+\epsilon +\|f\|/R_1^k0$. This implies that $H$
is proper on bounded
closed subsets of $[0,1]\times X$ with the norm
$\|(t,x)\|=$max $\{|t|,\|x\|\}$ for $(t,x)\in \mathbb{R}\times X$.
Since $H$ satisfies condition (+), it is proper on $[0,1]\times X$. Thus,
$H^{-1}(0)$ is compact and contained in $[0,1]\times B(0,R_2)$ for some $R_2\ge R_1$.
Since $B(0,R_2)$ is simply connected, $H(0,.)=T+C:B(0,R_2)\to Y$
is oriented by \cite[Proposition 3.7]{b3}. Hence, $H$ is oriented by
\cite[Proposition 3.5]{b3} and the homotopy
\cite[Theorem 6.1]{b5} implies that
\begin{align*}
\deg_{BCF}(H_1,B(R_1,0),0)
&=\deg_{BCF}(T+C+D-f,B(R_1,0),0) \\
&=\deg_{BCF}(T+C,B(R_1,0),0)\ne 0.
\end{align*}
Thus, the equation $Tx+Cx+Dx=f$ is solvable. The other conclusions
follow from Theorem \ref{thm2.1} since $H(0,.)= T+C+D$ satisfies
condition (+) and is therefore proper.
\end{proof}
\begin{remark} \label{rmk3.2} \rm
Theorems \ref{thm3.1} and \ref{thm3.4} are valid without the asymptotic assumption
if the k-positive homogeneity of $T+C$ is replaced by
$\|Tx+Cx\|\ge c\|x\|^k$ for all x outside some ball
(see Lemma \ref{lem2.2}), or by condition (+) for $T+C$ if $D=0$.
Moreover, the degree assumption in Theorem \ref{thm3.4}
holds if $T$ and $C$ are odd maps by the generalized Borsuk
theorem in Benevieri-Calamai \cite{b2}. In
particular, we have the following result.
\end{remark}
\begin{corollary} \label{coro3.1.}
Let $T:X\to Y$ be a Fredholm map of index zero and
$C:X\to Y$ be a continuous map such that $\alpha(C)< \beta (T)$, $T$
and $C$ be odd outside some ball and $T+C$ satisfy condition {\rm (+)}.
Then the equation $Tx+Cx=f$ is solvable for each $f\in Y$
with $(T+C)^{-1}(\{f\})$ compact and the cardinal number
$\mathop{\rm card}(T+C)^{-1}(\{f\})$ is constant, finite and
positive on each connected component of the set
$Y\setminus (T+C)(\Sigma)$.
\end{corollary}
\section{Applications to (quasi) linear elliptic nonlinear
boundary-value problems}
\subsection*{Potential problems with strongly nonlinear boundary-value
conditions}
Consider the nonlinear boundary-value problem
\begin{equation}
\begin{aligned}
\Delta \Phi=0\quad \text{in}\; Q\subset \mathbb{R}^2 \\
-\partial_n\Phi=b(x,\Phi(x))-f \quad\text{on }
\Gamma=\partial Q.
\end{aligned} \label{eq4.1}
\end{equation}
where $\Gamma$ is a simple smooth closed curve, $\partial_n$ is the
outer normal derivative on $\Gamma$. The nonlinearities appear only
in the boundary conditions.
Using the Kirchhoff transformation, more general quasilinear equations
can be transformed into this form. This kind of equations with various
nonlinearities arise in many applications like steady-state heat
transfer, electromagnetic problems with variable electrical conductivity
of the boundary, heat radiation and heat transfer (cf. \cite{r5} and the
references therein). Except for \cite{r4}, the earlier studies assume
that the nonlinearities have at most a linear growth and
were based on the boundary element method. We shall study \eqref{eq4.1}
using the theory in Sections 2-3. We note also that bifurcation
problems for quasilinear elliptic systems with nonlinearities in
the boundary conditions have recently been discussed by Shi and
Wang \cite{s1}.
Their study is based on the abstract global
bifurcation theorem of Pejsachowicz and Rabier \cite{p1} for Fredholm
maps of index zero.
Assume $b(x,u)=b_0(x,u)+b_1(x,u)$ satisfies
\begin{itemize}
\item[(1)] $b_0:\Gamma\times \mathbb{R}\to \mathbb{R}$ is
a Carath\'eodory function; i.e., $b_0(.,u)$ is measurable for all
$u\in \mathbb{R}$ and $b_0(x,.)$ is continuous for a.e. $x\in \Gamma$
\item[(2)] $b_0(x,)$ is strictly increasing on $\mathbb{R}$
\item[(3)] For $p\ge 2$, there exist constants $a_1>0$, $a_2\ge 0$,
$c_1>0$ and $c_2\ge 0$ such that
$$
|b_0(x,u)|\le a_1|u|^{p-1}+a_2,\;\;b_0(x,u)u\ge c_1|u|^p+c_2.
$$
\item[(4)] $b_1$ satisfies the Carath\'eodory conditions and
$|b_1(x,u)|\le M$ for all $(x,u)\in \Gamma\times \mathbb{R}$
and some $M>0$.
\end{itemize}
We shall reformulate \eqref{eq4.1} as a boundary integral equation.
