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\markboth{\hfil solvability of Hammerstein equations \hfil EJDE--2002/103}
{EJDE--2002/103\hfil Petronije S. Milojevi\'c \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 103?, pp. 1--15. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\vspace{\bigskipamount} \\
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On the unique constructive solvability of Hammerstein
equations
%
\thanks{ {\em Mathematics Subject Classifications:} 47H14, 35L70, 35L75, 35J40.
\hfil\break\indent
{\em Key words:} Unique approximation, solvability, Hammerstein equations,
A-proper maps, \hfil\break\indent
surjectivity, error estimates.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted July 13, 2002. Published December 10, 2002.} }
\date{}
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\author{Petronije S. Milojevi\'c}
\maketitle
\begin{abstract}
We study the unique constructive solvability of Hammerstein
operator equation in Banach spaces using iterative and projection
like methods. Some error estimates are also given. The linear part
is assumed to be either selfadjoint or non-selfadjoint.
Applications to Hammerstein integral equations are given.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\numberwithin{equation}{section}
\section{Introduction}
In this paper, we shall study the
unique constructive solvability of operator equation
$$ x-KFx=f\eqno (1.1)
$$
where $K$ is linear and $F$ is a nonlinear map. We first study
(1.1) in the operator form using an iterative process, the
$A$-proper mapping approach and the Brouwer degree theory. We
shall consider (1.1) in a general setting between two Banach
spaces. To that end, we shall use two approaches. One is based on
applying the Brouwer degree theory directly to the finite
dimensional approximations of the map $I-KF$, and the other one is
based on splitting first the map $K$ as a product of two suitable
maps and then use the Brouwer degree. Then we apply the obtained
results to Hammerstein integral equations. This work is a
continuation of our study of these equations in \cite{Mi-4}. There is
an extensive literature on Hammerstein equations and we refer to
\cite{Kr,K-Z,V}.
\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 $\mathop{\rm dist}(x,X_n)\to 0$ as $n\to\infty$ for each $x\in
X$. Let $P_n:X\to Y_n$ and $Q_n:Y\to Y_n$ be linear projections
onto $X_n$ and $Y_n$ respectively such that $P_nx\to x$ for each
$x\in X$ and $\delta=\max \|Q_n\|<\infty$. Then
$\Gamma=\{X_n,P_n;Y_n,Q_n\}$ is a projection scheme for $(X,Y)$.
\paragraph{Definition} % 2.1
A map $T:D \subset X \to Y$ is said to be {\em approximation-proper}
($A$-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 $f$ is given in
advance, 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{Mi-2,Mi-3,
Mi-5,Mi-7,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 non-compact\-ness} 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 non-compactness} of $D$ is defined as
$ \chi (D)= \inf \{ r>0|D \subset \cup _{ i=1} ^{ n}B(x _{ i},r),
\; x_i \in X, \; n \in N \}$.
Let $ \phi $ denote either the set or the ball-measure of non-compactness.
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
\cite{Mi-3,Mi-4,Mi-5,Mi-7}.
The following result states that ball-condensing perturbations of
stable $A$-proper maps are also $A$-proper.
\begin{theorem}(\cite{Mi-1}) \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 $n_0$
$$
\|Q_nTx-Q_nTy\|\ge c\|x-y\|\quad\mbox{for } x,y\in X_n , \; n\ge n_0
$$
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 $k\delta 0$ such that
$\|y\|\le m\|x\|$ for each $y\in Ax$, $x\in X$. It is
$c$-coercive if $\|y\|\ge c\|x\|$ for each $y\in Ax$, $x\in X$.
The following result from \cite{Mi-2} will be needed below.
