\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 54, pp. 1--25.\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 2004 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2004/54\hfil Solvability of Hammerstein equations]
{Solvability and the number of solutions of Hammerstein equations}
\author[Petronije S. Milojevi\'c\hfil EJDE-2004/54\hfilneg]
{Petronije S. Milojevi\'c}
\address{Department of Mathematical Sciences and CAMS,
New Jersey Institute of Technology, Newark, NJ 07102, USA}
\email{pemilo@m.njit.edu}
\date{}
\thanks{Submitted December 12, 2003. Published April 8, 2004.}
\subjclass[2000]{47H30, 35L70, 35L75, 35J40}
\keywords{Approximation solvability, Hammerstein equations, $A$-proper maps.}
\begin{abstract}
We study the solvability and the number of solutions to Hammerstein
operator equations in Banach spaces using a projection like method
and degree theory for corresponsing vector fields.
The linear part is assumed to be either selfadjoint or non-seladjoint.
We present also applications to Hammerstein integral equations.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
In this paper, we study the solvability and the number of solutions to
the Hammerstein operator equation
\begin{equation}
x-KFx=f\label{e1.1}
\end{equation}
where $K$ is linear and $F$ is a nonlinear map. We consider
\eqref{e1.1} in a general setting between two Banach spaces. To
that end, we use two approaches. One is based on using the
degree theory for $\phi$-condensing maps or applying the Brouwer
degree theory directly to the finite dimensional approximations of
the map $I-KF$ in conjunction with the (pseudo) $A$-proper mapping
approach. The other one is based on splitting first the map $K$ as
a product of two suitable maps and then using again these degree
theories. The linear part $K$ is assumed to be either selfadjoint
or non-selfadjoint. In the second case, we assume that $K$ is
either positive in the sense of Krasnoselskii, potentially
positive, $P$-positive (i.e., angle- bounded) or that it is
P-quasi-positive, which means that its selfadjoint part has at
most a finite number of negative eigenvalues of finite
multiplicity. The nonlinear part is assumed to be such that either
$I-KF$ is $A$-proper or $KF$ is $\phi$-condensing, or that the
corresponding map in an equivalent reformulation of \eqref{e1.1}
is a k-ball contractive perturbation of a strongly monotone map
and is therefore $A$-proper.
We begin with proving some continuation results on the number of
solutions of general nonlinear operator equations. Then we use
them to establish various results on the number of solutions of
\eqref{e1.1} assuming different conditions on the nonlinearity $F$
that imply a priori estimates on the solution set. In particular,
depending on the structure of the linear part $K$, we assume that
either $F$ has a linear growth, and/or $F$ satisfies a side
estimate of the form $(Fx,x)\le a(x)$ for a suitable functional
$a$. Unlike earlier studies, we also study \eqref{e1.1} with
nonlinearities that are the sum of a strongly monotone and k-ball
condensing maps. The last part of the paper is devoted to
applications of these abstract results to Hammerstein integral
equations. This work is a continuation of our study of these
equations in \cite{Mi9, Mi13}. There is an extensive literature on
Hammerstein equations and we refer to \cite{Kr,KZ,V}. In
particular, for the unique (approximation) solvability of these
equations we refer to \cite{V,APZ,Mi9,Mi13}.
\section{Some preliminaries on $A$-proper maps}
Let $\{X_n\}$ and $\{Y_n\}$ be finite dimensional subspaces of
Banach spaces $X$ and $Y$ respectively such that $\dim X_n=\dim Y_n$
for each $n$ and dist$ (x,X_n)\to 0$ as $n\to\infty$ for each
$x\in X$. Let $P_n:X\to X_n$ 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 $\Gamma=\{X_n,P_n;Y_n,Q_n\}$
is a projection scheme for $(X,Y)$.
{\bf Definition 2.1}. A map $T:D \subset X \to Y$ is said to be {\em
approximation-proper } ({\em $A$-proper} for short) with respect to $ \Gamma $
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 $ if in (ii) above
we do not require that a subsequence of $ \{ x_{ n_{ k}} \}$ converges to
$x$ for which $Tx=f$. If (ii) holds for a given $f$, we say that $T$ is
(pseudo) $A$-proper at $f$.
For the developments of the (pseudo) $A$-proper mapping theory and applications
to differential equations, we refer to \cite{Mi5, Mi12} and \cite{P}.
To demonstrate the generality and the unifying nature of the
(pseudo) $A$-proper mapping theory, we state now a number of examples of
$A$-proper and pseudo $A$-proper maps.
To look at $\phi$-condensing maps, 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 map $N:D \subset X \to X$ 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$).
Recall that $N:X\to Y$ is $K$-monotone for some $K:X\to Y^*$
if $(Nx-Ny,K(x-y))\ge 0$ for all $x$,
$y\in X$. It is said to be generalized pseudo-$K$-monotone (of type (KM))
if whenever $x_n\rightharpoonup x$ and $\limsup (Nx_n,K(x_n-x))\le 0$ then
$(Nx_n,K(x_n-x))\to 0$ and $Nx_n\rightharpoonup Nx$
(then $Nx_n\rightharpoonup Nx$).
Recall that $N$ is said to
be of type $(KS_+)$ if $x_n\rightharpoonup x$ and
$\limsup (Nx_n,K(x_n-x))\le 0$ imply that $x_n\to x$. If
$x_n\rightharpoonup x$ implies that $\limsup(Nx_n,K(x_n-x))\ge 0$, $N$ is said to be of type (KP).
If $Y=X^*$ and $K$ is the identity map, then these maps are called monotone,
generalized pseudo monotone, of type (M) and $(S_+)$ respectively. If $Y=X$
and $K=J$ the duality map, then $J$-monotone maps are called accretive. It is
known that bounded monotone maps are of type (M). We say that $N$
is demicontinuous if $x_n\to ~x$ in $X$ implies that $Nx_n\rightharpoonup Nx$.
It is well known that $I-N$ is $A$-proper if $N$ is ball-condensing and
that $K$-monotone like maps are pseudo $A$-proper under some conditions on $N$
and $K$.
Moreover, their perturbations
by Fredholm or hyperbolic like maps are $A$-proper or pseudo $A$-proper
(see \cite{Mi5, Mi6, Mi7, Mi10, Mi11, Mi12}).
The following result states that ball-condensing perturbations of
stable $A$-proper maps are also $A$-proper.
\begin{theorem}[\cite{Mi1}] \label{thm2.1}
Let $D\subset X$ be closed, $T:X\to Y$ be
continuous and $A$-proper with respect to a projectional scheme $\Gamma$
and a-stable,
i.e. for some $c>0$ and $n_0$
$$\|Q_nTx-Q_nTy\|\ge c\|x-y\|\quad\hbox{for }x,y\in X_n \hbox{ and }n\ge n_0$$
and $F:D\to Y$ be continuous. Then $T+F:D\to Y$ is $A$-proper with respect to $\Gamma$
if $F$ is $k$-ball contractive with $k\delta 0$ and $n_f\ge 1$ be such that
$$\deg (P_nH_0, B(0,r_f)\cap X_n,0)\ne 0\quad \hbox{for all }n\ge n_f.$$
Then the equation $H(1,x)=f$ is approximation solvable with respect to $\Gamma$ for each $f\in Y$. Moreover,
if $\Sigma=\{x\in X: H_1\;\hbox{is not invertible at}\;x\}$ and $H_1$
is continuous, then $H_1^{-1}(\{f\})$ is compact for each $f\in Y$ and the cardinal
number card$(H_1^{-1}(\{f\}))$ is constant, finite and positive on each
connected component of the set $Y\setminus H_1(\Sigma)$.
\end{theorem}
\begin{proof} The condition (+) implies
that for each $f\in Y$ there is an $r>R$ and $\gamma >0$ such that
$$\|H(t,x) - tf\|\ge \gamma \quad \hbox{for all }t\in [0,1],\;x\in \partial B(0,r).$$
Indeed, if this were not the case, there would exist $t_n\in [0,1]$ and
$x_n\in X$ such that $t_n\to t$ and $\|x_n\|\to \infty$ and
$H(t_n,x_n)-t_nf\to 0$ as $n\to \infty$. Hence, $H(t_n,x_n)\to tf$ and $\{x_n\}$ is
unbounded, in contradiction to condition (+).
Since $H_t$ is an $A$-proper homotopy, this implies that there is an
$n_0\ge 1$ such that
$$P_nH(t,x)\not=tP_nf\quad\hbox{for all }t\in [0,1],\; x\in \partial
B(0,r)\cap X_n, \;n\ge n_0.$$
By the Brouwer degree properties and the $A$-properness of
$H_1$, there is an $x\in X$ such that $H(1,x)=f$.
The other conclusions follow from Theorem \ref{thm3.1}.
\end{proof}
Next, we have the following homotopy theorem for $\phi$-condensing maps.
\begin{theorem} \label{thm3.3}
Let $F:[0,1]\times X\to X$ be a $\phi$-condensing homotopy
and $H=I-F$ satisfy condition (+). Let, for each $f\in X$, there be an $r_f>0$
such that
$$\deg (H_0, B(0,r_f),0)\ne 0.$$
Then the equation $H(1,x)=f$ is solvable for each $f\in X$. Moreover,
if $\Sigma=\{x\in X: H_1\;\hbox{is not invertible at}\;x\}$ and $H_1$
is continuous, then $H_1^{-1}(\{f\})$ is compact for each $f\in X$ and the cardinal
number card$(H_1^{-1}(\{f\}))$ is constant, finite and positive on each
connected component of the set $X\setminus H_1(\Sigma)$.
\end{theorem}
\begin{proof} As before, condition (+) implies
that for each $f\in X$ there is an $r>R$ and $\gamma >0$ such that
$$\|H(t,x) - tf\|\ge \gamma \quad \hbox{for all }t\in [0,1],\;x\in \partial B(0,r).$$
By the $\phi$ -condensing degree properties \cite{N}, there is an $x\in X$ such that $H(1,x)=f$.
The other conclusions follow from \cite[Theorem 3.2]{S} since its coercivity condition can
be replaced by condition (+).\end{proof}
The existence part of the following result can be found in \cite{Mi9}.
\begin{theorem} \label{thm3.4}
Let $X$ and $Y$ be Banach spaces, $K:Y\to X$
be linear and continuous and $F:X\to Y$ be nonlinear and such that
there are some constants $a$ and $b$ such that
$a\|K\|<1$ and
$$\|Fx\|\le a\|x\|+b\quad \hbox{for all }\|x\|\ge R.$$
a) Let
$H_t=I-tKF:X\to X$ be $A$-proper with respect to a projection scheme $\Gamma=\{X_n,P_n\}$ for $X$
for each $t\in [0,1]$, or $H_1$ is $A$-proper with respect to $\Gamma$ and $\delta a\|K\|<1$, where
$\delta =$max$\|P_n\|$. Then \eqref{e1.1}
is approximation solvable for each $f\in X$. Moreover, if
$\Sigma=\{x\in X: I-KF \mbox{ is not invertible at }x\}$
and $I-KF$ is continuous, then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in X$,
and the cardinal number card$(I-KF)^{-1}(\{f\})$ is constant, finite, and positive on
each connected component of the set $X\setminus (I-KF)(\Sigma)$.
\noindent b) If $I-KF:X\to X$ is pseudo $A$-proper with respect to $\Gamma$ and
$\delta a\|K\|<1$, then \eqref{e1.1} is solvable for each $f\in X$.
\end{theorem}
\begin{proof} (a) We shall show that the homotopy $H_t=I-tKF$ satisfies condition (+).
Indeed, let $H(t_n,x_n)=x_n-t_nKFx_n\to f$ in $Y$ for some $t_n\in [0,1]$ and
$x_n\in X$. It follows that for some $M>0$
$$ \|x_n\|\le \|H(t_n,x_n)\|+\|K\|\;\|Fx\|\le M+\|K\|(a\|x_n\|+b).
$$
Hence $\{x_n\}$ is bounded in $X$ since $a\|K\|<1$.
Moreover, for each $r>0$ and each $n\ge 1$,
$\deg (P_nH_0,B(0,r)\cap X_n,0)\ne 0$.
Hence, the conclusions follow from Theorem \ref{thm3.2}. If only $H_1$ is
$A$-proper, then it satisfies condition (+) as above and $x-KFx=f$ is
approximation solvable for each $f\in X$ (see part b) ). Hence,
Theorem \ref{thm3.1} applies.