Recall that the single layer operator $V$ is defined by
$$
Vu(x)=-1/(2\pi)\int_{\Gamma}u(y)log|x-y|ds_y,\quad x\in \Gamma
$$
and the double layer operator $K$ is defined by
$$
Ku(x)=1/(2\pi)\int_{\Gamma}u(y)\partial_n\log|x-y|ds_y,\quad
x\in \Gamma.
$$
We shall make the ansatz: Find a boundary distribution $u$
(in some space) such that
$$
\Phi(x)=-1/(2\pi)\int_{\Gamma}u(y)log|x-y|ds_y,\quad x\in Q.
$$
Then, by the properties of the normal derivative of the monopole
potential \cite{c4,v1},
we derive the nonlinear boundary integral equation \cite{r4}
$$
((1/2I-K^*)u+B(Vu)=f.
$$
This equation can be written in the form
\begin{equation}
Tu+Cu=f\label{eq4.2}
\end{equation}
where we set $T=1/2I-K^*+B_0V$, $Cu=B_1V$ with $B_iu=b_i(x,u)$, $i=0,1$.
\begin{theorem} \label{thm4.1}
Let {\rm (1)-(4)} hold and $q=p/(p-1)$. Then either
\begin{itemize}
\item[(i)] the BVP \eqref{eq4.1} is locally injective in $L_q(Q)$,
in which case it is uniquely solvable in $L_q(Q)$ for
each $h\in L_q(Q)$ and the solution depends continuously on h, or
\item[(ii)] the BVP \eqref{eq4.1} is not locally injective, in which case it is solvable in $L_q(Q)$ for each $h\in L_q(Q)$,
the solution set is compact and the cardinality
of the solution set is finite and constant on each connected component of
$L_q(Q)\setminus (T+C)(\Sigma)$, where
$\Sigma=\{u\in L_q(Q): \eqref{eq4.1} \text{ is not locally uniquely
solvable}\}$.
\end{itemize}
\end{theorem}
\begin{proof}
We have that $V,K,K^*:L_p(\Gamma)\to L_q(\Gamma)$ are compact maps
for each $p,q\in [1,\infty]$ \cite{e2} and therefore
such are the maps $B_0V, C:L_q(\Gamma)\to L_q(\Gamma)$.
By condition (2), it was shown in \cite{r4} that $T$ is strictly
$V$-monotone; i.e., for each $u,v\in L_p(\Gamma)$, $u\ne v$,
$$
(Tu-Tv,V(u-v))_{L_2(\Gamma)}>0.
$$
This implies that $T$ is injective.
Next, as in \cite{r3} but without the differentiability of $b_0$,
we shall show first that $T$ is surjective in $L_q(Q)$.
To that end, we shall show that the problem
\begin{equation}
\Delta \Phi=0,\quad \partial_n\Phi=B_0(\Phi)-f \label{e4.3}
\end{equation}
has a solution in $W_2^1(Q)$. For $\Phi\in W_2^1(Q)$, define
$$
j(\Phi(x))=\int_0^{\Phi(x)}b_0(x,s)ds.
$$
Since $b_0(x,.)$ is a strictly monotone proper function,
$j$ is a strictly convex and lower semicontinuous function
whose subgradient is given by
$\partial j(u)=B_0(u)$ \cite[Theorem 2.3]{b1}. Then the above problem
is equivalent to the minimization of the functional
$$
F(\Phi)=1/2\int_Q|\Delta\Phi|^2+G(\Phi)-\int_{\Gamma}f\Phi ds_{\Gamma}
$$
over $W_2^1(Q)$, where, for $\Phi\in W_2^1$,
$$
G(\Phi)=\int_{\Gamma}j(\Phi)ds_x, \quad \text{for }
j(\Phi)\in L_1(\Gamma)
$$
and $G(\Phi)=+\infty$ otherwise.
Since $W_2^1(Q)$ is continuously imbedded in $L_p(\Gamma)$,
condition (3) implies that for each $\Phi\in W_2^1(Q)$ there exist
constants $c>0$ and $c_1\in \mathbb{R}$
independent of $\Phi$ such that
$$
G(\Phi|_{\Gamma})\ge c\int_{\Gamma}|u(x)|^pds_x
+c_1\int_{\Gamma}|u(x)|ds_x,
$$
where $u=\Phi|_{\Gamma}$.
This implies that $F(\Phi)$ is coercive because
\begin{align*}
F(\Phi)&=1/2\int_Q|\Delta\Phi|^2+\int_{\Gamma}j(\Phi)ds_x
-\int_{\Gamma} f \Phi ds_{\Gamma} \\
&\ge 1/2\int_Q|\Delta\Phi|^2+c(p)\int_{\Gamma}|\Phi|^pds_x
-\|f\|_{L_p(\Gamma)}\|\Phi\|_{L_p(\Gamma)}.
\end{align*}
Hence, there is a unique function $\Phi\in W_2^1(Q)$ that minimizes $F$.
Next, if $\Phi\in W_2^1(Q)$ is the unique minimizer of F, then by
the properties of V there is a unique
boundary function $u\in W_2^{-1/2}(\Gamma)$ such that
$Vu=\Phi|_{\Gamma}$. As in \cite{r4}, we get
that $u\in L_q(\Gamma)$ and is a solution of $Tu=f$ for each
$f\in L_q(\Gamma)$.
Thus, $T: L_q(\Gamma)\to L_q(\Gamma)$ is bijective.
Since $T$ is a compact perturbation of the identity and injective,
it is an open map by the Shauder invariance of domain theorem.
Hence, $T$ is a homeomorphism.
Since $b_1(x,.)$ is bounded and $C$ is compact by the compactness
of $V$, the conclusions follow from
Corollary \ref{coro2.1}.
\end{proof}
\begin{example} \label{exa4.1} \rm
If $b_0(u)=|u|u^{p-2}$ and $b_1=0$, or if $b_0(u)=|u|u^{p-2}$ and
$b_1(u)=arctan\;u$ with $p$ even,
then part (i) of Theorem \ref{thm4.1} holds. Note that $b(u)$ is strictly
increasing in either case
and therefore \eqref{eq4.1} is injective as in the proof of
Theorem \ref{thm4.1}.