\begin{theorem} \label{thm2.2}
Let $T:\overline U\subset X\to Y$ be A-proper with respect to $\Gamma$
and $x_0$ be a solution of $Tx=f$. Suppose that $A$ is
an odd multivalued derivative of $T$ at $x_0$ and there exist
constants $c_0>0$ and $n_0\ge 1$ such that
$$
\|Q_nu\|\ge c_0\|x\|\quad\mbox{for }x\in X_n,\quad u\in Ax,
\;\;n\ge n_0.\eqno(2.1)
$$
(a) If $x_0$ is an isolated solution, then the equation $Tx=f$ is strongly
approximation solvable in $B_r(x_0)$ for some $r>0.$
\noindent (b) If, in addition, $A$ is $c_1$-coercive for some $c_1>0$,
then $x_0$ is an isolated solution, the conclusion of (a) holds
and, for $\epsilon\in (0,c_0)$, approximate solutions
$x_n$ satisfy
$$
\|x_n-x_0\|\le (c_0-\epsilon)^{-1}\|Tx_n-f\| \quad
\mbox{for } n\ge n_1\ge n_0.\eqno(2.2)
$$
(c) If $x_0$ is an isolated solution in $B_r(x_0)$,
$A$ is $c_2$-bounded for some $c_2$ and
$$
Tx-Ty\in A(x-y)+R(x-y)\quad \mbox{whenever } x-y\in B_r \eqno(2.3)
$$
and $z/\|x-y\|\to 0$ as $x\to x_0$ and $y\to x_0$ for each
$z\in R(x-y)$, then the equation $Tx=f$ is uniquely approximation solvable
in $B_r(x_0)$ and the unique solutions
$x_n\in B_r(x_0)\cap X_n$ of $Q_nTx=Q_nf$ satisfy
$$
\|x_n-x_0\|\le k\|P_nx_0-x_0\|\le c\mathop{\rm dist}(x_0,X_n),\eqno(2.4)
$$
where $k$ depends on $c_0$, $c_2$, $\epsilon$ and
$\delta$ and $c=2k\delta_1$, $\delta_1=\sup\|P_n\|$.
\end{theorem}
\section{Unique constructive solvability of Hammerstein operator equations}
In this section, we shall prove a number of constructive solvability results for Eq. (1.1)
using an iterative process and finite dimensional approximations and give the error
estimates in the second case.
We shall consider (1.1) in a general setting between two
Banach spaces. To that end, we shall use two approaches. One is based on
using the contraction mapping principal or
applying the Brouwer degree theory directly to the finite
dimensional approximations of the map $I-KF$,
and the other one is based on splitting first the map $K$ as
a product of two suitable maps and then use the Brouwer degree.
We study (1.1) with A selfadjoint as well as non selfadjoint. Our
first result is an extension of a theorem of Dolph \cite{Do}.
\begin{theorem} \label{thm3.1}
Let $K:X\to X$ be a continuous linear map,
$\lambda^{-1}\notin \sigma(K)$, $d=\|(I-\lambda K)^{-1}K\|^{-1}$ and
$F: X\to X$ be nonlinear and continuous.
\noindent a) If for some $k\in (0,d)$,
$$ \|Fx-\lambda x-(Fy-\lambda y)\|\le k\|x-y\|\quad\mbox{for all}
\;x,y\in H.\eqno (3.1)
$$
then (1.1) is uniquely solvable for each $f\in X$ and the solution
is the limit of the iteration process
$$
x_n -\lambda Kx_n=KFx_{n-1}-\lambda Kx_{n-1}+f.\eqno (3.2)
$$
b) If, in addition, either $K$ is compact or $\delta =\max\|P_n\|=1$ and
$k\|(I-\lambda K)^{-1}\|\;\|K\|<1$ and $P_nKx=Kx$ for $x\in X_n$, then
(1.1) is approximation solvable with respect to $\Gamma $ for each
$f\in X$ and the approximate solutions
$\{x_n\in X_n\}$ of $x-P_nKFx=P_nf$ satisfy
$$
\|x_n-x\|\le c\|x_n-KFx_n-f\|\quad\mbox{for some } c
\mbox{ and all large } n.\eqno (3.3)
$$
and
$$
\|x_n-x\|\le c\|P_nx-x\|\le c_1\mathop{\rm dist}(x,X_n).\eqno (3.4)
$$
c) If condition (3.1) holds with $k=d$, $X$ is a uniformly
convex space with $\delta =1$ and
$$
\|Fx-\lambda x\|\le a\|x\|+b\quad\mbox{for some } a0, x\in X.
\eqno (3.5)
$$
then (1.1) is solvable for each $f\in X$.
\end{theorem}
\paragraph{Proof.} Equation (1.1) is equivalent to $Ax-Nx=f$ with
$A=I-\lambda K$ and $N=K(F-\lambda I)$. Hence, it is easy to show
that $A^{-1}N$ is $k_1=k\|A^{-1}K\|$-contractive with $k_1<1$.
Thus, part a) follows from
the contractive fixed point principle and c) follows from
Theorem 3.3 in \cite{Mi-4}.