\noindent (b) If $I-KF$ is pseudo $A$-proper,
then condition
$$P_nH(t,x)\not=tP_nf\quad \hbox{for all }t\in [0,1],\; x\in \partial
B(0,r)\cap X_n, \;n\ge n_0.$$
holds since $\delta a\|K\|<1$ and the solvability follows from the
pseudo $A$-propernes of $I-KF$.\end{proof}
Since a ball condensing perturbation of the identity map is an
$A$-proper map, we have the following special case.
\begin{corollary} \label{coro3.1}
Let $K:Y\to X$
be linear and continuous and $F:X\to Y$ be nonlinear and such that
$KF$ is a continuous $\phi$-condensing map and there are some constants $a$ and $b$ such that
$a\|K\|<1$, and
$$\|Fx\|\le a\|x\|+b\quad \hbox{for all }\|x\|\ge R.$$
Then \eqref{e1.1}
is approximation solvable for each $f\in X$ with respect to a projection scheme $\Gamma=\{X_n,P_n\}$
for $X$ with $\delta =$max$\|P_n\|=1$ if $\phi =\chi$. It is solvable if $\phi =\gamma$. Moreover, if
$\Sigma=\{x\in X: I-KF \hbox{ is not locally invertible at }x\}$,
then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in X$,
and the cardinal number card$(I-KF)^{-1}(\{f\})$ is constant, finite, and positive on
each connected component of the set $X\setminus (I-KF)(\Sigma)$.
\end{corollary}
Next, we shall discuss other sets of conditions on $K$ and $F$ that imply
the $A$-properness of an operator in an equivalent formulation of our equation.
Recall that a map K acting in a Hilbert
space H is called positive in the sense of Krasnoselski if there exists a
number $\mu >0$ for which
$$(Kx,Kx)\le \mu (Kx,x)\quad x\in H.$$
The infimum of all such numbers $\mu$ is called the positivity constant of $K$ and is denoted by $\mu (K)$.
The simplest example of a positive map is provided by any bounded selfadjoint
positive definite map $K$ on $H$. Then $\mu (K)=\|K\|$
for such maps. A compact normal map $K$ in a Hilbert space is positive on $H$ if and only if ( cf. \cite{K} ) the number
$$[\inf_{\lambda \in\sigma(K),\lambda\ne 0}Re(\lambda ^{-1})]^{-1}$$
is well defined and positive. In that case, it is equal to $\mu (K)$.
Let $X$ be a reflexive embeddable Banach space, that
is, there is a Hilbert space $H$ such that $X\subset H\subset X^*$ so that
$=(y,x)$ for each $y\in H, x\in X$, where $<,>$ is the duality pairing
of $X$ and $X^*$. Let $K:X^*\to X$
be a positive semidefinite bounded selfadjoint map in the sense that
$=$ for all $x,y\in X^*$. Then the positive semidefinite
square root $K_H^{1/2}$ of the restriction $K_H$ of $K$
to $H$ can be extended to a bounded linear map $T:X^*\to H$ such that $K=T^*T$, where the adjoint map
$T^*=K_H^{1/2}$ of $T$ is a bounded map from $H$ to $X$ ( see \cite{V} ).
We shall look at the following equivalent formulation of \eqref{e1.1}
\begin{equation}
y-TFCy=h,\quad h\in H. \label{e3.1}
\end{equation}
We need the following lemma (cf. \cite{V}).
\begin{lemma} \label{lm3.1}
Equations \eqref{e1.1} and \eqref{e3.1} are equivalent with $f$
restricted to $C(H)$; each solution $y$ of \eqref{e3.1} determines a solution
$x=Cy$ of \eqref{e1.1} and each solution $x$ of \eqref{e1.1} with $f\in C(H)$ determines a
solution $y=TFx+h$ of \eqref{e3.1} with $f=Ch$ and $x=Cy$. Moreover, the map
$C:S(h)=(I-TFC)^{-1}(\{h\})\to S=(I-KF)^{-1}(\{Ch\})$ is bijective.
\end{lemma}
\begin{proof} Let $y_1$ and $y_2$ be distinct solutions of \eqref{e3.1}.
Applying $C$ to
$y_i-TFCy_i=h$ and using the fact that $K=CT$, we get that
$x_1=Cy_1$ and $x_2=Cy_2$ are solutions of \eqref{e1.1}. They are distinct since
$$0<\|y_1-y_2\|^2=(TFCy_1-TFCy_2,y_1-y_2)=$$
$$(FCy_1-FCy_2,C(y_1-y_2))=(Fx_1-Fx_2,x_1-x_2).$$
Conversely, let $f\in C(H)$ and $x_1$
and $x_2$ be distinct solutions of \eqref{e1.1}.
Let $f=Ch$ for some $h\in H$. Set $y_i=TFx_i+h$. Then $Cy_i= CTFx_i+h=KFx_i+f$
and so $x_i=Cy_i$. Hence, $y_i=TFCy_i+h$, i.e., $y_i$ are
solutions of \eqref{e3.1}. They are distinct since $y_1=y_2$ implies that $x_1=Cy_1=Cy_2=x_2$.
These arguments show that $C:S(h)\to S$ is a bijection.
\end{proof}
\begin{corollary} \label{coro3.2}
Let $X$ be a reflexive embeddable Banach space
($X\subset H\subset X^*$), $K:X^*\to X$ be a positive semidefinite bounded
selfadjoint map, and $C=K_H^{1/2}$, where $K_H$ is the restriction
of $K$ to $H$, $\mu (K)=\|C\|^2$ and $T:X^*\to H$ be a bounded linear extension of $K_H^{1/2}$.
Let $F=F_1+F_2:X\to X^*$ be a nonlinear map, $a$ and $b$ be
constants and $c$ be the smallest number such that
\begin{itemize}
\item[(i)] $(F_1x-F_1y,x-y)\le c\|x-y\|^2$ for all $x,y\in X$,
and either
\item[(ii)] $a\|K\|<1$ and $\|Fx\|\le a\|x\|+b$ for all $\|x\|\ge R$, or
\item[(iii)] $(a+c)\mu (K)<1$ and $\|F_2x\|\le a\|x\|+b$ for all $\|x\|\ge R$,
\end{itemize}
and $TF_2C$ is a continuous $k$-ball contraction with $k< 1-c\mu(K)$.
Then \eqref{e1.1} is approximation solvable in X for each
$f\in C(H)\subset X$ with respect to a projection scheme
$\Gamma=\{X_n, P_n\}$ for $X$, $\delta=$max $\|P_n\|=1$.
Moreover, if
$\Sigma_H=\{h\in H : I-TFC \hbox{ is not locally invertible at }h\}$,
then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in C(H)$,
and the cardinal number card$(I-KF)^{-1}(\{f\})$ is constant, finite, and positive on
each connected component of the set $H\setminus (I-TFC)(\Sigma_H)$
intersected by $C(H)$.
\end{corollary}
\begin{proof} Equation \eqref{e1.1} is equivalent to \eqref{e3.1}.
Hence, we shall consider this more suitable formulation. We claim that the map $I-TF_1C:H\to H$
is $1-c\mu (K)$-strongly monotone. Indeed, for $x,y\in H$, we have
\begin{align*}
(x-TF_1Cx-y+TF_1Cy,x-y)&=\|x-y\|^2- (TF_1Cx-TF_1Cy,x-y)\\
&=\|x-y\|^2-(F_1Cx-F_1Cy,Cx-Cy)\\
&\ge (1-c\mu (K))\|x-y\|^2.
\end{align*}
Since $TF_2C$ is $k$-ball contractive with $k<1-c\mu (K)$ , we see that
$I-tTFC$ is $A$-proper with respect to $\Gamma=\{H_n,P_n\}$ for $H$ by
Corollary \ref{coro2.1}. Moreover,
$I-tTFC$ satisfies condition (+). Indeed, let $H(t_n,x_n)=x_n-t_nTFCx_n\to g$. If (ii)
holds, then
$$\|x_n\|\le \|H(t_n,x_n)\|+\|T\|(a\|Cx_n\|+b)\le M+a\|K\|\|x_n\|$$
for some constant $M$ since $\|T\|=\|C\|=\|K\|^{1/2}$. It follows that $\{x_n\}$
is bounded. Next, let (iii) hold. Then
\begin{align*}
&(H(t_n,x_n),x_n)\\
&=(x_n-t_nTFCx_n,x_n)\\
&=(x_n-t_nTF_1Cx_n+t_nTF_10,x_n)-t_n(TF_10,x_n)-t_n(TF_2Cx_n,x_n)\\
&\ge (1-c\mu (K))\|x_n\|^2-\|TF_10\|\;\|x_n\|-a\|Cx_n\|^2-b\|Cx_n\|\\
&\ge (1-(a+c)\mu (K))\|x_n\|^2-(\|TF_10\|+b\|C\|)\|x_n\|.
\end{align*}
It follows that $\{x_n\}$ is bounded, for otherwise dividing by $\|x_n\|^2$ and passing to the limit
we get that $(a+c)\mu (K)\ge 1$, a contradiction. Hence, condition (+) holds
in either case.
By Theorem \ref{thm3.2}, we have that the equation $y-TFCy=h$ is solvable for each $h\in H$,
$S(h)=(I-TFC)^{-1}(\{h\})\ne \emptyset$ and compact, and card$S(h)$
is constant, positive and finite on each connected component of the open set $H\setminus (I-TFC)(\Sigma_H)$,
where $\Sigma_H=\{h\in H: I-TFC\;\hbox{is not locally invertible at}\; h\}$.
Next, applying $C$ to $y-TFCy=h$ and using the fact that $K=CT$, we get that $x-KFx=f$ with
$x=Cy\in X$. By Lemma \ref{lm3.1}, we get that card$S=(I-KF)^{-1}(\{Ch\})=$card$S(h)$. Hence,
card$(I-KF)^{-1}(\{f\})$ is constant, finite and positive
on each connected component of $H\setminus (I-TFC)(\Sigma_H)$ intersected by $C(H)$.
\end{proof}
Next, let us look at the case when $K$ is not selfadjoint. We begin by
describing the setting of the problem. Let $X$ be an embeddable Banach space,
$X\subset H\subset X^*$.
Let $K:X^*\to X$ be a linear map and $K_{H}$ be the restriction of $K$ to $H$
such that $K_{H}:H\to H$.
Let $A=(K+K^*)/2$ denote the selfadjoint part of $K$ and $B=(K-K^*)/2$ be
the skew-adjoint part of $K$. Assume that $A$ is positive definite. Under
our assumptions on $K$, both $A$ and $B$ map $X^*$ into $X$. We know that
$A$ can be represented in the form $A=CC^*$, where $C=A^{1/2}$ is the
square root of $A$, $C:H\to X$, and the adjoint map $C^*:X^*\to H$.
As in \cite{APZ} and \cite{Mi13}, we say that $K$ is $P$-positive if $C^{-1}K(C^*)^{-1}$ exists and
is bounded in $H$. It is $S$-positive if $K(C^*)^{-1}$ exists and is
bounded in $H$. Clearly, the $P$-positivity implies the $S$-positivity
but not conversely. It is easy to see that $K$ is $P$-positive if and only if
$C^{-1}B(C^*)^{-1}$ is bounded in $H$, and is $S$-positive if and only if
$B(C^*)^{-1}$ is bounded in $H$. Moreover, $K$ is $P$-positive if and
only if $K$ is angle-bounded, i.e.,
$$ |(Kx,y)-(y,Kx)|\le a(Kx,x)^{1/2}(Ky,y)^{1/2},\quad x,y\in H.$$
Denote by $M$ and $N$ the closure of the maps $C^{-1}K(C^*)^{-1}$ and
$K(C^*)^{-1}$, respectively, in $H$. Note that $M$ and $N$ are defined
on the closure ( in $H$ ) of the range of $C=A^{1/2}$ and suppose that
their domains coincide with $H$. We require that
the following decompositions hold
$$K=CMC^*,\quad K=NC^*.$$
Note that $K$, $M$ and $N$ are related as: $N=CM, N^*=M^*C^*$ and we
have $(Mx,x)=\|x\|^2$ for all $x\in H$. Hence, both $M$ and $M^*$ have trivial nullspaces.
Denote by $\mu (K)=\|N\|^2$, which is the positivity constant of $K$
in the sense of Krasnoselski.