Part (ii) of Theorem \ref{thm4.1} is valid
if e.g., $b_1(u)=a \sin u+b \cos u$.
\end{example}
When the nonlinearities have a linear growth, we have the
following result.
\begin{theorem} \label{thm4.2}
Let $b(x,u)=b_0(x,u)+b_1(x,u)$ be such that $b_0(x,u)$ is
a Carath\'eodory function,
$b_0(x,.)$ is strictly increasing on $\mathbb{R}$ and
\begin{itemize}
\item[(1)] There exist constants $a_1>0$ and $a_2\ge 0$ such that
$$|b_0(x,u)|\le a_1|u|+a_2$$
\item[(2)] $b_1$ satisfies the Carath\'eodory conditions and for some
positive constants $c_1$ and $c_2$ with $c_1$ sufficiently small
$$
|b_1(x,u)|\le c_1|u|+c_2\quad \text{for all }
(x,u)\in \Gamma\times \mathbb{R}.
$$
\end{itemize}
Assume that $I-K$ is injective. Then either
\begin{itemize}
\item[(i)] the BVP \eqref{eq4.1} is locally injective in $L_2(Q)$,
in which case it is uniquely solvable in $L_2(Q)$ for
each $h\in L_2(Q)$ and the solution depends continuously on $h$, or
\item[(ii)] the BVP \eqref{eq4.1} is not locally injective, in which
case it is solvable in $L_2(Q)$ for each $h\in L_2(Q)$,
the solution set is compact and the cardinality of the solution set
is finite and constant on each connected component of
$L_2(Q)\setminus (T+C)(\Sigma)$, where
$\Sigma=\{u\in L_2(Q): \eqref{eq4.1}\text{ is not locally uniquely
solvable}\}$ and $T=I-K+B_0V$.
\end{itemize}
\end{theorem}
\begin{proof}
The BVP \eqref{eq4.1} is equivalent to the operator equation
$$
(I-K)u+B(Vu)=f
$$
(cf. \cite{r5}). This equation can be written in the form
\begin{equation}
Tu+Cu=f\label{eq4.4}
\end{equation}
where we set $T=I-K+B_0V$, $Cu=B_1V$ with $B_iu=b_i(x,u)$, i=0,1.
Since $K$ is compact and $I-K$ is injective, it is a homeomorphism
by the Fredholm alternative.
As above, $T$ is a compact perturbation of the identity and injective.
It satisfies condition (+).
Indeed, let $y_n=(I-K+B_0V)u_n\to y$
in $L_2(\Gamma)$. Then $c\|u_n\|\le \|(I-K)u_n\|=
\|y_n-B_0Vu_n\|\le c_1+a\|V\|\|u_n\|+b$. Since $a$ is sufficiently small,
this implies that $\{u_n\}$ is bounded and $T$ satisfies condition (+).
Since $T$ is a compact perturbation
of the identity, it is proper on bounded closed subsets of
$L_2(\Gamma)$. Condition (+) implies that $T$ is proper on
$L_2(\Gamma)$. Hence, the range $R(T)$ is closed and therefore
$T$ is a homeomorphism by Theorem \ref{thm2.2}(i)(b) applied to $I-K$
and $B_0V$. By condition 2), it follows as above that
that $T+tC$ satisfies condition (+) and the conclusions follow
from Theorem \ref{thm2.2}.
\end{proof}
\begin{remark} \label{rmk4.1} \rm
Theorem \ref{thm4.2} is also valid when $b_0=0$. The injectivity of $I-K$
has been studied in \cite{c4,v1}.
\end{remark}
\subsection*{Semilinear elliptic equations with nonlinear
boundary-value conditions}
Consider the nonlinear BVP
\begin{gather}
\Delta u=f(x,u,\nabla u)+g\quad \text{in } Q\subset
\mathbb{R}^n, \label{eq4.5}\\
-\partial_n u=b(x,u(x))-h \quad \text{on }
\Gamma=\partial Q,. \label{e4.6}
\end{gather}
where $Q\subset \mathbb{R}^n$, (n=2 or 3), is a bounded domain
with smooth boundary $\Gamma$ satisfying a scaling assumption
$\mathop{\rm diam}(Q)<1$ for $n=2$, $\partial_n$ is the outer normal
derivative on $\Gamma$.
Let $b=b_0+b_1$. As in \cite{r5}, assume that
\begin{itemize}
\item[(1)] $b_0(x,u)$ is a Carath\'eodory function such that
$\frac{\partial}{\partial u} b_0(x,u)$ is Borel
measurable and satisfies
$$
00$ sufficiently small.
\end{itemize}
Define the Nemytskii maps $B_i:L_2(\Gamma)\to L_2(\Gamma)$ by
$B_iu(x)=b_i(x,u(x))$, $i=0,1$,
and $F:H^1(Q)\to L_2(Q)$ by $Fu(x)=f(x,u(x),\nabla u(x))$.
Denote by $H^s(Q)$ and $H^s(\Gamma)$ the
Sobolev spaces of order s in Q and on $\Gamma$, respectively.
In particular, $H^{-s}(Q)=(\tilde {H}^s(Q))^*$,
where $\tilde{H}^s$ is the completion of $C_0^{\infty}(Q)$
in $H^s(\mathbb{R}^n)$. We also have that \cite{r5}
for each $0\le s\le 1$, $B_0:H^s(\Gamma)\to H^s(\Gamma)$ is bounded.
Denote by $(.,.)$ the $L_2$ inner product.