Regarding part b), we need only to show that condition (2.1)
of Theorem 2.3 holds. Assume first that $K$ is compact. Then $I-KF$
is $A$-proper with respect to $\Gamma$.
Set $B_1x=\{K(y-\lambda x): \|y-\lambda x\|\le k\|x\|\}$ and
$Bx=Ax-B_1x$ for $x\in X$. Then $B$ is
homogeneous with $Bx$ convex for each $x\in X$
and $A(x-y)-(Nx-Ny)\in B(x-y)$ for each $x,y\in X$. Moreover, if
$0\in Bx$, then $Ax=K(y-\lambda x)$ for some $y$ and
$$
\|x\|\le \|A^{-1}K\|\,\|y-\lambda x\|<\|x\|.
$$
Hence, $x=0$. Since $B_1$ is upper semicontinuous and compact,
$B=A-B_1$ is a multivalued $A$-proper map with respect to $\Gamma$
(cf. \cite{Mi-7}). This implies that
condition (2.1) holds. Since also $Nx-Ny\in B_1(x-y)$, the conclusions
follow from Theorem 2.3.
Next, let $\delta =1$ and $l=k\|(I-\lambda K)^{-1}\|\;\|K\|<1$. Then $I-KF$ is
$A$-proper with respect to $\Gamma=\{X_n,P_n\}$. Indeed, let $\{x_n\in X_n\}$
be bounded and $x_n-P_nKFx_n\to f$. Set $y_n=(I-\lambda K)x_n$. Then
$y_n-P_nK(F-\lambda I)(I-\lambda K)^{-1}y_n\to f$ and the map
$F_1=K(F-\lambda I)(I-\lambda K)^{-1}$ is an $l$-contraction with $l<1$.
Hence, $I-F_1$ is $A$-proper with respect to $\Gamma$ and therefore, a
subsequence $y_{n_k}\to y$ and $y-F_1y=f$. Hence, $x-KFx=f$ with
$x=(I-\lambda K)^{-1}y$, proving that $I-KF$ is $A$-proper.
Now, let $y\in P_n(Ax-B_1x)$
for some $x\in X_n$. Then $y=P_n(Ax-Kv)=Ax-P_nKv$ for some $v$ with
$\|v\|\le k\|x\|$ and $x=A^{-1}P_n(y+Kv)$. Hence,
$$
\|x\|\le \delta \|A^{-1}\|\big(\|y\|+k\|K\|\,\|x\|\big)
$$
and
$$ \big(1-k\|A^{-1}\|\,\|K\|\big)\|x\|\le \|A^{-1}\|\,\|y\|.
$$
Hence, (2.1) holds and similarly we show that $B$ is c-coercive.
Thus, Theorem 2.3 applies. \hfill$\square$
Let $\Sigma (K)$ be the set of characteristic values of $K$, i.e.,
$\Sigma (K)=\{\mu\;|\; 1/\mu \in \sigma (K)\}$,
and $\mu^* =inf \{\mu\;|\;\mu \in \Sigma (K)\cap (0,\infty) \}$.
For $c\in \Sigma(K)\cap (-\infty,\mu^*]$,
define $d^-_c=\mathop{\rm dist}(c,\Sigma(K)\cap (-\infty,c))$.
We have (cf. also \cite{Mi-4})
\begin{theorem} \label{thm3.2}
Let $K:H\to H$ be a selfadjoint map, $F: H\to H$ be nonlinear and continuous.
Assume that\\
(i) $(Fx-Fy,x-y)\ge \alpha \|x-y\|^2$ for all $x,y\in H$,
\\
(ii) $\|Fx-Fy\|\le \beta \|x-y\|$ for all $x,y\in H$.
\noindent
(a) If (i)-(ii) hold and $\beta ^2<\alpha d^-_c+c(d^-_c-c-2\alpha)$
for some $c\le \mu^*$, then (1.1) is uniquely solvable for each
$f\in X$ and the solution is the limit of the iteration process (3.2).
Moreover, if also $P_nKx=Kx$ on $X_n$ and either $K$ is compact or
$\beta^2+\lambda ^2$ is sufficiently small, then (1.1) is uniquely
approximation solvable
for each $f\in H$ and (3.3)-(3.4) hold.
\noindent
(b) If $\beta^2\le \alpha d^-_c+c(d^-_c-c-2\alpha)$ and, for some
$a<\lambda=c-d^-_c/2$ and $b>0$,
$$
\|Fx-\lambda x\|\le a\|x\|+b\quad\mbox{for all }x\in H
$$
then (1.1) is solvable for each $f\in H$.