Let $F:X\to X^*$ be a nonlinear map and consider the Hammerstein equation
\begin{equation}
x - KFx = f \label{e3.2}
\end{equation}
For $f\in N(H)$, let $h\in H$ be a solution of
\begin{equation}
M^*h-N^*FNh=M^*k \label{e3.3}
\end{equation}
where $f=Nk$ for some $k\in H$. Then $M^*(h-C^*FNh-k)=0$ since $N=CM$ and
$N^*=M^*C^*$. Hence, $h=C^*FNh+k$ since $M^*$ is injective and therefore
$$Nh=NC^*FNh+Nk=KFNh+f$$
since $K=NC^*$. Thus $x=Nh$ is a solution of \eqref{e3.2}. So the solvability of
\eqref{e3.2} is reduced to the solvability of \eqref{e3.3}. Actually these two equations
are equivalent.
\begin{lemma} \label{lm3.2}
Equations \eqref{e3.2} and \eqref{e3.3} are equivalent with $f$
restricted to $N(H)$; each solution $h$ of \eqref{e3.3} determines a solution
$x=Nh$ of \eqref{e3.2} and each solution $x$ of \eqref{e3.2} with $f\in N(H)$ determines a
solution $h=C^*Fx+k$ of \eqref{e3.3} with $f=Nk$ and $x=Nh$. Moreover, the map
$N:S(M^*k)=(M^*-N^*FN)^{-1}(\{M^*k\})\to S=(I-KF)^{-1}(\{Nk\})$ is bijective.
\end{lemma}
\begin{proof} Let $h_1$ and $h_2$ be distinct solutions of \eqref{e3.3}.
We have seen above that
$x_1=Nh_1$ and $x_2=Nh_2$ are solutions of \eqref{e3.2}. They are distinct since
\begin{align*}
0<\|h_1-h_2\|^2&=(M(h_1-h_2),h_1-h_2)\\
&=(N^*FNh_1-N^*FNh_2,h_1-h_2)\\
&=(FNh_1-FNh_2,N(h_1-h_2))\\
&=(Fx_1-Fx_2,x_1-x_2).
\end{align*}
Conversely, let $f\in N(H)$ and $x_1$ and $x_2$ be distinct solutions of \eqref{e3.2}.
Let $f=Nk$ for some $k\in H$. Set $h_i=C^*Fx_i+k$. Then $Nh_i= NC^*Fx_i+Nk=KFx_i+f$
and so $x_i=Nh_i$. Hence, $M^*h_i=M^*C^*FNh_i+M^*k=N^*FNh_i+M^*k$, i.e., $h_i$ are
solutions of \eqref{e3.3}. They are distinct since $h_1=h_2$ implies that $x_1=Nh_1=Nh_2=x_2$.
These arguments show that $N:S(M^*k)\to S$ is bijective. \end{proof}
\begin{corollary} \label{coro3.3}
Let $X$ be a reflexive embeddable Banach
space ($X\subset H\subset X^*$) and $K:X^*\to X$ be a linear continuous
$P$-positive map.
Let $F=F_1+F_2:X\to X^*$ be a nonlinear map,
$N^*F_2N$ be continuous and k-ball contractive with $k<1-c\mu (K)$ and
there are positive constants $a$ and $b$, $R>0$ and $c$ be
the smallest number such that
\begin{itemize}
\item[(i)] $(F_1x-F_1y,x-y)\le c\|x-y\|^2$ for all$x,y\in X$, and either
\item[(ii)] $a\|K\|<1$ and $\|Fx\|\le a\|x\|+b$ for all $\|x\|\ge R$, or
\item[(iii)] $(a+c)\mu (K)<1$ and $\|F_2x\|\le a\|x\|+b$ for all $\|x\|\ge R$.
\end{itemize}
Then \eqref{e1.1} is solvable in $X$ for each $f\in N(H)\subset X$.
Moreover, if
$\Sigma_H=\{h\in H: M^*-N^*FN \hbox{ is not locally invertible at }h\}$
then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in N(H)$,
and the cardinal number card$(I-KF)^{-1}(\{f\})$ is constant, finite, and positive
on each connected component of the set $H\setminus (M^*-N^*FN)(\Sigma_H)$
intersected by $N(H)$.
\end{corollary}
\begin{proof} Define the homotopy $H(t,x)=M^*x-tN^*FNx$ on $[0,1]\times H$.
It suffices to show that the map $H_t=M^*-tN^*FN:H \to H$ is $A$-proper with respect to $\Gamma=\{H_n,P_n\}$ for each $t\in [0,1]$
and satisfies condition (+). Set $H_{1t}=M^*-tN^*F_1N:H \to H$.
Then, for each $x,y\in H$ we have that
\begin{align*}
(H_1(t,x)-H_1(t,y),x-y)&= \|x-y)\|^2-t(N^*(F_1Nx-F_1Ny),x-y)\\
&=\|x-y\|^2-t(F_1Nx-F_1Ny,Nx-Ny)\\
&\ge (1-c\mu (K))\|x-y\|^2.
\end{align*}
Since $N^*F_2N$ is $k$-ball contraction, $H_t$ is
$A$-proper with respect to $\Gamma$ by Corollary \ref{coro2.1}.
Next, let $f\in N(H)\subset X$, $f=Nk$, be fixed. We claim that
$H(t,x)-tM^*h$ satisfies condition (+).
If not, then there would exist $x_n\in H$, $t_n\in [0,1]$ such
that $\|x_n\|\to \infty$ and
$$y_n=H(t_n,x_n)-t_nM^*k\to g$$
as $n\to \infty.$ Let (ii) hold. Then
$$M^*x_n=y_n+t_nN^*FNx_n-t_nM^*k$$
and
\begin{align*}
\|x_n\|^2&=(M^*x_n,x_n) = (y_n,x_n)+t_n(FNx_n,Nx_n)-t_n(M^*k,x_n)\\
&\le (\|y_n\|+\|M^*k\|+b\|N\|)\|x_n\|+a\mu (K)\;\|x_n\|^2.
\end{align*}
Dividing by $\|x_n\|^2$ and passing to the limit, we get that $1\le a\mu (K)$, a cotradiction.
Hence, condition (+) holds.
Next, let (iii) hold. Then, as above,
\begin{align*}
&(H(t_n,x_n),x_n)\\
&=(M^*x_n-t_nN^*FNx_n,x_n)\\
&=(M^*x_n-t_nN^*F_1Nx_n+t_nN^*F_10,x_n)-t_n(N^*F_10,x_n)-t_n(N^*F_2Nx_n,x_n)\\
&\ge (1-c\mu (K))\|x_n\|^2-\|N^*F_10\|\;\|x_n\|-a\|Nx_n\|^2-b\|Nx_n\| \\
&\ge (1-(a+c)\mu (K))\|x_n\|^2-(\|N^*F_10\|+b\|N\|)\|x_n\|.
\end{align*}
It follows that $\{x_n\}$ is bounded, for otherwise dividing by $\|x_n\|^2$ and
passing to the limit we get that $(a+c)\mu (K)\ge 1$, a contradiction.
Hence, condition (+) holds in either case.
This and the $A$-properness of $M^*-N^*FN$ imply
that $M^*h-N^*FNh=M^*k$ for some $h\in H$ by Theorem \ref{thm3.2}.
As before, we get that $Nh=NC^*FNh+Nk=KFNh+f$ since $K=NC^*$. Thus, $x-KFx=f$ with $x=Nh\in X$.
Next, we have that $Y=N(H)$ is a Banach subspace of X and
$I-KF:Y\to Y$, since $N:H\to X$ is continuous and therefore it is
closed. Moreover, $S(M^*k)$ is nonempty and compact, and
$\mathop{\rm card}S(M^*k)$ is constant and finite on each
connected component of the open set $H\setminus (M^*-N^*FN)(\Sigma_H)$ by
Theorem \ref{thm3.2}. By Lemma \ref{lm3.2}, we get
$\mathop{\rm card}S=(I-KF)^{-1}(\{f\})=\mathop{\rm card}S(M^*k)$ with $f=Nk$.
Hence, $\mathop{\rm card}(I-KF)^{-1}(\{f\})$
is constant, positive and finite on each connected component of
$H\setminus (M^*-N^*FN)(\Sigma_H)$ intersected by $N(H)$.\end{proof}
Next, we shall look at the case when the selfadjoint part $A$ of $K$ is not positive
definite. Suppose that $A$ is quasi-positive definite in $H$, i.e., it has at most a finite
number of negative eigenvalues of finite multiplicity.
Let $U$ be the subspace spanned by the eigenvectors of $A$ corresponding to these negative eigenvalues
of $A$ and $P:H\to U$ be the orthogonal projection onto $U$. Then $P$ commutes with $A$,
but not necessarily with $B$, and acts both in $X$ and $X^*$. The operator $|A|=(I-2P)A$
is easily seen to be positive definite. Hence, we have the decomposition $|A|=DD^*$, where
$D=|A|^{1/2}:H\to X$ and $D^*:X^*\to H$.
Following \cite{APZ,Mi13}, we call the map $K$ $P$-quasi-positive if the map $D^{-1}K(D^*)^{-1}$ exists and is
bounded in $H$, and $S$-quasi-positive if the map $K(D^*)^{-1}$ exists and is bounded in $H$.
Let $M$ and $N$ denote the closure in $H$ of the the bounded maps $D^{-1}K(D^*)^{-1}$ and
$K(D^*)^{-1}$ respectively. Assume that they are both defined on the whole space $H$.
We assume that we have the following decompositions
$$K=DMD^*,\quad K=ND^*.$$
Then we have $N=DM$, $N^*=M^*D^*$, and
$\langle Mh,h\rangle=\|h\|^2-2\|Ph\|^2$ for all $h\in H$.
Define the number
$$\nu (K)=\sup\{\nu: \nu >0,\;\|Nh\|\ge (\nu)^{1/2}\|Ph\|,\;h\in H\}.$$
Note that for a selfadjoint map $K$, $\nu (K)$ is the absolute value of the largest
negative eigenvalue of $K$.
\begin{lemma} \label{lm3.3}
Equations \eqref{e3.2} and \eqref{e3.3} are equivalent with $f$
restricted to $N(H)$; each solution $h$ of \eqref{e3.3} determines a solution
$x=Nh$ of \eqref{e3.2} and each solution $x$ of \eqref{e3.2} with $f\in N(H)$ determines a
solution $h=D^*Fx+k$ of \eqref{e3.3} with $f=Nk$ and $x=Nh$. Moreover, the map
$N:S(M^*k)\to S=(I-KF)^{-1}(\{Nk\})$ is bijective.
\end{lemma}
\begin{proof} Let $h_1$ and $h_2$ be distinct solutions of \eqref{e3.3}.
Since $N=DM$, $K=ND^*$ and $M$ is injective, we get as before that
$x_1=Nh_1$ and $x_2=Nh_2$ are solutions of \eqref{e3.2}. They are distinct since
\begin{align*}
0&\ne \|h_1-h_2\|^2-2\|P(h_1-h_2)\|^2\\
&=(M(h_1-h_2),h_1-h_2)=(N^*FNh_1-N^*FNh_2,h_1-h_2)\\
&=(FNh_1-FNh_2,N(h_1-h_2))=(Fx_1-Fx_2,x_1-x_2).
\end{align*}
Conversely, let $f\in N(H)$ and $x_1$ and $x_2$ be distinct solutions of \eqref{e3.2}.
Let $f=Nk$ for some $k\in H$. Set $h_i=D^*Fx_i+k$. Then $Nh_i= ND^*Fx_i+Nk=KFx_i+f$
and so $x_i=Nh_i$. Hence, $M^*h_i=M^*D^*FNh_i+M^*k=N^*FNh_i+M^*k$, i.e., $h_i$ are
solutions of \eqref{e3.3}. They are distinct since $h_1=h_2$ implies that $x_1=Nh_1=Nh_2=x_2$.
These arguments show that $N:S(M^*k)\to S$ is bijective.\end{proof}
We have the following result when $K$ is $P$-quasi-positive.
\begin{corollary} \label{coro3.4}
Let $X$ be a reflexive embeddable Banach
space ($X\subset H\subset X^*$) and $K:X^*\to X$ be a linear continuous
P-quasi-positive map with $c\nu (K)<-1$.