As in \cite{e1}, inserting \eqref{eq4.5}-\eqref{e4.6} into Green's formula
$$
\int_Q\Delta u.v\,dx + \int_Q\Delta u.\Delta v\,dx
-\int_{\Gamma}\frac{\partial u}{\partial n} v\,ds_{\Gamma}=0
$$
we obtain the weak formulation of \eqref{eq4.5}-\eqref{e4.6}:
for a given $g\in \tilde{H}^{-1}$, find $u\in H^1(Q)$ such that
for all $v\in H^1(Q)$,
\begin{equation}
\begin{aligned}
(Au,v)_{H^1(Q)}
&=(\nabla u,\nabla v)_Q+(B_0u|_{\Gamma},v|_{\Gamma})_{\Gamma}
-(B_1u|_{\Gamma},v|_{\Gamma})_{\Gamma}\\
&\quad -(h,v|_{\Gamma})_{\Gamma}-(Fu,v)_Q-(g,v)_Q=0. \label{eq4.7}
\end{aligned}
\end{equation}
Define $T,C:H^1(Q)\to H^1(Q)$ by
\begin{gather*}
(Tu,v)_{H^1(Q)}=(\nabla u,\nabla v)_Q
+(B_0u|_{\Gamma},v|_{\Gamma})_{\Gamma},\\
(Cu,v)_{H^1(Q)}=-(B_1u|_{\Gamma},v|_{\Gamma})_{\Gamma}
-(h,v|_{\Gamma})_{\Gamma}-(Fu,v)_Q-(g,v)_Q.
\end{gather*}
\begin{theorem} \label{thm4.3}
Let {\rm (1)-(2)} hold. Then either
\begin{itemize}
\item[(i)] The problem \eqref{eq4.5}-\eqref{e4.6} is locally injective
in $H^1(Q)$, in which case it is uniquely solvable in
$H^1(Q)$ for each $g\in \tilde {H}^{-1}(Q)$ and $h\in H^{-1/2}(\Gamma)$
and the solution depends continuously on $(g,h)$, or
\item[(ii)] the problem \eqref{eq4.5}-\eqref{e4.6} is not
locally injective in $H^1(Q)$, in which case it is
solvable in $H^1(Q)$ for each
$(g,h)\in \tilde H^{-1}(Q)\times H^{-1/2}(\Gamma)$,
the solution set is compact and the cardinality
of the solution set is finite and constant on each connected component of
$H^1(Q)\setminus (T+C)(\Sigma)$, where
$\Sigma=\{u\in H^1(Q):\text{\eqref{eq4.5}-\eqref{e4.6}
is not locally uniquely solvable}\}$.
\end{itemize}
\end{theorem}
\begin{proof}
By (1), $B_0$ satisfies the Lipschitz condition and is $l$-strongly
monotone in $L_2(\Gamma)$. This implies that $T$ also satisfies
the Lipschitz condition and that
$$
(Tu-Tv,u-v)_{H^1(Q)}\ge \|\nabla(u-v)\|^2_{L_2(Q)}
+l\|u-v\|_{L_2(\Gamma)}\ge k\|u-v\|_{H^1(Q)}^2.
$$
Hence, $T$ is $k$-strongly monotone and is therefore a homeomorphism
in $H^1(Q)$ with $\|Tu-Tv\|\ge k\|u-v\|$.
Moreover, $C:H^1(Q)\to H^1(Q)$ is compact since $F:H^1(Q)\to L_2(Q)$
is continuous and the embedding
of $L_2(Q)$ into $\tilde{H}^{-1}(Q)$ is compact, and
$B_1:L_2(\Gamma)\to L_2(\Gamma)$ is continuous and
the embeddings $H^{1/2}(\Gamma)\to L_2(\Gamma)\to H^{-1/2}(\Gamma)$
are also compact.
Since c is sufficiently small, $\|Tu_n+Cu_n\|\to \infty$
as $\|u_n\|\to \infty$.
Hence, being proper on bounded closed subsets,
$T+C$ is proper. Moreover, $T+tC$ satisfies condition (+) and
the conclusions of the theorem follow
from Theorem \ref{thm2.2}.
\end{proof}
\begin{remark} \label{rmk4.2} \rm
The solvability of problem \eqref{eq4.5}-\eqref{e4.6} with sublinear
nonlinearities was proved by Efendiev, Schmitz and Wendland
\cite{e1} using a degree theory for compact perturbations of
strongly monotone maps.
\end{remark}
\subsection*{Semilinear elliptic equations with strong nonlinearities}
Consider the problem
\begin{equation}
\begin{gathered}
\Delta u+\lambda_1u -f(u)+g(u)=h,\quad (h\in L_2)\\
u\big|_{\partial Q}=0,
\end{gathered} \label{eq4.8}
\end{equation}
where $Q$ is a bounded domain in $\mathbb{R}^n$ and $\lambda_1$
is the smallest positive eigenvalue of $\Delta$ on $Q$.
Assume that $f$ and $g=g_1+g_2$ are Carath\'eodory functions such that
\begin{itemize}
\item[(1)] For $p\ge 2$ if $n=2$ and $p\in [2,2n/(n-2))$ if $n\ge 3$,
there exist constants
$a_1>0$, $a_2\ge 0$, $c_1>0$ and $c_2\ge 0$ such that
$$
|f(u)|\le a_1|u|^p+a_2,\quad f(u)u\ge c_1|u|^{p+1}+c_2.
$$
\item[(2)] $f$ is differentiable
\item[(3)] $|g_1(u)|\le b_1|u|^p+b_2$ with $b_1\le a_1$ and
$g_1(u)u\ge c_1|u|^{p+1}+c_2$.
\item[(4)] $|g_2(u)|\le c_1|u|^p+c_2$ for all u with $c_1$
sufficiently small.