\end{theorem}
\paragraph{Proof.}
Let $\lambda =c-d^-_c/2$. Then $\lambda \notin \Sigma (K)$
and $d=\mathop{\rm dist}(\lambda, \Sigma (K))>0$.
Since $(I-\lambda K)^{-1}K=-1/\lambda+1/\lambda (I-\lambda K)^{-1}$,
we have that (\cite{Ka})
$\|(I-\lambda K)^{-1}K\|=\sup_{\mu \in \sigma (K)}|-1/\lambda+1/\lambda
(1-\lambda \mu)^{-1}|=\sup_{\mu \in \Sigma (K)}|(\mu -\lambda )^{-1}|=
d^{-1}$.
Using conditions (i)-(ii), we get
$$
\|Fx+\lambda x-(Fy+\lambda y)\|\le (\beta ^2+\lambda ^2
+2\alpha \lambda)^{1/2}\|x-y\|.
$$
By our choice of $\lambda$ and the condition on $\beta$, we get
$$
\beta ^2+\lambda ^2+2\alpha \lambda=\beta^2+\alpha d^-_c+c(d^-_c-c-2\alpha)+
(d^-_c/2)^2<(d^-_c/2)^2=d^{2}.
$$
Hence, the conclusions follow from Theorem 3.1. \hfill$\square$
\begin{theorem} \label{thm3.3}
Let $K:H\to H$ be selfadjoint, $F:H\to H$ be a gradient map and
$B^{\pm}:H\to H$ be selfadjoint maps such that \\
(i)$(B^-(x-y),x-y)\le (Fx-Fy,x-y)\le (B^+(x-y),x-y)$ for all
$x,y\in H$.\\
(ii) $\delta \|B^{\pm}-\lambda I\|\le d
=\min\{|\mu| : \mu \in \sigma(I-\lambda K)^{-1}K\}$
for some $\lambda $.
\noindent
(a) If the inequality is strict in (ii), then (1.1)
is uniquely solvable for each $f\in X$ and the solution is the limit of the
iteration process (3.2). Moreover, if also $P_nKx=Kx$ on $X_n$ and
$\|B^{\pm}-\lambda I\|$ is sufficiently small , then (1.1) is uniquely
approximation solvable with respect to $\Gamma $
for $H$ for each $f\in H$ and the approximate solutions satisfy (3.3)-(3.4).
\noindent
(b) If, in addition, there are $0\mu ^*$, and $b>0$,
$$ \|Fx-\lambda x\|\le a\|x\|+b\;\;\mbox{for all}\;x\in H$$
then (1.1) is solvable for each $f\in H$.
\end{theorem}
\paragraph{Proof.} As above, we have that
$$
\|Fx+\lambda x-Fy-\lambda y\|\le \max(|\alpha+\lambda|,|\beta+\lambda|)
\|x-y\|.
$$
By our choice of $\lambda $ as given in b), we conclude that
$|\alpha+\lambda|\le d=\mathop{\rm dist}(\lambda,\Sigma(K))=d^{\pm}_c/2$ and
$|\beta+\lambda|\le d$ with the inequalities being strict in part a).
Hence, Theorem 3.1 is applicable. \hfill$\square$ \smallskip
\paragraph{Remark} The unique solvability results of the type in
Theorems 3.2-3.4 for semilinear equations have been
obtained in \cite{BM,Mi-2,Mi-6}.
\vskip 2mm
Using a suitable splitting of K, we can still prove the unique
approximation solvability of Eq. (1.1) without assuming condition (ii)
in Theorem 3.2. Let us first describe the setting for this approach
(cf. \cite{V}).
Recall that a map $K$ acting in a Hilbert space $H$ is called positive in
the sense of Krasnoselskii 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 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{Kr}) the number
$$
[\inf_{\lambda \in \sigma (K),\lambda\ne 0}Re(\lambda ^{-1})]^{-1}
$$
is well defined and positive. It that case, it is equal to $\mu (K)$.