Let $F=F_1+F_2:X\to X^*$ be a nonlinear map,
$N^*F_2N$ be continuous and k-ball contractive with $k<-(1+c\nu (K))$ and
there are positive constants $a$ and $b$, $R>0$ and let $c$ be
the smallest number such that $1+(a+c)\nu (K)<0$ and
\begin{itemize}
\item[(i)] $(F_1x-F_1y,x-y)\le c\|x-y\|^2\;\;\hbox{for all }\;\;x,y\in X$
\item[(ii)] $\|F_2x\|\le a\|x\|+b$ for all $\|x\|\ge R$,
\end{itemize}
Then \eqref{e1.1} is solvable in $X$ for each $f\in N(H)\subset X$.
Moreover, if
$\Sigma_H=\{h\in H: M^*-N^*FN \mbox{ is not invertible at h}\}$
then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in N(H)$,
and the cardinal number $\mathop{\rm card}(I-KF)^{-1}(\{f\})$ is constant, finite, and positive on
each connected component of the set $H\setminus (M^*-N^*FN)(\Sigma_H)$
intersected by $N(H)$.
\end{corollary}
\begin{proof} Define the
homotopy $H(t,x)=M^*x-tN^*FNx$ on $[0,1]\times H$.
Again, it suffices to show that he map $H_t=M^*-tN^*FN:H
\to H$ is $A$-proper with respect to $\Gamma=\{H_n,P_n\}$ for each $t\in [0,1]$
and satisfies condition (+). Set $H_{1t}=M^*-tN^*F_1N:H
\to H$. Then, for each $x,y\in H$ we have that
\begin{align*}
&(H_1(t,x)-H_1(t,y),x-y)= (M^*x-M^*y,x-y)-t(N^*(F_1Nx-F_1Ny),x-y)\\
&=\|x-y\|^2-2\|P(x-y)\|^2-t(F_1Nx-F_1Ny,Nx-Ny)\\
&\ge \|x-y\|^2-2\|P(x-y)\|^2-tc\|Nx-Ny\|^2)\\
&\ge\|x-y\|^2-2\|P(x-y)\|^2-c\nu (K)\|P(x-y)\|^2\\
&\ge \|x-y\|^2-(2+c\nu (K))\|P(x-y)\|^2=-(1+c\nu (K))\|x-y\|^2.
\end{align*}
Since $N^*F_2N$ is a $k$-ball contraction, $H_t$ is
$A$-proper with respect to $\Gamma$ by Corollary \ref{coro2.1}.
Next, let $f\in N(H)\subset X$, $f=Nk$, be fixed. We claim that
$H(t,x)-tM^*h$ satisfies condition (+).
If not, then there would exist $x_n\in H$, $t_n\in [0,1]$ such
that $\|x_n\|\to \infty$ and
$$y_n=H(t_n,x_n)-t_nM^*k\to g$$
as $n\to \infty.$ Then, as above,
\begin{align*}
(H(t_n,x_n),x_n)
&=(M^*x_n-t_nN^*FNx_n,x_n)\\
&=(M^*x_n,x_n)-t_n(N^*F_1Nx_n,x_n)-t_n(N^*F_2Nx_n,x_n)\\
&=\|x_n\|^2-2\|Px_n\|^2-t_n(N^*F_1Nx_n+t_nN^*F_10,x_n)\\
&\quad -t_n(F_10,Nx_n) -t_n(F_2Nx_n,Nx_n)\\
&\geq -(1+c\nu (K))\|x_n\|^2-\|N^*F_10\|\;\|x_n\|-a\|Nx_n\|^2-b\|Nx_n\|\\
&\geq -(1+(a+c)\nu (K))\|x_n\|^2-\|N^*F_10\|\;\|x_n\|-b\nu (K)^{1/2}\|Px_n\|)\\
&\geq -(1+(a+c)\nu (K))\|x_n\|^2-(\|N^*F_10\|-b\nu (K)^{1/2})\|x_n\|.
\end{align*}
Since
$$(H(t_n,x_n),x_n)=(y_n,x_n)+t_n(M^*k,x_n)\le C\|x_n\|$$
for some constant $C$, we get that
$$-(1+(a+c)\nu (K))\|x_n\|^2-(\|N^*F_10\|-b\nu (K)^{1/2})\|x_n\|\le C\|x_n\|.$$
It follows that $\{x_n\}$ is bounded, for otherwise dividing by $\|x_n\|^2$ and
passing to the limit we get that $1+(a+c)\nu (K)\ge 0$, a contradiction.
Hence, condition (+) holds in either case.
\end{proof}
Next, we shall continue our study of \eqref{e1.1} assuming that the nonlinearity
has a one sided estimate and the linear map $K$ is either positive or
$P$-(quasi)-positive.
\begin{theorem} \label{thm3.5}
Let $X$ be a reflexive embeddable Banach
space ($X\subset H\subset X^*$), $K:X^*\to X$ be a map such that the restriction
of $K$ to $H$, $K_H$, is selfadjoint and positive semidefinite and $F:X\to X^*$
be such that $I-tTFC$ is $A$-proper in $H$ for each $t\in [0,1]$,
and for some constants $a$, $b$, $d$, $R>0$ and $\gamma \in (0,2]$,
$$(Fx,x)\le a\|x\|^2+b\|x\|^{2-\gamma}+d,\quad x\in X\setminus B(0,R)
$$
and $a\lambda<1$, where $\lambda $ is the leading eigenvalue of $K$.
Then \eqref{e1.1} is solvable for each $f\in C(H)\subset X$.
Further, if
$\Sigma_H=\{h\in H : I-TFC \hbox{ is not locally invertible at }h\}$
then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in C(H)$,
and the cardinal number $\mathop{\rm card}(I-KF)^{-1}(\{f\})$ is constant, finite,
and positive on each connected component of the set $H\setminus (I-TFC)(\Sigma_H)$
intersected by $C(H)$.
\end{theorem}
\begin{proof} The map $K$ can be represented as $K=CC^*$, where $C:H\to X$ and
$C^*:X^*\to H$. The restriction of $C$ to $H$ coincides with the selfadjoint positive
semidefinite square root of $K$. Moreover, $\|C\|=\lambda ^{1/2}$ when considered
as a map in $H$. Consider the homotopy $H(t,x)=x-tTFCx$ on $[0,1]\times H$. We claim
that $H(t,x)-tf$ satisfies condition (+). Indeed, let $(t_n,x_n)$ be such that
$H(t_n,x_n)-t_nf\to g$. If $\|x_n\|\to \infty$, then for some $M$,
\begin{align*}
\|x_n\|^2&=(H(t_n,x_n)-t_nf,x_n)+t(TFCx,x)\\
&\le M\|x_n\|+(TFCx,x)\\
&\le M\|x_n\|+(FCx,Cx)\\
&\le M\|x_n\|+a\|Cx_n\|^2+b\|x_n\|^{2-\gamma}+d\le M\|x_n\|
+ a\lambda\|x_n\|^2+b\|x_n\|^{2-\gamma}+d.
\end{align*}
Dividing by $\|x_n\|^2$, we get
$$ 1\le a\lambda+M\|x_n\|^{-1}+b\|x_n\|^{-\gamma}+d\|x_n\|^{-2}.$$
Passing to the limit, we get that $1\le a\lambda$, a contradiction.
Hence, $\{x_n\}$ is bounded and condition (+) holds.
By Theorem \ref{thm3.2}, we get a solution $y$ of $y-C^*FCy=h$ for each $h\in H$ and
$x=Cy$ is a solution of $x-KFx=f$. The other conclusions
follow as in Corollary \ref{coro3.2}. \end{proof}
\begin{remark} \label{rmk3.1} \rm
The one sided condition on $F$ in Theorem \ref{thm3.5}, as well as in other results
below where it appears, can be replaced by
$$ (Fx,x)\le a(x)\quad \hbox{for all } x\in X\setminus B(0,R)
$$
for a suitable function $a:X\to R^+$.
\end{remark}
An easy consequence of Theorems \ref{thm3.3} and \ref{thm3.5} is the following
result.
\begin{corollary} \label{coro3.5}
Let $X$ be a reflexive embeddable Banach space ($X\subset H\subset X^*$),
$K:X^*\to X$ be a map such that the restriction of $K$ to $H$, $K_H$,
is selfadjoint and positive semidefinite and $F:X\to X^*$ be such that $TFC$
is $\phi$-condensing, and for some constants $a$, $b$, $d$ and $\gamma \in (0,2]$
$$(Fx,x)\le a\|x\|^2+b\|x\|^{2-\gamma}+d,\quad x\in X\setminus B(0,R)
$$
and $a\lambda<1$, where $\lambda $ is the leading eigenvalue of $K$.
Then \eqref{e1.1} is solvable for each $f\in C(H)\subset X$.
Further, if
$\Sigma_H=\{h\in H: I-TFC \hbox{ is not locally invertible at }h\}$
then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in C(H)$,
and the cardinal number $\mathop{\rm card}(I-KF)^{-1}(\{f\})$ is constant, finite, and positive
on each connected component of the set $H\setminus (I-TFC)(\Sigma_H)$
intersected by $C(H)$.
\end{corollary}
\begin{corollary} \label{coro3.6}
Let $X$ be a reflexive embeddable Banach space ($X\subset H\subset X^*$),
$K:X^*\to X$ be a positive semidefinite bounded selfadjoint map in $H$, and
$C=K_H^{1/2}$, where $K_H$ is the restriction
of $K$ to $H$, $\mu (K)=\|C\|^2$ and $T:X^*\to H$ be a bounded linear extension of $K_H^{1/2}$.
Let $F=F_1+F_2:X\to X^*$ be a nonlinear map, $a$ $b$, $d$ and $\gamma \in (0,2]$ be
constants such that $a\lambda <1$, $R>0$ and $c$ be
the smallest number such that
\begin{gather*}
(F_1x-F_1y,x-y)\le c\|x-y\|^2 \quad \hbox{for all } x,y\in X\\
(Fx,x)\le a\|x\|^2+b\|x\|^{2-\gamma}+d,\quad x\in X\setminus B(0,R)
\end{gather*}
and $TF_2C$ is a continuous $k$-ball contraction with $k< 1-c\mu(K)$.
Then \eqref{e1.1} is approximation solvable in $X$ for each
$f\in C(H)\subset X$ with respect to a projection scheme
$\Gamma=\{X_n, P_n\}$ for $X$, $\delta=$max $\|P_n\|=1$.
Moreover, if
$\Sigma_H=\{h\in H: I-TFC \hbox{ is not locally invertible at }h\}$
then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in C(H)$,
and the cardinal number $\mathop{\rm card}(I-KF)^{-1}(\{f\})$ is constant, finite, and positive
on each connected component of the set $H\setminus (I-TFC)(\Sigma_H)$
intersected by $C(H)$.
\end{corollary}
\begin{proof}
We have shown before that $I-tTFC:H\to H$ is $A$-proper with respect to
$\Gamma =\{H_n,P_n\}$, $t\in [0,1]$. Then the conclusions follow from
Theorem \ref{thm3.5}. \end{proof}
Next, we shall give an extension of Theorem \ref{thm3.5} to non-selfadjoint $K$.
\begin{theorem} \label{thm3.6}
Let $X$ be a reflexive embeddable Banach
space ($X\subset H\subset X^*$), $K:X^*\to X$ be a linear continuous
$P$-positive map and $F:X\to X^*$ be such that $M^*-tN^*FN$ is $A$-proper
with respect to $\Gamma$ for each $t\in [0,1]$,
and for some constants $a$, $b$, $d$, $\gamma \in (0,2]$ and $R>0$,
$$(Fx,x)\le a\|x\|^2+b\|x\|^{2-\gamma}+d,\;\;x\in X\setminus B(0,R)$$
and $a\mu (K)<1$.
Then \eqref{e1.1} is solvable for each $f\in N(H)\subset X$.
Moreover, if
$\Sigma_H=\{h\in H: M^*-N^*FN \hbox{ is not locally invertible at }h\}$
then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in N(H)$,
and the cardinal number $\mathop{\rm card}(I-KF)^{-1}(\{f\})$ is constant, finite, and
positive on each connected component of the set $H\setminus (M^*-N^*FN)(\Sigma_H)$
intersected by $N(H)$.
\end{theorem}
\begin{proof} The homotopy $H(t,x)=M^*x-tN^*FNx$ on $[0,1]\times H$ is
$A$-proper with respect to $\Gamma=\{H_n,P_n\}$. By Theorem \ref{thm3.2}, it is left
to show that it satisfies condition (+).