\end{itemize}
Define $X=\{u\in W_2^1(Q):u=0\text{ on }\partial Q\}$. Note that
$X$ is compactly embedded into $L_p(Q)$ for each $p$ as in (1)
by the Sobolev embedding theorem.
We shall look at weak solutions of \eqref{eq4.8}; i.e., $u\in X$
such that $Tu+Cu=h$, where
$$
(Tu,v)_{1,2}=(\nabla u,\nabla v)-\lambda_1(u,v)+(f(u),v),\quad
(C_iu,v)=(g_i(u),v),\quad i=1,2
$$
$C=C_1+C_2$ and $(.,.)$ is the $L_2$ inner product.
In X, the derivative of $T$ is $T'(u)v=\Delta v-\lambda_1v-
f'(u)v$. Since $T'(u)$ is a selfadjoint elliptic map in X, $T$
is Fredholm of index zero in $X$.
Next, we shall show that $\|Tu+C_1u\|\to \infty$ if
$\|u\|\to \infty$ in $X$.
Suppose that $\|u_n\|_{2,1}=\|\nabla u_n\|_2^2+\|u_n\|_2^2\to \infty$.
Thus
\begin{align*}
((T+C_1)u_n,u_n)
&=\|\nabla u_n\|_2^2-\lambda_1\|u_n\|_2^2+(f(u_n)+g_1(u_n),u_n)\\
&\ge \|\nabla u_n\|_2^2-\lambda_1\|u_n\|_2^2+c\|u_n\|_{p+1}^{p+1}
-c'\|u\|_2.
\end{align*}
If $\|\nabla u_n\|_2^2\to \infty$ and $\|u_n\|_2^2\le k$, then
\[
((T+C_1)u_n,u_n)\ge \|\nabla u_n\|_2^2-\lambda_1\|u_n\|_2^2
+c\|u_n\|_2^{p+1}-c'\|u_n\|_2
\ge\|\nabla u_n\|_2^2-k_1
\]
since $\|\nabla u\|_2^2\ge c_1\|u\|_2^2$.
If also $\|u_n\|_2^2\to \infty$, then
\[
((T+C_1)u_n,u_n)\ge \|\nabla u_n\|_2^2+(k_2\|u_n\|_2^{p-1}
-\lambda_1-c'/\|u_n\|_2)\|u_n\|_2^2
\]
since $L_{p+1}\subset L_2$.
Thus, in either case $ ((T+C_1)u_n,u_n)/\|u_n\|_{2,1}^2\ge k_3>0$ as
$\|u_n\|_{2,1}^2\to \infty $. Hence, $\|(T+C_1)u_n\|\to \infty$ as
$\|u_n\|_{2,1}^2\to \infty $ by the Cauchy-Schwartz
inequality. In a similar way, we can show that
$\|Tu_n\|\to \infty $ as $\|u\|\to \infty$ in X.
To show that $T$ is proper, let $K\subset X$ be compact and note
that $T^{-1}(K)$ is bounded by the coercivity of $T$.
Let $\{u_n\}\subset T^{-1}(K)$. We may assume that
it converges weakly to u and $Tu_n\to v$ in $X$. Since $X$ is
compactly embedded in $L_p$ for each $p$ as in (1), we have that
$u_n\to u$ in $L_2$ and $L_p$ and $(Tu_n,u_n)\to (v,u)$. Since
$$
(Tu_n,u_n)=\|\nabla u_n\|_2^2-\lambda_1\|u_n\|_2^2+(T_1u_n,u_n)
$$
and $T_1u=f(u)$ is compact in X, we get that $\|\nabla u_n\|_2$
converges. Hence, we have that $\{u_n\}$ converges weakly in $X$ and
$\|u_n\|_{1,2}$ converges. Since $X$ is a Hilbert space,
$\{u_n\}$ converges in $X$.
Hence, $T$ is proper.
\begin{theorem} \label{thm4.4}
Let {\rm (1)-(4)} hold. Then either
\begin{itemize}
\item[(i)] BVP \eqref{eq4.8} is locally injective in $X$, in which
case it is uniquely solvable in $X$ for
each $h\in L_2(Q)$ and the solution depends continuously on h, or
\item[(ii)] BVP \eqref{eq4.8} is not locally injective, in which case
it is solvable in X for each $h\in L_2(Q)$ and its solution set
is compact. Moreover, the cardinality of the solution set is finite
and constant on each connected component of
$X\setminus (T+C)(\Sigma)$, where
$\Sigma=\{u\in X: \text{\eqref{eq4.8} is not locally uniquely solvable}\}$.
\end{itemize}
\end{theorem}
\begin{proof} Since $c_1$ is sufficiently small,
$\|C_2u\|\le a\|Tu+C_1u\| + b$ for all u and $a<1$.
This and $\|Tu+C_1u\|\to \infty$ as $\|u\|\to \infty$ imply
that $\|Tu+C_1u+tC_2u\|\to \infty$ as $\|u\|\to \infty$.
Since $T$ is proper and C is compact,
it follows that $T+C$ is proper. Indeed, let $K\subset Y$ be compact
and $x_n\in (T+C)^{-1}(K)$.
Then $y_n=(T+C)x_n\to y$ since $y_n\in K$ and $\{x_n\}$ is
bounded by the preceding remark.
Since $\{y_n-Cx_n\}$ is compact, $T$ is proper and $Tx_n=y_n-Cx_n$
is compact we get that a subsequence $x_{n_k}\to x$. By the continuity
of $T+C$, $Tx+Cx=y$ in $Y$.
Hence, $T+C$ is proper and the conclusions of the theorem follow
from Theorem \ref{thm2.3}.
\end{proof}
\begin{theorem} \label{thm4.5}
Let {\rm (1)-(2)} hold with $f'(0)=0$, $f'(u)>0$ and $|g(u)|\le M$
for some $M>0$ and all $u$.