Let $X$ be a reflexive embeddable Banach space
( $X\subset H\subset X^*$), $K:X^*\to X$ be a positive definite bounded
selfadjoint map in $H$, and $C=K_H^{1/2}:H\to X$, where $K_H$ is the restriction
of $K$ to $H$. We know that the positive square root can be extended to a
bounded linear map $T:X^*\to H$ such that
$K=T^*T$, where the adjoint of $T$ is $T^*=K^{1/2}_H=C:H\to X$
and $C^*=T$ (cf. \cite{V}). Hence, we
can write $K=CT$. Set $\mu (K)=\|C\|^2$. We have the following
extension of Theorem 3.2 as well as of a result of Vainberg \cite{V}.
\begin{theorem} \label{thm3.5}
Let $X$ be a reflexive embeddable Banach space
( $X\subset H\subset X^*$), $K:X^*\to X$ be a positive definite bounded
selfadjoint map in $H$, and $C=K_H^{1/2}:H\to X$, where $K_H$ is the restriction
of $K$ to $H$, and $T:X^*\to H$ be a bounded linear extension of $K_H^{1/2}$.
Suppose that $F:X\to X^*$and $c$ is
the smallest number such that
$$
(Fx-Fy,x-y)\le c\|x-y\|^2\quad\mbox{for all }x,y\in X
$$
a) If $c\mu(K)<1$, then (1.1) is uniquely approximation solvable
in $X$ for each $f\in C(H)\subset X$.
\\
b) If $c\mu (K)=1$ and $I-TFC$ satisfies condition (+) in $H$,
then Eq. (1.1) is solvable in X for each $f\in
C(H)\subset X$.
\end{theorem}
\paragraph{Proof.} a) We have that $K=T^*T$, where the adjoint of $T$ is
$T^*=K^{1/2}_H=C:H\to X$ and $C^*=T$. Hence, we can write $K=CT$.
Let $f\in C(H)$ so that $f=Ch$ for some $h\in H$.
We note that (1.1) is equivalent to the following equation (cf. \cite{V})
$$ y-TFCy=h.\;\;\eqno (3.6)
$$
The map $I-TFC:H\to H$ is $1-c\mu (K)$-strongly
monotone. Indeed, for each $x,y\in H$ we have
\begin{align*}
(x-TFCx-y+TFCy,x-y)
&= \|x-y)\|^2- (TFCx-TFCy,x-y)\\
&=\|x-y\|^2- (FCx-FCy,Cx-Cy)\\
&\ge \|x-y\|^2- c\|Cx-Cy\|^2\\
&\ge (1-c\mu (K))\|x-y\|^2.
\end{align*}
Hence, $I-TFC$ is $A$-proper with respect to $\Gamma=\{H_n,P_n\}$ for $H$ and coercive. It follows that Eq. (3.6)
is uniquely approximation solvable for each $h\in H$. Therefore, (1.1) is
uniquely approximation solvable for each $f\in C(H)$.
\noindent
b) If $c\mu(K)=1$, then as in part a) we get that $I-TFC$ is monotone and satisfies
condition (+). Hence, (3.6) is solvable for each $h\in H$ and therefore
(1.1) is solvable for each $f\in C(H)$.
\hfill$\square$
\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 definite bounded
selfadjoint map in $H$, and $C=K_H^{1/2}$, where $K_H$ is the restriction
of $K$ to $H$, and $T:X^*\to H$ be a bounded linear extension of $K_H^{1/2}$.
Let $F=F_1+F_2:X\to X^*$ and $c$ be
the smallest number such that
$$(F_1x-F_1y,x-y)\le c\|x-y\|^2\;\;\mbox{for all }\;\;x,y\in X$$
and $F_2$ is a Lipshitz map with constant k such that $1-(c+k)\mu(K)>0$.
Then (1.1) is uniquely approximation solvable
in $X$ for each $f\in C(H)\subset X$.
\end{corollary}
\paragraph{Proof.}
It suffices to show that $I-TFC:H\to H$ is $A$-proper.
For $x,y\in H$, we have
\begin{align*}
&(x-TFCx-y+TFCy,x-y)\\
&= \|x-y)\|^2- (TFCx-TFCy,x-y)\\
&=\|x-y\|^2- (F_1Cx-F_1Cy,Cx-Cy)-(F_2Cx-F_2Cx,Cx-Cy)\\
&\ge \|x-y\|^2- (c+k)\|Cx-Cy\|^2\\
&\ge(1-(c+k)\mu(K))\|x-y\|^2.
\end{align*}
Hence, $I-TFC$ is strongly monotone and therefore $A$-proper with respect
to $\Gamma=\{H_n, P_n\}$. \hfill$\square$ \smallskip
Next, let us look at case when $K$ is not selfadjoint. Our study of
Eq. (1.1) in this case is motivated by the work of
Appel-Pascale-Zabrejko \cite{APZ} for nonlinear Hammerstein integral
equations. Following their arguments about the existence of solutions,
we present an abstract version of their results and also prove
the constructive solvability of Eq. (1.1).