Let $f\in N(H)\subset X$, $f=Nk$, be fixed. We claim that
$H(t,x)-tM^*h$ satisfies condition (+).
If not, then there would exist $x_n\in H$, $t_n\in [0,1]$ such
that $\|x_n\|\to \infty$ and
$$y_n=H(t_n,x_n)-t_nM^*k\to g$$
as $n\to \infty$. Then
$$M^*x_n=y_n+t_nN^*FNx_n-t_nM^*k$$
and
\begin{align*}
\|x_n\|^2&=(M^*x_n,x_n) \\
&= (y_n,x_n)+t_n(FNx_n,Nx_n)-t_n(M^*k,x_n)\\
&\le (\|y_n\|+\|M^*k\|)\|x_n\|+b(\|N\|\;\|x_n\|)^{2-\gamma}+d+a\mu (K)\;\|x_n\|^2.
\end{align*}
Dividing by $\|x_n\|^2$ and passing to the limit, we get that $1\le a\mu (K)$, a
contradiction. Hence, condition (+) holds.
This and the $A$-properness of $M^*-N^*FN$ imply
that $M^*h-N^*FNh=M^*k$ for some $h\in H$. The rest of the proof follows as in
Corollary \ref{coro3.3}.\end{proof}
\begin{corollary} \label{coro3.7}
Let $X$ be a reflexive embeddable Banach
space ($X\subset H\subset X^*$), $K:X^*\to X$ be a linear $P$-positive map
and $F:X\to X^*$ be such that $N^*FN$ is ball condensing,
and for some constants $a$, $b$, $d$, $\gamma \in (0,2]$ and $R>0$
$$(Fx,x)\le a\|x\|^2+b\|x\|^{2-\gamma}+d,\;\;x\in X\setminus B(0,R)$$
and $a\mu (K)<1$.
Then \eqref{e1.1} is solvable for each $f\in N(H)\subset X$.
Moreover, if
$\Sigma_H=\{h\in H: M^*-N^*FN \mbox{ is not invertible at }h\}$
then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in N(H)$,
and the cardinal number $\mathop{\rm card}(I-KF)^{-1}(\{f\})$ is constant, finite, and
positive on each connected component of the set $H\setminus (M^*-N^*FN)(\Sigma_H)$
intersected by $N(H)$.
\end{corollary}
\begin{proof} It suffices to observe that $M^*-N^*FN$ is $A$-proper with respect
to $\Gamma$ by Corollary \ref{coro2.1}.\end{proof}
\begin{corollary} \label{coro3.8}
Let $X$ be a reflexive embeddable Banach space
($X\subset H\subset X^*$) and $K:X^*\to X$ be a linear continuous
$P$-positive map. Let $F=F_1+F_2:X\to X^*$ be a nonlinear map,
$N^*F_2N$ be continuous and k-ball contraction with $k<1-c\mu (K)$ and
there are positive constants $a$, $b$, $d$, $\gamma \in (0,2]$ and $R>0$
with $a\mu (K)<1$, and let $c$
be the smallest number such that
\begin{itemize}
\item[(i)] $(F_1x-F_1y,x-y)\le c\|x-y\|^2\;\;x,y\in X$
\item[(ii)] $(Fx,x)\le a\|x\|^2+b\|x\|^{2-\gamma}+d,\;\;x\in X\setminus B(0,R)$
\end{itemize}
Then \eqref{e1.1} is solvable in $X$ for each $f\in N(H)\subset X$.
Moreover, if
$\Sigma_H=\{h\in H: M^*-N^*FN \mbox{ is not invertible at }h\}$
then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in N(H)$,
and the cardinal number $\mathop{\rm card}(I-KF)^{-1}(\{f\})$ is constant, finite, and positive
on each connected component of the set $H\setminus (M^*-N^*FN)(\Sigma_H)$
intersected by $N(H)$.
\end{corollary}
\begin{proof} As in the proof of Corollary \ref{coro3.4}, we have that $M^*-tN^*FN$
is $A$-proper with respect to $\Gamma$ for each $t\in [0,1]$. Hence, the conclusions
follow from Theorem \ref{thm3.6}.\end{proof}
For $P$-quasi-positive $K$ we have the following statement.
\begin{theorem} \label{thm3.7}
Let $X$ be a reflexive embeddable Banach
space ($X\subset H\subset X^*$), $K:X^*\to X$ be a linear continuous
$P$-quasi-positive map and $F:X\to X^*$ be such that $M^*-tN^*FN$ is $A$-proper
with respect to $\Gamma$ for each $t\in [0,1]$,
and for some constants $a$, $b$, $d$, $\gamma \in (0,2]$ and $R>0$
$$(Fx,x)\le a\|x\|^2+b\|x\|^{2-\gamma}+d,\quad x\in X\setminus B(0,R)$$
and $-(1+a\nu (K)) >0$.
Then \eqref{e1.1} is solvable for each $f\in N(H)\subset X$.
Moreover, if
$\Sigma_H=\{h\in H: M^*-N^*FN \mbox{ is not invertible at }h\}$
then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in N(H)$,
and the cardinal number $\mathop{\rm card}(I-KF)^{-1}(\{f\})$ is constant, finite, and
positive on each connected component of the set $H\setminus (M^*-N^*FN)(\Sigma_H)$
intersected by $N(H)$.
\end{theorem}
\begin{proof} The homotopy $H(t,x)=M^*x-tN^*FNx$ on $[0,1]\times H$ is
$A$-proper with respect to $\Gamma=\{H_n,P_n\}$. By Theorem \ref{thm3.2}, it is left
to show that it satisfies condition (+).
Let $f\in N(H)\subset X$, $f=Nk$, be fixed. We claim that
$H(t,x)-tM^*h$ satisfies condition (+).
If not, then there would exist $x_n\in H$, $t_n\in [0,1]$ such
that $\|x_n\|\to \infty$ and
$$y_n=H(t_n,x_n)-t_nM^*k\to g$$
as $n\to \infty$. Then
$$ M^*x_n=y_n+t_nN^*FNx_n-t_nM^*k $$
and
\begin{align*}
&(y_n,x_n)\\
&=(M^*x_n,x_n)-t_n(FNx_n,Nx_n)+t_n(M^*k,x_n)\\
&\ge \|x_n\|^2-2\|Px_n\|^2 -t_na\|Nx_n\|^2
-t_nb\|Nx_n\|^{2-\gamma}-t_nd-t_n\|M^*k\|\;\|x_n\|\\
&\ge \|x_n\|^2-(2+a\nu(K))\|Px_n\|^2
-b(\nu (K)^{1/2}\|P_nx_n\|)^{2-\gamma}-\|M^*k\|\;\|x_n\|-d\\
&\ge -(1+a\nu (K))\|x_n\|^2-b(\nu (K) ^{1/2}\|x_n\|)^{2-\gamma}-\|M^*k\|\;\|x_n\|-d.
\end{align*}
Dividing by $\|x_n\|^2$ and passing to the limit, we get that $1+a\nu (K)\ge 0$,
a contradiction. Hence, condition (+) holds.
This and the $A$-properness of $M^*-N^*FN$ imply
that $M^*h-N^*FNh=M^*k$ for some $h\in H$. The rest of the proof follows as in
Corollary \ref{coro3.3}.\end{proof}
\begin{corollary} \label{coro3.9}
Let $X$ be a reflexive embeddable Banach
space ($X\subset H\subset X^*$), $K:X^*\to X$ be a linear $P$-quasi-positive map
and $F:X\to X^*$ be such that $N^*FN$ is ball condensing,
and for some constants $a$, $b$, $d$, $\gamma \in (0,2]$ and $R>0$
$$(Fx,x)\le a\|x\|^2+b\|x\|^{2-\gamma}+d,\quad x\in X\setminus B(0,R)$$
and $-(1+a\nu (K))>0$.
Then \eqref{e1.1} is solvable for each $f\in N(H)\subset X$.
Moreover, if
$\Sigma_H=\{h\in H: M^*-N^*FN \mbox{ is not invertible at }h\}$
then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in N(H)$,
and the cardinal number $\mathop{\rm card}(I-KF)^{-1}(\{f\})$ is constant, finite, and
positive on each connected component of the set $H\setminus (M^*-N^*FN)(\Sigma_H)$
intersected by $N(H)$.
\end{corollary}
\begin{proof} We have that $M^*-tN^*FN+2P$ is $A$-proper with respect to
$\Gamma$ for each $t\in [0,1]$ by Corollary \ref{coro2.1} since
$(M^*x+2Px,x)=\|x\|^2$. But,
$P$ is a compact map and therefore the map $M^*-tN^*FN$ is $A$-proper
as a compact perturbation of an $A$-proper map. Hence, the conclusions follow
by Theorem \ref{thm3.7}.\end{proof}
\begin{corollary} \label{coro3.10}
Let $X$ be a reflexive embeddable Banach
space ($X\subset H\subset X^*$) and $K:X^*\to X$ be a linear continuous
P-quasi-positive map.
Let $F=F_1+F_2:X\to X^*$ be a nonlinear map,
$N^*F_2N$ be continuous and k-ball contractive with $k<1-c\mu (K)$ and
there are positive constants $a$, $b$, $d$, $\gamma \in (0,2]$ and $R>0$ with $-(1+a\nu (K))<0$, and let $c$
be the smallest number such that
\begin{itemize}
\item[(i)] $(F_1x-F_1y,x-y)\le c\|x-y\|^2\;\;x,y\in X$
\item[(ii)] $(Fx,x)\le a\|x\|^2+b\|x\|^{2-\gamma}+d,\;\;x\in X\setminus B(0,R)$.
\end{itemize}
Then \eqref{e1.1} is solvable in $X$ for each $f\in N(H)\subset X$.
Moreover, if
$\Sigma_H=\{h\in H: M^*-N^*FN \mbox{ is not invertible at }h\}$
then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in N(H)$,
and the cardinal number $\mathop{\rm card}(I-KF)^{-1}(\{f\})$ is constant, finite, and positive on
each connected component of the set $H\setminus (M^*-N^*FN)(\Sigma_H)$
intersected by $N(H)$.
\end{corollary}
\begin{proof} As in the proof of Corollary \ref{coro3.4}, we have that $M^*-tN^*FN$ is
$A$-proper with respect to $\Gamma$ for each $t\in [0,1]$.
Hence, the conclusions follow by Theorem \ref{thm3.7}.\end{proof}
For our next result, assume that a Hilbert space $H$ is in a duality with
a Banach space $Y$ with $H\subset Y$ and $K:H\to H$ is positive. Then
its selfadjoint part $A=1/2(K+K^*)$ is positive semidefinite
and therefore its square root $C=A^{1/2}:H\to H$ is also positive and semidefinite.
Assume that $K$ has a decomposition of the form $K=NC:H\to Y$ for some continuous linear
map $N:H\to Y$.
\begin{lemma} \label{lm3.4}
Equations \eqref{e1.1} and $y-FKy=h$ with $h\in H$ and $f\in K(H)$ are equivalent;
each solution $y$ of $y-FKy=h$ determines a solution $x=Ky$ of \eqref{e1.1} and
each solution $x$ of \eqref{e1.1} with $f\in K(H)$ determines a solution
$y=Fx+h$ of $y-FKy=h$ with $f=Kh$ and $x=Kh$.
Moreover, the map $K:S(h)=(I-FK)^{-1}(\{h\})\to S=(I-KF)^{-1}(\{Kh\})$ is bijective.
\end{lemma}
\begin{proof} Let $y_1$ and $y_2$ be distinct solutions of $y-FKy=h$.
Applying $K$ to $y_i-FKy_i=h$, we get that
$x_1=Ky_1$ and $x_2=Ky_2$ are solutions of \eqref{e1.1}. They are distinct since
$$0<\|y_1-y_2\|^2=(FKy_1-FKy_2,y_1-y_2)$$
implies that $FKy_1\ne FKy_2$ and therefore $x_1=Ky_1\ne x_2=Ky_2$.
Conversely, let $f\in K(H)$ and $x_1$
and $x_2$ be distinct solutions of \eqref{e1.1}.
Let $f=Kh$ for some $h\in H$. Set $y_i=Fx_i+h$. Then $Ky_i= KFx_i+f$
and so $x_i=Ky_i$. Hence, $y_i=FKy_i+h$, i.e., $y_i$ are
solutions of $y-FKy=h$. They are distinct since $y_1=y_2$ implies that $x_1=Ky_1=Ky_2=x_2$.