Then either
\begin{itemize}
\item[(i)] BVP \eqref{eq4.8} is locally injective in $X$, in
which case it is uniquely solvable in $X$ for
each $h\in L_2(Q)$ and the solution depends continuously on $h$, or
\item[(ii)] BVP \eqref{eq4.8} is not locally injective, in which case
it is solvable in X for each $h\in L_2(Q)$
and the solution set is compact. Moreover, the cardinality
of the solution set is finite
and constant on each connected component of
$X\setminus (T+C)(\Sigma)$, where
$\Sigma=\{u\in X: \text{\eqref{eq4.8} is not locally uniquely solvable}\}$.
\end{itemize}
\end{theorem}
\begin{proof}
$T$ is a proper Fredholm map of index zero by the above remarks.
Next, we shall show that the singular set of $T$ consists only of 0.
Suppose that
$T'(u)v=0$. Using the variational characterization of $\lambda_1$,
we get that $u=0$ since
$0=(T'(u)v,v)=\int_Q|\nabla v|^2-\lambda_1 v^2+\int_Qf'(u)v^2>0$. Hence,
since $f'(0)=0$, $T'(0)v=0$, i.e., $\Delta v+\lambda_1 v=0$ and
$v=0$ on the boundary of Q. Thus,
this problem has nontrivial solutions and therefore the singular set
of $T$ consists only of 0. Since $T$ is Fredholm of index zero and
0 is an isolated singular point, $T$ is a local homeomorphism and
therefore a homeomorphism by its properness and the Banach-Mazur
theorem.
Set $Cu=g(u)$. Since $\|Cu\|\le M_1<\infty $ for all $u$, the
conclusions of the theorem follow from
Corollary \ref{coro2.3}, since $T+C$ is proper by the compactness of $C$ as
shown in Theorem \ref{thm4.4}.
\end{proof}
\begin{theorem} \label{thm4.6}
Let {\rm (1)-(4)} hold and $f$ and $g_1$ be odd. Then \eqref{eq4.8}
is solvable in $X$ for each $h\in L_2(Q)$ and its solution set
is compact. Moreover, the cardinality of the solution set is finite
and constant on each connected component of $X\setminus (T+C)(\Sigma)$,
where $\Sigma=\{u\in X: \text{\eqref{eq4.8} is not locally
uniquely solvable}\}$.
\end{theorem}
\begin{proof}
Note that $T+C_1$ is odd. As in the proof of Theorem \ref{thm4.4},
we have that $\|Tu+C_1u+tC_2u\|\to \infty$ as $\|u\|\to \infty$. Hence,
the conclusions of the theorem follow from
Theorem \ref{thm3.3}.
\end{proof}
\section{Solvability of quasilinear elliptic BVP's with asymptotically
positive homogeneous nonlinearities}
Let $Q\subset \mathbb{R}^n$ be a bounded domain with the smooth boundary and consider
the boundary value problem in a divergent form
\begin{equation}
\sum_{|\alpha|\le m}(-1)^{|\alpha|}D^{\alpha}
A_{\alpha}(x,u,\dots,D^mu)+
k\sum_{|\alpha|\le m}(-1)^{|\alpha|}D^{\alpha}
B_{\alpha}(x,u,\dots,D^mu)=f\label{eq5.1}
\end{equation}
Let $X$ be the closed subspace of $W_p^m(Q)$ corresponding to
the Dirichlet conditions.
Define the maps $T,D:X\to X^*$ by
\begin{gather*}
(Tu,v)=\sum_{|\alpha|\le m}
\int_QA_{\alpha}(x,u,\dots,D^mu)D^{\alpha}v dx,\\
(Du,v)=\sum_{|\alpha|\le m}\int_QB_{\alpha}(x,u,\dots,D^mu)
D^{\alpha}v dx.
\end{gather*}
Then weak solutions of \eqref{eq5.1} are solutions of the operator
equation
\begin{equation}
Tu+Du=f,\;u\in X.\label{eq5.2}
\end{equation}
We impose the following conditions on $A_{\alpha}$.
\begin{itemize}
\item[(A1)] For each $\alpha$, let
$A_{\alpha}:Q\times \mathbb{R}^{s_m}\to \mathbb{R}$ be such that
$A_{\alpha}(x,\xi)$ is measurable in $x$ and for each fixed
$\xi$, and has continuous derivatives in $\xi$ for a.e. $x$;
\item[(A2)] Assume that for $p>2$, $x\in Q$, $\xi\in \mathbb{R}^{s_m}$,
$\eta\in \mathbb{R}^{s_m-s_{m-1}}$,
$|\alpha|, |\beta|\le m$ the $A_{\alpha}$'s satisfy
$$
|A_{\alpha}(x,\xi)|\le g_0(|\xi_0|)(h(x)+\sum_{m-n/p\le\gamma|\le m}
|\xi_{\gamma}|^{p-1});
$$
\item[(A3)] $ |A_{\alpha,\beta}(x,\xi)|\le g_1(|\xi_0|)(b(x)
+\sum_{m-n/p\le\gamma|\le m} |\xi_{\gamma}|^{p-2})$,
\item[(A4)] $ \sum_{|\alpha|=|\beta|=m}
A_{\alpha,\beta}(x,\xi)\eta_{\alpha}\eta_{\beta}\ge
g_2(1+\sum_{|\gamma|=m}|\xi_{\gamma}|)^{p-2}\;\sum_{|\alpha|=m}
\eta_{\alpha}^2,$
where $A_{\alpha,\beta}(x,\xi)=\partial/\partial\xi_{\beta}
A_{\beta}(x,\xi)$,
$h,b\in L_q(Q)$, $g_0, g_1$ are continuous positive
nondecreasing functions and $g_2>0$ is a constant.