We begin by
describing the setting of the problem. Let $X$ be an embeddable Banach space, that
is, there is a Hilbert space $H$ such that $X\subset H\subset X^*$ so that
$\langle y,x\rangle =(y,x)$ for each $y\in H, x\in X$, where $\langle,\rangle$
is the duality pairing of $X$ and $X^*$.
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 $A$ and $C:H\to X$, and the adjoint map $C^*:X^*\to H$.
We say that $K$ is $P$-positive definite 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, $P$-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.
$$|(Kx,y)-(y,Kx)|\le a (Kx,x)^{1/2}(Ky,y)^{1/2}\;\;x,y\in H.$$
Denote by $M$ and $N$ the closure of the maps $C^{-1}K(C^*)^{-1}$ and
$K(C^*)^{-1}$, respectively, in $H$. The mappings $M$ 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 the
following decompositions to hold
$$
K=CMC^*,\;K=NC^*.
$$
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 Krasnoselskii.
Let $F:X\to X^*$ be a nonlinear map and consider the Hammerstein equation
$$ x-KFx=f.\eqno (3.7)
$$
As in \cite{APZ}, we reduce the solvability of Eq. (3.7) to the solvability
of an equivalent equation.
For $f\in N(H)$, let $h\in H$ be a solution of
$$ M^*h-N^*FNh=M^*k \eqno(3.8)
$$
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 (3.7). So the solvability of
(3.7) is reduced to the solvability of (3.8). We have the
following extension of Theorem 3.5. Its unique solvability part is an
abstract extension of a result of Appel-Pascale-Zabrejko \cite{APZ}
for Hammerstein integral equations.
\begin{theorem} \label{thm3.7}
Let $K:X^*\to X$ be P-positive in H, $F:X\to X^*$ and $c$ be
the smallest number such that
$$ (Fx-Fy,x-y)\le c\|x-y\|^2\quad\mbox{for all }x,y\in X
$$
a) If $c\mu(K)<1$, then (3.7) is uniquely approximation solvable
in $X$ for each $f\in N(H)\subset X$.
\noindent
b) If $c\mu (K)=1$ and $M^*-N^*FN$ satisfies condition (+) in $H$,
then (3.7) is solvable in X for each $f\in
N(H)\subset X$.
\end{theorem}
\paragraph{Proof.} a) As in \cite{APZ}, we shall
prove that the map $M^*-N^*FN:H\to H$ is $1-c\mu (K)$-strongly
monotone. Indeed, for each $x,y\in H$ we have
\begin{align*}
&(M^*x-N^*FNx-M^*y+N^*FNy,x-y)\\
&=\|x-y\|^2- (FNx-FNy,Nx-Ny)\\
&\ge \|x-y\|^2- c\|Nx-Ny\|^2\\
&\ge (1-c\mu (K))\|x-y\|^2.
\end{align*}
Hence, $M^*-N^*FN$ is $A$-proper with respect to $\Gamma=\{H_n,P_n\}$ for $H$ and coercive.
Hence, (3.8) is uniquely approximation solvable for each $k\in H$ and
therefore so is (3.7) for each $f\in N(H)$.
\noindent
b) If $c\mu(K)=1$, then as in part a) we get that $M^*-N^*FN$ is monotone
and satisfies condition (+). Hence, (3.8) is solvable and therefore so
is (3.6) for each $f\in N(H)$. \hfill$\square$
\begin{corollary} \label{coro3.8}
Let $K:X^*\to X$ be P-positive in $H$,
$F=F_1+F_2:X\to X^*$ and $c$ be
the smallest number such that
$$
(F_1x-F_1y,x-y)\le c\|x-y\|^2\quad\mbox{for all }x,y\in X
$$
and $F_2$ is a Lipshitz map with constant $k$ such that $1-(c+k)\mu(K)>0$.
Then (3.7) is uniquely approximation solvable
in $X$ for each $f\in N(H)\subset X$.
\end{corollary}
\paragraph{Proof.}
It suffices to show that $M^*-N^*FN:H\to H$ is $A$-proper and coercive.