These arguments show that $K:S(h)\to S$ is a bijection.\end{proof}
\begin{theorem} \label{thm3.8}
Let a Hilbert space $H$ be in a duality with
a Banach space $Y$ with $H\subset Y$ and $K:H\to H$ be positive and $K=NC:H\to Y$
for some continuous linear map $N:H\to Y$. Let
$F:Y\to H$ be a bounded nonlinear map such that $I-FK:H\to H$ is $A$-proper and
$$(Fx,x)\le a\|x\|^2+b\|x\|^{2-\gamma}+d,\quad x\in Y\setminus B(0,R)$$
for some constants $a, b, d, \gamma\in (0,2], R>0$ and $a\mu (K)<1$.
Then \eqref{e1.1} is solvable for each $f\in K(H)$. Moreover,if
$\Sigma=\{x\in H: I-FK \mbox{ is not invertible at }x\}$
and $I-FK$ is continuous, then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in K(H)$,
and the cardinal number $\mathop{\rm card}(I-KF)^{-1}(\{f\})$ is constant, finite, and positive on
each connected component of the set $H\setminus (I-FK)(\Sigma)$ intersected
by $K(H)$.
\end{theorem}
\begin{proof} Let $h\in H$ and and $f=Kh$. Consider the homotopy $H(t,y)=y-tFKy$
on $[0,1]\times H$. We claim that $H(t,y)-th$ satisfies condition (+)
for each $h\in H$. Let $t_n\in [0,1]$ and
$y_n\in H$ be such that $u_n=y_n-t_nFKy_n-t_nh\to g$. Then the positivity of $K$ implies
\begin{align*}
0\le (Ky_n,y_n)&\le (u_n,Ky_n)+t_n(FKy_n+h,Ky_n)\\
&\le \|u_n\|\;\|Ky_n\|+a\|Ky_n\|^2+b\|Ky_n\|^{2-\gamma}+\|h\|\;\|Ky_n\|+d\\
&\le a\mu (K)(y_n,Ky_n)+b\|Ky_n\|^{2-\gamma}+(\|h\|+|u_n\|)\|Ky_n\|+d.
\end{align*}
Hence,
$$(Ky_n,y_n)\le (1-a\mu (K))^{-1}(b\|Ky_n\|^{2-\gamma}+(\|h\|+\|u_n\|)\|Ky_n\|+d).$$
Moreover,
\begin{align*}
(Ky_n,y_n)&=(Ay_n,y_n)=(Cy_n,Cy_n)\\
&\le (1-a\mu(K))^{-1}(b\|Ky_n\|^{2-\gamma}+(\|h\|+\|u_n\|)\|Ky_n\|+d).
\end{align*}
But, $K=NC$ and therefore,
$$\|Ky_n\|\le \|N\|\;\|Cy_n\|\le c_1\|Ky_n\|^{1-\gamma/2}+c_2\|Ky_n\|^{1/2}+c_3
$$
for some constants $c_1$, $c_2$ and $c_3$. Since the real function
$f(t)=t-c_1t^{1-\gamma/2}-c_2t^{1/2}$ tends to infinity as $t\to \infty$, and
for each $n$
$$\|Ky_n\|-c_1\|Ky_n|^{1-\gamma/2}-c_2\|Ky_n\|^{1/2}\le c_3
$$
it follows that $\{\|Ky_n\|: n=1,2,\dots \}$ is a bounded set. Thus
$$\|y_n\|\le \|u_n\|+\|FKy_n\|+\|h\|\le c_4
$$
for some constant $c_4$ and all $n$ by the boundedness of $F$. This shows that
$H_t$ satisfies condition (+). By Theorem \ref{thm3.2}, we have that the equation $y-FKy=h$
is solvable for each $h\in H$,
$S(h)=(I-FK)^{-1}(\{h\})\ne \emptyset$ and compact, and $\mathop{\rm card}S(h)$
is constant and finite on each connected component of the open set
$H\setminus (I-FK)(\Sigma)$.
Next, applying $K$ to $y-FKy=h$ we get that $x-KFx=f$ with
$x=Ky\in H$. By Lemma \ref{lm3.4}, we get that
$\mathop{\rm card}S=(I-KF)^{-1}(\{Kh\})=\mathop{\rm card}S(h)$. Hence,
$\mathop{\rm card}(I-KF)^{-1}(\{f\})$ is constant, finite and positive
on each connected component of $H\setminus (I-FK)(\Sigma)$
intersected by $K(H)$. \end{proof}
\begin{corollary} \label{coro3.11}
Let a Hilbert space $H$ be in a duality with
a Banach space $Y$ with $H\subset Y$ and $K:H\to H$ be positive and $K=NC:H\to Y$
for some continuous linear map $N:H\to Y$. Let
$F:Y\to H$ be a bounded nonlinear map such that $FK:H\to H$ is continuous
and $\phi$-condensing, and
$$
(Fx,x)\le a\|x\|^2+b\|x\|^{2-\gamma}+d,\quad x\in Y\setminus B(0,R)$$
for some constants $a, b, d, \gamma\in (0,2], R>0$ and $a\mu (K)<1$.
Then \eqref{e1.1} is solvable for each $f\in K(H)$. Moreover,if
$\Sigma=\{x\in H: I-FK \mbox{ is not invertible at }x\}$,
then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in K(H)$,
and the cardinal number $\mathop{\rm card}(I-KF)^{-1}(\{f\})$ is constant, finite, and positive on
each connected component of the set $H\setminus (I-FK)(\Sigma)$
intersected by $K(H)$.
\end{corollary}
\begin{proof} If $KF$ is ball condensing, Theorem \ref{thm3.8} applies. If $KF$ is
set condensing, then arguing as in Theorem \ref{thm3.8} we can prove this case again.
\end{proof}
\begin{remark} \label{rmk3.2} \rm
If $K:H\to H$ is a positive, normal and compact map, then there is a map
$N:H\to Y$ such that $K=NC$ (cf. \cite{K}). In this case $FK$ is compact and
Corollary \ref{coro3.11} is applicable.
\end{remark}
Next, assuming only the positivity of $K$, we can still prove the solvability
of \eqref{e1.1} by requiring additionally that $F$ has a linear growth.
\begin{theorem} \label{thm3.9}
Let $X$ be a reflexive embeddable Banach
space ($X\subset H\subset X^*$), $K:X^*\to X$ be a continuous map such
that the restriction $K_H$ of $K$ to $H$
is positive and $F:X\to X^*$ be such that $I-KF$ is $A$-proper,
\begin{gather*}
\|Fx\|\le a\|x\|+b,\quad x\in X \\
(Fx,x)\le c\|x\|^2+d,\quad x\in X\setminus B(0,R)
\end{gather*}
for some constants $a, b, c, d, R>0$ and $c\mu (K)<1$.
Then \eqref{e1.1} is approximation solvable for each $f\in X$. Moreover, if
$\Sigma=\{x\in X: I-KF \mbox{ is not invertible at }x\}$
and $I-KF$ is continuous, then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in X$,
and the cardinal number card$(I-KF)^{-1}(\{f\})$ is constant, finite, and
positive on each connected component of the set $X\setminus (I-KF)(\Sigma)$.
\end{theorem}
\begin{proof} Consider the homotopy $H(t,x)=x-tKFx$ on $[0,1]\times X$. We claim
that $H(t,x)-tf$ satisfies condition (+)
for each $f\in X$. Indeed, let $t_n\in [0,1]$ and
$x_n\in X$ be such that $u_n=x_n-t_nKFx_n-t_nf\to g$. Then $Fx_n=F(u_n+t_nKFx_n+t_nf)$ and
set $y_n=t_nFx_n$. Then $y_n=t_nF(u_n+Ky_n+t_nf)$ and
\begin{align*}
&(y_n,Ky_n)\\
&=(t_nF(u_n+Ky_n+t_nf),Ky_n)\\
&=t_n(F(u_n+Ky_n+t_nf),u_n+Ky_n+t_nf)- (F(u_n+Ky_n+t_nf),u_n+t_nf) \\
&\le c\|u_n+Ky_n+t_nf\|^2+\|F(u_n+Ky_n+t_nf)\|\;\|u_n+t_nf\|+d\\
&\le c\|Ky_n\|^2+(a+2c)\|u_n+t_nf\|\;\|Ky_n\|+(a+c)\|u_n+t_nf\|^2\\
&\quad +b\|u_n+t_nf\|+c_1).
\end{align*}
Since $(y,Ky)\ge 1/\mu(K)\|Ky\|^2$ for all $y\in H$ and $H$ is dense in $X^*$,
we have that $(y,Ky)\ge 1/\mu(K)\|Ky\|^2$ for all $y\in X^*$. Hence,
\begin{align*}
\|Ky_n\|^2&\le \mu(K)(y_n,Ky_n)\\
&\le \mu(K)(c\|Ky_n\|^2+(a+2c)\|u_n+t_nf\|\;\|Ky_n\|+
(a+c)\|u_n+t_nf\|^2\\
&\quad +b\|u_n+t_nf\|+c_1).
\end{align*}
Next, we have that
$$\|x_n\|=\|u_n+Ky_n+t_nf\|\le \|u_n\|+\|Ky_n\|+\|f\|\le M+\|Ky_n\|
$$
for some constant $M$. If $Ky_n\to 0$, it follows that $\{x_n\}$ is bounded.
If $\{Ky_n\}$ does not converge to zero, then after dividing the above inequality
by $\|Ky_n\|$ we get
\begin{align*}
\|Ky_n\|&\le (1-c\mu(K))^{-1}\mu(K)[(a+2c)\|u_n+t_nf\|\\
&\quad +\big((a+c)\|u_n+t_nf\|^2+b\|u_n+t_nf\|+c_1\big)/\|Ky_n\|]\le M_1
\end{align*}
for all $n$ and some constant $M_1$. Hence, $\{x_n\}$ is bounded
in either case and condition (+) holds.
The conclusions now follow from Theorem \ref{thm3.2} since
$$\deg (P_nH_0,B(0,r)\cap H_n,0)=\deg (I,B(0,r)\cap H_n,0)\ne 0$$
for all $n\ge n_0$. \end{proof}
\begin{corollary} \label{coro3.12}
Let $X$ be a reflexive embeddable Banach
space ($X\subset H\subset X^*$), $K:X^*\to X$ be a map such that the restriction
$K_H$ of $K$ to $H$ is positive and $F:X\to X^*$ be nonlinear and such that
$KF$ is a continuous $\phi$-condensing map and
\begin{gather*}
\|Fx\|\le a\|x\|+b,\quad x\in X\\
(Fx,x)\le c\|x\|^2+d,\quad x\in X\setminus B(0,R)
\end{gather*}
for some constants $a, b, c, d$ and $c\mu(K)<1$. Then \eqref{e1.1} is solvable
for each $f\in X$. Moreover, if
$\Sigma=\{x\in X: I-KF \mbox{ is not invertible at }x\}$,
then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in X$,
and the cardinal number card$(I-KF)^{-1}(\{f\})$ is constant, finite, and positive on
each connected component of the set $X\setminus (I-KF)(\Sigma)$.
\end{corollary}
\begin{proof} Consider the homotopy
$H(t,x)=x-tKFx-tf$. Then the conclusions follow
from Theorems \ref{thm3.3} and \ref{thm3.9}. \end{proof}
Next, we shall extend Theorem \ref{thm3.9} to potentially positive maps. Recall that
a map $K:H\to H$ is called potentially positive (from below) (cf. \cite{K})
if there exists a $\lambda \in R$ such that the map $I-\lambda K$ is
continuously invertible and the map $K_{\lambda}=(I-\lambda K)^{-1}K$ is
positive on $H$. Clearly, any positive map is potentially positive.
Moreover, a completely continuous selfadjoint map is potentially positive
if and only if it has a finite number of negative eigenvalues.