\end{itemize}
For $|\alpha|\le m$, there are Carath\'eodory functions
$a_{\alpha}$ such that
\begin{itemize}
\item[(a1)] $a_{\alpha}(x,t\xi)=t^{p-1}a_{\alpha}(x,\xi)$
for all $t>0$, $\xi\in \mathbb{R}^{s_m}$
\item[(a2)] $|1/t^{p-1}A_{\alpha}(x,t\xi)-a_{\alpha}(x,\xi)|\le c(t)
(1+|\xi|)^{p-1}$
\end{itemize}
for each $t>0$, $x\in Q$, and $\xi\in \mathbb{R}^{s_m}$,
where $0<\lim c(t)$ is sufficiently small as $t\to \infty$.
\begin{proposition} \label{prop5.1}
Assume {\rm (A1)-(A4)}. Then $T:X\to X^*$
is Fredholm of index zero and is proper on bounded closed subsets
of $X$.
\end{proposition}
\begin{proof}
The map $T:X\to X^*$ is continuous and of type $(S_+)$ \cite{s2} and,
as shown before, it is proper on bounded closed subsets of $X$.
By \cite[Lemma 3.1]{s2}, the Fr\'echet
derivative $T'(u)$ of $T$ at $u\in X$ is given by
\begin{equation}
(T'(u)v,w)=\sum_{|\alpha|,|\beta|\le m}\int_Q A_{\alpha,\beta}
(x,u,\dots,D^mu)D^{\beta}v D^{\alpha}w dx\label{eq5.3}
\end{equation}
Next, we shall show that $T'(u)$ is Fredholm of index zero for each
$u\in X$.
First, we shall show that $T'(u)$ satisfies condition $(S_+)$.
We can write it in the form $T'(u)=T_1'(u)+T_2'(u)$, where
\begin{gather}
(T_1'(u)v,w)=\sum_{|\alpha|=|\beta|= m}\int_Q A_{\alpha,\beta}
(x,u,\dots,D^mu)D^{\beta}v D^{\alpha}w dx, \label{eq5.4}\\
(T_2'(u)v,w)
=\sum_{|\alpha|,|\beta|\le m,\, |\alpha+\beta|<2m}
\int_Q A_{\alpha,\beta}
(x,u,\dots,D^mu)D^{\beta}v D^{\alpha}w dx. \label{eq5.5}
\end{gather}
It is easy to see that
$|(T_2'(u)v,w)|\le c\|v\|_{m,p}\;\|w\|_{m-1,p}$ for some
constant $c>0$. Hence, $T_2'(u):X\to X^*$ is compact
since the embedding of $W_p^m$ into $W_q^k$ is compact for
$0\le k\le m-1$ if
$1/q>1/p-(m-k)/n>0$, or if $q<\infty$ and $1/p=(m-k)/n$.
Next, we shall show that $T_1'(u):X\to X^*$ is of type $(S_+)$.
Let $v_n\in X$ be such that $v_n\rightharpoonup v$ in $X$ and
$$
\limsup_{n\to \infty}(T_1'(u)v_n,v_n-v)\le 0.
$$
It follows that $D^{\alpha}v_n\to D^{\alpha}v$ in $L_p$ for each
$|\alpha|2$ there exist a constant
$c>0$ and $h_{\alpha}(x)\in L_q(Q)$, $1/p+1/q=1$, such that
$$
|B_{\alpha}(x,\xi)|\le c(h_{\alpha}(x)+|\xi|^{p-1})
$$
\item[(B2)] There is a sufficiently small $k_1>0$ such that
$$
\sum_{|\alpha|=m}|B_{\alpha}(x,\eta,\xi_{\alpha})
- B_{\alpha}(x,\eta,\xi_{\alpha}')|\le k_1\sum_{|\alpha|=m}|
\xi_{\alpha}-\xi_{\alpha'}|
$$
for each a.e. $x\in Q$, $\eta\in \mathbb{R}^?$ and
$\xi_{\alpha},\xi_{\alpha'}\in \mathbb{R}^?$.
\end{itemize}
Note that if $B_{\alpha}$'s are differentiable for
$|\alpha|=m$ and
$B_{\alpha \alpha}(x,\xi)=\partial/\partial\xi_{\alpha}B_{\alpha}(x,\xi)$
are sufficiently small, then (B2) holds.
In view of Proposition \ref{prop5.1}, the results in the form of
Theorems \ref{thm4.1}--\ref{thm4.3} are valid for \eqref{eq5.1} as well as the
corresponding ones involving maps that are asymptotically close
to positively k-homogeneous maps. A sample of such a theorem
is given next.
Consider also the equation
\begin{equation}
\sum_{|\alpha|\le m}(-1)^{|\alpha|}D^{\alpha}a_{\alpha}(x,u,\dots,D^m)
=f\label{eq5.8}
\end{equation}
in $X$. Define the map $A:X\to X^*$ by
\begin{equation}
(Au,v)=\sum_{|\alpha|\le m}\int_Qa_{\alpha}(x,u,\dots,D^mu)
D^{\alpha}v dx.\label{eq5.9}
\end{equation}
Then weak solutions of \eqref{eq5.8} are solutions of the operator
equation
\begin{equation}
Au=f,\quad u\in X.\label{eq5.10}
\end{equation}
\begin{theorem} \label{thm5.1}
Assume that {\rm (A1)--(A4), (a1)--(a2), (B1)--(B2)} hold and that
for all large $r$, $\deg_{BCF}(T,B(0,r),0)\ne 0$.
Suppose that $A$ is of type $(S_+)$ and $Au=0$ has no a nontrivial
solution.