Equation \eqref{e1.1} can be written in the following equivalent form
$$x-K_{\lambda}F_{\lambda}x=(I-\lambda K)^{-1}(f)$$
where $F_{\lambda}=F-\lambda I$. Clearly,
$$
S(f)=(I-KF)^{-1}(\{f\})=S_{\lambda}((I-\lambda K)^{-1}f)
=(I-K_{\lambda}F_{\lambda})^{-1}(\{(I-\lambda K)^{-1}f\}).$$
Moreover, $I-KF:X\to X$ is locally invertible at $x_0\in X$ if and only if
$(I-\lambda K)^{-1}(I-KF):X\to X$ is locally invertible at $x_0\in X$ since
$I-\lambda K:H\to H$ is a homeomorphism. Hence,
$\Sigma=\{x\in X: I-KF \mbox{ is not invertible at }x\}=\Sigma_{\lambda}=
\{x\in X: (I-\lambda K)^{-1}(I-KF)\hbox{ is not locally invertible at } x\}=
\{x\in X: I-(I-\lambda K)^{-1}K(F-\lambda I)\hbox{ is not locally invertible at }x\}$.
We have the following extension of Theorem \ref{thm3.9} to potentially positive maps.
\begin{theorem} \label{thm3.10}
Let $X$ be a reflexive embeddable Banach space
($X\subset H\subset X^*$), $K:X^*\to X$ be a map such that the restriction $K_H$
of $K$ to $H$ is potentially positive and $F:X\to X^*$ be such that
$I-K_{\lambda}F_{\lambda}$ is $A$-proper,
and
\begin{gather*}
\|Fx\|\le a\|x\|+b,\quad x\in X\\
(Fx,x)\le c\|x\|^2+c_1,\quad x\in X
\end{gather*}
for some constants $a, b, c, c_1$ and $(c-\lambda)\mu(K)<1$. Then \eqref{e1.1}
is approximation solvable for each $f\in X$. Moreover, if
$\Sigma=\{x\in X: I-KF \mbox{ is not invertible at }x\}$
and $I-KF$ is continuous, then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in X$,
and the cardinal number card$(I-KF)^{-1}(\{f\})$ is constant, finite, and positive on
each connected component of the set $X\setminus (I-KF)(\Sigma)$.
\end{theorem}
\begin{proof} Let $F_{\lambda}=F-\lambda I$ and consider the homotopy
$H_{\lambda}(t,x)=x-tK_{\lambda}F_{\lambda}x-tf$. Then arguing as in
Theorem \ref{thm3.9}, we
get that $S(f)=(I-KF)^{-1}(\{f\})=S_{\lambda}((I-\lambda K)^{-1}f)$ is not empty and
compact, and card$S(f)$ is constant, finite and positive on each connected component $U_i$
of the open set $X\setminus (I-K_{\lambda}F_{\lambda})(\Sigma)=\cup_i U_i$.
Since $(I-\lambda K)(X\setminus (I-K_{\lambda}F_{\lambda})(\Sigma))=X\setminus (I-\lambda K)
(I-K_{\lambda}F_{\lambda})(\Sigma)=X\setminus (I-KF)(\Sigma)$, we get that
$X\setminus (I-KF)(\Sigma)=\cup_i(I-\lambda K)U_i$. Hence, $f\in (I-\lambda K)U_i$ if
and only if $f=(I-\lambda K)g$ with $g\in U_i$. Therefore, card$(I-KF)^{-1}(\{f\})$ is constant,
finite, and positive on each connected component of the set
$X\setminus (I-KF)(\Sigma)$. \end{proof}
\begin{corollary} \label{coro3.13}
Let $X$ be a reflexive embeddable Banach space
($X\subset H\subset X^*$), $K:X^*\to X$ be a map such that the restriction $K_H$
of $K$ to $H$ is potentially positive and $F:X\to X^*$ be such that
$K_{\lambda}F_{\lambda}$ is $\phi$-condensing,
and
\begin{gather*}
\|Fx\|\le a\|x\|+b,\;\;x\in X\\
(Fx,x)\le c\|x\|^2+c_1,\;\;x\in X
\end{gather*}
for some constants $a, b, c, c_1$ and $(c-\lambda)\mu(K)<1$. Then \eqref{e1.1}
is solvable for each $f\in X$. Moreover, if
$\Sigma=\{x\in X: I-KF \mbox{ is not invertible at }x\}$
and $I-KF$ is continuous, then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in X$,
and the cardinal number card$(I-KF)^{-1}(\{f\})$ is constant, finite, and positive on
each connected component of the set $X\setminus (I-KF)(\Sigma)$.
\end{corollary}
\begin{proof} Let $F_{\lambda}=F-\lambda I$ and consider the homotopy
$H_{\lambda}(t,x)=x-tK_{\lambda}F_{\lambda}x-tf$. Then the conclusions
follow from Theorems \ref{thm3.3} and \ref{thm3.10}. \end{proof}
We say that $T$ satisfies condition (++)
if whenever $\{x_n\}$ is bounded and $Tx_n\to f$, then $Tx=f$ for
some $x\in X$.
Let $\sigma (K)$ denote the spectrum of $K$.
Our next result involves a suitable Leray-Schauder type of
condition.
\begin{theorem} \label{thm3.11}
Let $K:X\to X$ be a continuous linear map,
$\lambda^{-1}\notin \sigma(K)$, $F: X\to X$ be nonlinear,
$T_p=pI-(I-\lambda K)^{-1}K(F-\lambda I)
:X\to X$ for $p\ge 1$, $T_1$ satisfy
condition (+) and either $F$ is odd or, for some $R>0$,
\begin{equation}
K(F-\lambda I)x\ne t(I-\lambda K)x\quad \hbox{for } \|x\|\ge R, t>1. \label{e3.4}
\end{equation}
a) If $T_1$ is $A$-proper with respect to $\Gamma$, then \eqref{e1.1}
is approximation solvable for each $f\in X$. Moreover, if
$\Sigma=\{x\in X: I-KF \mbox{ is not invertible at }x\}$
and $I-KF$ is continuous, then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in X$,
and the cardinal number card$(I-KF)^{-1}(\{f\})$ is constant, finite, and positive on
each connected component of the set $X\setminus (I-KF)(\Sigma)$.
\noindent
b) If $T_p$ is $A$-proper with respect to $\Gamma$ for each $p>1$
and $T_1$ satisfies condition (++), then \eqref{e1.1} is
solvable for each $f\in X$.
\end{theorem}
\begin{proof} Equation \eqref{e1.1} is equivalent to
\begin{equation}
Ax-Nx=f\label{e3.5}
\end{equation}
where $A=I-\lambda K$ and $N=K(F-\lambda I)$.
It is easy to see that (3.4) implies that
$$Nx\ne tAx\;\;\hbox{for}\; \|x\|\ge R, t>1.$$
Hence, the (approximate) solvability of \eqref{e1.1}
follows from \cite[Theorem 4.1]{Mi3}.
Next, set $\Sigma_1=\{x\in X: I-A^{-1}N\mbox{ is not invertible at }x\}$.
Then $\{(I-A^{-1}N)^{-1}(\{h\})\}$ is compact for each $h\in X$ and the cardinal
number card$(I-A^{-1}N)^{-1}(\{h\})$ is constant, finite and positive on each connected
component of $X\setminus (I-A^{-1}N)(\Sigma_1)$ by Theorem \ref{thm3.1}.
Since $A$ is a homeomorphism and
$\Sigma = \Sigma_1$, we have that card$((I-KF)^{-1}(\{f\}))=$card$((I-A^{-1}N)^{-1}(\{A^{-1}(f)\})$
on each connected component $U_i$ of $X\setminus (I-A^{-1}N)(\Sigma)$. As before, we get that
card$(I-KF)^{-1}(\{f\})$ is constant, finite, and positive on
each connected component of the set $X\setminus (I-KF)(\Sigma)$. \end{proof}
An easy consequence of Theorem \ref{thm3.11} is the following result.
\begin{corollary} \label{coro3.14}
Let $K:X\to X$ be a continuous linear map,
$\lambda^{-1}\notin \sigma(K)$, $F: X\to X$ be nonlinear,
$T_p=pI-(I-\lambda K)^{-1}K(F-\lambda I):X\to X$ for $p\ge 1$, and
\begin{equation}
|F-\lambda I|=\limsup_{\|x\|\to \infty}\|Fx-\lambda x\|/\|x\|<
\|(I-\lambda K)^{-1}K\|^{-1}.\label{e3.6}
\end{equation}
a) If $T_1$ is $A$-proper with respect to $\Gamma$, then \eqref{e1.1}
is approximation solvable for each $f\in X$.
Moreover, if
$\Sigma=\{x\in X: I-KF \mbox{ is not invertible at }x\}$
and $T_1$ is continuous, then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in X$,
and the cardinal number card$(I-KF)^{-1}(\{f\})$ is constant, finite, and positive on
each connected component of the set $X\setminus (I-KF)(\Sigma)$.
\noindent b) If $T_p$ is $A$-proper with respect to $\Gamma$ for each $p>1$
and $T_1$ satisfies condition (++), then \eqref{e1.1} is
solvable for each $f\in X$.
\end{corollary}
\begin{corollary} \label{coro3.15}
Let $X$ be a uniformly convex space with a
scheme $\Gamma=\{X_n,P_n\}$, max$\|P_n\|=1$, $K:X\to X$
be a continuous linear map,
$\lambda^{-1}\notin \sigma(K)$ and $F: X\to X$ be nonlinear such that
$ (I-\lambda K)^{-1}K(F-\lambda I)
:X\to X$ is nonexpensive and \eqref{e3.6} hold.
Then \eqref{e1.1} is solvable for each $f\in X$.
\end{corollary}
Let us now look at some special cases.
\begin{corollary} \label{coro3.16}
Let $K:H\to H$ be a positive, compact and normal linear map,
$\lambda^{-1}\notin \sigma(K)$, $F: X\to X$ be a nonlinear map such that
\begin{equation}
(|F-\lambda I|+\lambda )\mu (K)<1. \label{e3.7}
\end{equation}
Then \eqref{e1.1} is approximation solvable for each $f\in H$.
Moreover, if
$\Sigma=\{x\in X: I-KF \mbox{ is not invertible at }x\}$,
then $(I-KF)^{-1}(\{f\})$ is compact for each $f\in X$,
and the cardinal number card$(I-KF)^{-1}(\{f\})$ is constant, finite, and positive on
each connected component of the set $X\setminus (I-KF)(\Sigma)$.
\end{corollary}
\begin{proof} Since $K$ is compact, it suffices to show that \eqref{e3.7} implies
\eqref{e3.6}. We know that \eqref{e3.7} is equivalent to
$$
|F-\lambda I|+\lambda <\inf_{\gamma \in \sigma (K),\gamma \ne 0}
\mathop{\rm Re}(\gamma ^{-1})
$$
and the spectrum is $\sigma (K_{\lambda})=\{\gamma /(1-\lambda \gamma): \gamma \in \sigma (K)\}.$
Hence
$$|F-\lambda I|< \inf_{\gamma \in \sigma (K),\gamma \ne 0}\mathop{\rm Re}(\gamma ^{-1})
-\lambda=\inf_{\gamma \in \sigma (K),\;\gamma \ne 0}\mathop{\rm Re}((1-\lambda \gamma)/\gamma). $$
Thus,
$$|F-\lambda I|^2<\inf_{\gamma \in \sigma (K),\;\gamma \ne 0}
\{[\mathop{\rm Re}(\gamma ^{-1})-\lambda]^2+(Im\lambda ^{-1})^2\}.$$
which is equivalent to $|F-\lambda I|\;\|K_{\lambda}\|<1$.
Hence, \eqref{e3.6} holds. \end{proof}
Let $\Sigma (K)$ be the set of characteristic values of $K$, i.e.,
$\Sigma (K)=\{\mu: 1/\mu \in \sigma (K)\}$.
\begin{theorem} \label{thm3.12}
Let $K:H\to H$ be a selfadjoint map, $\lambda \notin \Sigma (K)$, $F: H\to H$
be nonlinear and continuous and
$T_p=pI-(I-\lambda K)^{-1}K(F-\lambda I):H\to H$ for $p\ge 1$.
Suppose that for some $k$ with $k\delta 1$
and $T_1$ satisfies condition (++), then \eqref{e1.1} is
solvable for each $f\in X$.
\end{theorem}
\begin{proof} Equation \eqref{e1.1} is equivalent to
$x=(I-\lambda K)^{-1}K(F-\lambda)x +(I-\lambda K)^{-1}f$. Since
$(I-\lambda K)^{-1}K=-1/\lambda+1/\lambda (I-\lambda K)^{-1}$, we have
that, \cite{K},
$$
\|(I-\lambda K)^{-1}K\|=\sup_{\mu \in \sigma (K)}|-1/\lambda+1/\lambda
(1-\lambda \mu)^{-1}|=\sup_{\mu \in \Sigma (K)}|(\mu -\lambda )^{-1}|=d^{-1}.
$$
Then the conclusions follow from Corollary \ref{coro3.14}.
\end{proof}
\section{Hammerstein integral equations}
Let $Q\subset R^n$ be a bounded domain, $k(t,s):Q\times Q\to R$ be measurable
and $f(s,u):Q\times R\to R$ be a Caratheodory function. We consider the problem
of finding a solution $u\in L_2(Q)$ of the Hammerstein integral equation
\begin{equation}
u(t)=\int_Qk(t,s)f(s,u(s))ds+g(t)\label{e4.1}
\end{equation}
where $g$ is a measurable function. There is a vast literature on
the solvability of \eqref{e4.1} and we just mention the books by
Krasnoselskii \cite{Kr} and Vainberg \cite{V}. Define the linear map
$$Ku(t)=\int_Qk(t,s)u(s)\,ds $$
in $H=L_2(Q)$. Define $Fu=f(s,u(s))$ and note that \eqref{e4.1} can be
written in the form $u-KFu=g$.
\begin{theorem} \label{thm4.1}
Let $K:H\to H$ be compact and selfadjoint,
$\Sigma (K)=\{\lambda : \lambda ^{-1}\in \sigma (K)\}$ and assume that either
one of the following conditions holds
\begin{itemize}
\item[(i)] Let $\lambda \notin \Sigma (K)$ and $a<\mathop{\rm dist}(\lambda ,
\Sigma (K))$ be such that for some $h\in L_2(Q)$,
$$|f(s,u)-\lambda u|\le a|u|+h(s)\quad \hbox{for all }s\in Q,\;u\in R,
$$
\item[(ii)] There are $\lambda , \mu \in \Sigma (K)$ such that
$(\lambda ,\mu )\cap \Sigma (K)=\emptyset$ and $\lambda <\alpha <\beta <\mu $ and
$\epsilon >0$ such that for $s\in Q$
$$\alpha +\epsilon \le f_-(s)=\liminf_{|u|\to \infty}(f(s,u)/u)\le f_+(s)
=\limsup_{|u|\to \infty}(f(s,u)/u)\le \beta -\epsilon .
$$
\item[(iii)] Let $K$ be positive and compact normal map in $H$ with
\begin{gather*}
|f(s,u)|\le c|u|+c(s),\quad s\in Q, u\in R, c(s)\in L_2(Q),\\
uf(s,u)\le ku^2+b(s),\quad s\in Q, u\in R, b(s)\in L_1(Q),\\
\|(I-\gamma K)^{-1}K\|(c+k)/2<1\,.
\end{gather*}
\end{itemize}
Then \eqref{e4.1} is approximation solvable in $L_2$ for each $g\in L_2$ and the
number of its solutions is constant and finite on each connected component
of $L_2(Q)\setminus (I-KF)(\Sigma)$, where
$\Sigma =\{ u\in L_2(Q):I-KF\hbox{ is not invertible at }u\}$.
\end{theorem}
\begin{proof} We shall show first that (ii) implies (i). From (ii),
we get that there is $R>0$ such that
$$ \alpha 0$ and $c>k$, we see that (i) holds with
$\lambda =(k-c)/2$ and $a=(c+k)/2$. Hence, the conclusion holds by
Theorem \ref{thm3.12} and Corollary \ref{coro3.14}.
\end{proof}
Let us now look at the case when $K$ is not selfadjoint nor compact. Suppose
first that $K$ is $P$-positive in $H=L_2(Q)$. Suppose that $K$ acts from $L_q$
into $L_p$ for $2\le p\le \infty$ and $q=p/(p-1)$ with $q=1$ if $p=\infty$.
As before, let $A=1/2(K+K^*)$ be the selfadjoint part of $K$ and $B=1/2(K-K^*)$
be the skew-adjoint part of $K$. They both act from $L_q$ into $L_p$. Assume
that $A$ is positive definite. Then it can be represented in the form $A=CC^*$,
where $C=A^{1/2}:L_2\to L_p$ and the adjoint operator $C^*:L_q\to L_2$. Assume
that $K$ is $P$-positive operator in $L_2$. Denote by $M$ and $N$ the closure of the maps
$C^{-1}K(C^*)^{-1}$ and $K(C^*)^{-1}$, respectively, in $L_2$. Note that $M$ and $N$ are defined
on the closure (in $L_2$) of the range of $C=A^{1/2}$. This closure
coincides with $L_2$ in our case. Since $K$ is $P$-positive,
the following decompositions hold (cf. \cite{APZ})
$$K=CMC^*,\;K=NC^*.$$
Note that $K$, $M$ and $N$ are related as: $N=CM, N^*=M^*C^*$ and we
have $(Mh,h)=\|h\|^2$ for all $h\in L_2$. Hence, both $M$ and $M^*$ have trivial
nullspaces.
Denote by $\mu (K)=\|N\|^2$, which is the positivity constant of $K$
in the sense of Krasnoselski. Set $Fx(s)=f(s,x(s))$.
\begin{theorem} \label{thm4.2}
Suppose that $K$ is $P$-positive in $L_2(Q)$, $f=f_1+f_2$ satisfies
the Caratheodory condition, $F:L_p\to L_q$, and there are constants $a$, $b$
$c$ and $k$ such that $a\|K\|<1$, $k<1-c\mu (K)$ and
\begin{itemize}
\item[(i)] $|f(s,u)|\le a|u|+b$ ($s\in Q,\;u\in \mathbb{R}$)
\item[(ii)] $(f_1(s,u)-f_1(s,v),u-v)\le c|u-v|^2$ ($s\in Q,\;u,v\in \mathbb{R}$)
\item[(iii)] $|f_2(s,u)-f_2(s,v)|\le k|u-v|\;\;(s\in Q,\;u,v\in \mathbb{R})$.
\end{itemize}
Then \eqref{e4.1} is approximation solvable in $L_2$ for each $g\in N(L_2)$ and the
number of its solutions is constant and finite on each connected component
of $L_2(Q)\setminus (M^*-N^*FN)(\Sigma_{L_2})$ intersected by $N(L_2)$,
where \\
$\Sigma_{L_2} =\{ u\in L_2: M^*-N^*FN\hbox{ is not invertible at }u\}$.
\end{theorem}
The above theorem follows from Corollary \ref{coro3.3}.
\begin{theorem} \label{thm4.3}
Suppose that $K$ is $P$-positive in $L_2(Q)$, $f=f_1+f_2$ satisfies
the Caratheodory condition, $F:L_p\to L_q$, and there are constants $a$, $b$, $d$,
$\gamma \in (0,2]$,
$c$ and $k$ such that $a\mu (K)<1$, $k<1-c\mu(K)$ and
\begin{itemize}
\item[(i)] $(f(s,u),u)\le a|u|^2+b|u|^{2-\gamma}+d$ $(s\in Q,\;u\in \mathbb{R})$
\item[(ii)] $(f_1(s,u)-f_1(s,v),u-v)\le c|u-v|^2$ $(s\in Q,\;u,v\in \mathbb{R})$
\item[(iii)] $|f_2(s,u)-f_2(s,v)|\le k|u-v|$ $(s\in Q,\;u,v\in \mathbb{R})$.
\end{itemize}
Then \eqref{e4.1} is approximation solvable in $L_2$ for each $g\in N(L_2)$ and the
number of its solutions is constant and finite on each connected component
of $L_2(Q)\setminus (M^*-N^*FN)(\Sigma_{L_2})$ intersected by $N(L_2)$,
where \\
$\Sigma_{L_2} =\{ u\in L_2: M^*-N^*FN\hbox{ is not invertible at }u\}$.
\end{theorem}
The above theorem follows from Corollary \ref{coro3.8}.
Next, we shall look at the case when the selfadjoint part $A$ of $K$ is not positive
definite. Suppose that $A$ is quasi-positive definite in $L_2$, i.e., it has at most a finite
number of negative eigenvalues of finite multiplicity.
Let $U$ be the subspace spanned by the eigenvectors of $A$ corresponding to these negative eigenvalues
of $A$ and $P:L_2\to U$ be the orthogonal projection onto $U$. Then $P$ commutes with $A$,
but not necessarily with $B$, and acts both in $L_p$ and $L_q$.
The operator $|A|=(I-2P)A$ is easily seen to be positive definite. Hence,
we have the decomposition $|A|=DD^*$, where
$D=|A|^{1/2}:L_2\to L_p$ and $D^*:L_q\to L_p$.
As before, the map $K$ $P$-quasi-positive if the map $D^{-1}K(D^*)^{-1}$ exists
and is bounded in $L_2$, and $S$-quasi-positive if the map $K(D^*)^{-1}$ exists
and is bounded in $H$. Let $M$ and $N$ denote the closure in $L_2$ of the the
bounded maps $D^{-1}K(D^*)^{-1}$ and $K(D^*)^{-1}$ respectively.
They are both defined on the whole space $L_2$ (cf. \cite{APZ})
and have the following decompositions
$$K=DMD^*,\quad K=ND^*.$$
Then we have $N=DM$, $N^*=M^*D^*$, and $\langle Mh,h\rangle =\|h\|^2-2\|Ph\|^2$
for all $h\in H$.
Define the number
$$\nu (K)=\sup\{\nu: \nu >0,\;\|Nh\|\ge (\nu)^{1/2}\|Ph\|,\;h\in H\}.$$
Note that for a selfadjoint map $K$, $\nu (K)$ is the absolute value of the largest
negative eigenvalue of $K$.
We have the following result when $K$ is P-quasi-positive.
\begin{theorem} \label{thm4.4}
Suppose that $K$ is P-quasi-positive in $L_2(Q)$, $f=f_1+f_2$ satisfies
the Caratheodory condition, $F:L_p\to L_q$, and there are constants
$a$, $b$, $d$, $\gamma\in (0,2]$, $c$ and $k$ such that
$a+c\nu (K)<-1$, $k<-(1+c\nu (K))$ and
\begin{itemize}
\item[(i)] $|f(s,u)|\le a|u|+b$ ($s\in Q,\;u\in \mathbb{R}$)
\item[(ii)] $(f_1(s,u)-f_1(s,v),u-v)\le c|u-v|^2$ ($s\in Q,\;u,v\in \mathbb{R}$)
\item[(iii)] $|f_2(s,u)-f_2(s,v)|\le k|u-v|$ ($s\in Q,\;u,v\in \mathbb{R}$).
\end{itemize}
Then \eqref{e4.1} is approximation solvable in $L_2$ for each $g\in N(L_2)$ and the
number of its solutions is constant and finite on each connected component
of $L_2(Q)\setminus (M^*-N^*FN)(\Sigma_{L_2})$ intersected by $N(L_2)$,
where \\
$\Sigma_{L_2} =\{ u\in L_2: M^*-N^*FN \hbox{ is not invertible at }u\}$.
\end{theorem}
The above theorem follows from Corollary \ref{coro3.4}.
\begin{theorem} \label{thm4.5}
Suppose that $K$ is P-quasi-positive in $L_2(Q)$, $f=f_1+f_2$ satisfies
the Caratheodory condition, $F:L_p\to L_q$, and there are constants
$a$, $b$, $d$, $\gamma\in (0,2]$,
$c$ and $k$ such that $a+c\nu (K)<-1$, $k<-(1+c\nu (K))$ and
\begin{itemize}
\item[(i)] $(f(s,u),u)\le a|u|^2+b|u|^{2-\gamma}+d$ ($s\in Q,\;u\in \mathbb{R}$)
\item[(ii)] $(f_1(s,u)-f_1(s,v),u-v)\le c|u-v|^2$ ($s\in Q,\;u,v\in \mathbb{R}$)
\item[(iii)] $|f_2(s,u)-f_2(s,v)|\le k|u-v|$ ($s\in Q,\;u,v\in \mathbb{R}$).
\end{itemize}
Then \eqref{e4.1} is approximation solvable in $L_2$ for each $g\in N(L_2)$ and the
number of its solutions is constant and finite on each connected component
of $L_2(Q)\setminus (M^*-N^*FN)(\Sigma_{L_2})$ intersected by $N(L_2)$,
where \\
$\Sigma_{L_2} =\{ u\in L_2: M^*-N^*FN\hbox{ is not invertible at }u\}$.
\end{theorem}
The above theorem follows from Corollary \ref{coro3.4}.
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