Then \eqref{eq5.1} is solvable for each $f\in X^*$, has a compact
set of solutions whose cardinal number is constant, finite and
positive on each connected component of the set
$X^*\setminus (T+D)(\Sigma)$, where
$\Sigma=\{u\in X: T+D \text{ is not locally invertible at } u\}$.
\end{theorem}
\begin{proof}
In view of our discussion above, $T$ is a Fredholm map of index
zero that is proper on bounded closed subsets of $X$.
It is proper on $X$ since it satisfies
condition (+), and therefore $\beta(T)>0$.
We need to show that $\alpha(D)<\beta(T)$.
We note that the boundedness and continuity of D follow from
(A1)--(A2), the Sobolev embedding theorem and the continuity of
the Nemytskii maps
in $L_p$ spaces. We can write $D=D_1+D_2$, where
\begin{gather*}
(D_1u,v)=\sum_{|\alpha|= m}\int_QB_{\alpha}
(x,u,\dots,D^mu)D^{\alpha}v dx,\\
(D_2u,v)=\sum_{|\alpha|< m}\int_QB_{\alpha}(x,u,\dots,D^mu)
D^{\alpha}v dx,
\end{gather*}
The map $D_1:X\to X^*$ is $k_1$-set contractive by (B2),
while $D_2:X\to X^*$ is compact
by the Sobolev embedding theorem. Hence,
$\alpha(D)=\alpha(D_1)\le k_1<\beta (T)$ since $k_1$ is sufficiently
small. Finally, condition (a2) implies that $T$ is asymptotically
close to the $(p-1)-$ positive homogeneous map $A$ given by \eqref{eq5.9}.
Hence, Theorem \ref{thm3.4} applies with $C=0$.
\end{proof}
\begin{example} \label{exa5.1} \rm
Let $s>0$, $k$ be sufficiently small, and look at
\begin{equation}
-\Delta u-\mu u \frac{|u|^s}{1+|u|^s} +kF(x,u,\nabla u)=f\label{eq5.11}
\end{equation}
and
\begin{equation}
-\Delta u-\mu u =0\label{eq5.12}
\end{equation}
Let $A_0(x,\xi_0,\xi_1,\dots,\xi_n)=\xi_0
\frac{|\xi_0|^s}{1+|\xi_0|^s}$
and $A_i(x,\xi_0,\xi_1,\dots,\xi_n)=a_i(x,\xi_0,\xi_1,\dots,\xi_n)=\xi_i$
and $a_0(x,\xi_0,\xi_1,\dots,\xi_n)=\xi_0$.
Then $A_0, A_i, a_0$ and $a_i$ satisfy (a1)-(a2). Let $F$ satisfy (B1).
Then \eqref{eq5.11} has a solution $u\in W_2^1(Q)$, $u=0$ on
$\partial Q$, for each $f\in L_2(Q)$
if $\mu$ is not an eigenvalue of \eqref{eq5.12}.
\end{example}
\begin{example} \label{exa5.2} \rm
Let $p>2$, $k$ be sufficiently small, and look at
\begin{equation}
-\sum_{i=1}^n \frac{\partial}{\partial x_i}
[(1+\sum_{j=1}^n|\frac{\partial u}{\partial x_j}|^2)^{p/2-1}
\frac{\partial u}{\partial x_i}]
+\mu(1+|u|^2)^{p/2-1}u+kDu=f\label{eq5.13}
\end{equation}
and
\begin{equation}
-\sum_{i=1}^n\frac{\partial}{\partial x_i}
(\sum_{j=1}^n|\frac{\partial u}{\partial x_j}|^2)^{p/2-1}
\frac{\partial u}{\partial x_i})
+\mu (|u|^2)^{p/2-1}u=0.
\label{eq5.14}
\end{equation}
where $D:W_p^1\to W_p^1$ is $k$-set contractive; e.g.,
$Du=F(x,u,\nabla u)$ in which case it is compact,
or $Du=\sum_{i=1}^n\partial/\partial x_ic_i(x,u,\nabla u)$ with
the $c_i$ $k_i$-contractive in $\nabla u$ with $k_i$ small.
Let $A_0(x,\xi_0,\xi_1,\dots\xi_n)=(1+\xi_0^2)^{p/2-1}\xi_0$,
$A_i(x,\xi_0,\xi_1,\dots,\xi_n)=(1+\sum_{j=1}^n\xi_j^2)^{p/2-1}\xi_i$,
$a_0(x,\xi_0,\xi_1,\dots\xi_n)= (\xi_0^2)^{p/2-1}\xi_0$ and
$a_i(x,\xi_0,\xi_1,\dots,\xi_n)=\sum_{j=1}^n(\xi_j^2)^{p/2-1}\xi_i$.
Then the matrix $(A_{ij}(x,\xi_0,\xi_1,\dots\xi_n))$
is symmetric. Let $n=2$ for simplicity. Then the eigenvalues of
the matrix are $\lambda_1=(p/2-1)(1+\xi_1^2+\xi_2^2)^{p/2-1}$ and
$\lambda_2=\lambda_1+(p-2)(1+\xi_1^2+\xi_2^2)^{p/2-2}(\xi_1^2+\xi_2^2)$.
Hence, $A_0, A_i, a_0$ and $a_i$ satisfy
conditions (A1)--(A4) and (a1)--(a2), respectively.
Then \eqref{eq5.13} has a compact set of solution $u\in W_p^1(Q)$, $u=0$
on $\partial Q$, for each
$f\in X^*$, if $\mu$ is not an eigenvalue of \eqref{eq5.14} and $n=2$.
The solution set is finite for all f as in Theorem \ref{thm5.1}
since the corresponding map $T$ is odd.
\end{example}
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