For $x,y\in H$, we have
\begin{align*}
&(M^*x-N^*FNx-M^*y+N^*FNy,x-y)\\
&\ge \|x-y\|^2- (c+k)\|Nx-Ny\|^2\\
&\ge (1-(c+k)\mu (K))\|x-y\|^2.
\end{align*}
Hence, $M^*-N^*FN$ is A-proper with respect to $\Gamma=\{H_n,P_n\}$
and coercive. \hfill$\square$ \smallskip
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 $X^*$. Then 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 $D^*:X^*\to H$.
As in \cite{APZ}, 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 bounded maps
$D^{-1}K(D^*)^{-1}$ and $K(D^*)^{-1}$ respectively. Assume that they are both defined on the
whole space $H$. Suppose that we have the following decompositions
$$K=DMD^*,\;\;\;K=ND^*.$$
Then we have $N=DM$, $N^*=M^*D^*$, and $=\|h\|^2-2\|Ph\|^2$ 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 extension of
Theorem \ref{thm3.7}. Its unique solvability part is an abstact extension
of a result in \cite{APZ} for Hammerstein integral equations.
\begin{theorem} \label{thm3.9}
Let $K:X^*\to X$ be P-quasi-positive in H, $F:X\to X^*$ and $c$ be
the smallest number such that
$$
(Fx-Fy,x-y)\le c\|x-y\|^2\quad\mbox{for all }x,y\in X
$$
a) If $c\nu(K)<-1$, then (1.1) is uniquely approximation solvable
in $X$ for each $f\in N(H)\subset X$.
\\
b) If $c\nu (K)=-1$ and $M^*-N^*FN$ satisfies condition (+) in $H$,
then (1.1) is solvable in X for each $f\in
N(H)\subset X$.
\end{theorem}
\paragraph{Proof.} a) As in \cite{APZ}, we have that
the map $M^*-N^*FN:H\to H$ is $-(1+c\nu (K))$-strongly
monotone. Indeed, for each $x,y\in H$ we have
\begin{align*}
&(M^*x-N^*FNx-M^*y+N^*FNy,x-y)\\
&=\|x-y\|^2-2\|P(x-y)\|^2-(FNx-FNy,Nx-Ny)\\
&\ge \|x-y\|^2-(2+c\nu (K))\|P(x-y)\|^2\ge -(1+c\nu (K))\|x-y\|^2.
\end{align*}
Hence, $M^*-N^*FN$ is $A$-proper with respect to $\Gamma=\{H_n,P_n\}$ for
$H$ and coercive.
Hence, (3.8) is uniquely approximation solvable for each $k\in H$ and
therefore so is (3.7) for each $f\in N(H)$.
\noindent
b) If $c\nu(K)=-1$, then as in part a) we get that $M^*-N^*FN$ is monotone
monotone and satisfies
condition (+). Hence, Eq. (3.8) is solvable and therefore so is (3.6)
for each $f\in N(H)$. \hfill$\square$
\begin{corollary} \label{coro3.10}
Let $K:X^*\to X$ be P-quasi-positive in $H$
and $F=F_1+F_2:X\to X^*$ and $c$ be the smallest number such that
$$
(F_1x-F_1y,x-y)\le c\|x-y\|^2\quad\mbox{for all }x,y\in X
$$
and $F_2$ is a Lipshitz map with constant $k$ such that $1-(c+k)\mu(K)>0$.
Then (1.1) is uniquely approximation solvable
in $X$ for each $f\in N(H)\subset X$.
\end{corollary}
\paragraph{Proof.}
It suffices to show that $M^*-N^*FN:H\to H$ is $A$-proper and coercive.
For $x,y\in H$, we have
\begin{align*}
&(M^*x-N^*FNx-M^*y+N^*FNy,x-y)\\
&=\|x-y\|^2- (F_1Nx-F_1Ny,Nx-Ny)-(F_2Nx-F_2Ny,Nx-Ny)\\
&\ge (1-(c+k)\mu (K))\|x-y\|^2.
\end{align*}
Hence, $M^*-N^*FN$ is A-proper with respect to $\Gamma=\{H_n,P_n\}$
and coercive. \hfill$\square$ \smallskip
\section{Hammerstein integral equations}
Let $Q\subset R^n$ be a bounded domain, $k(t,s):Q\times Q\to R$
be measurable and $f(s,u):Q\times R\to R$ is a Caratheodory
function. We consider the problem of a solution $u\in L_2(Q)$ of the
Hammerstein integral equation
$$
u(t)=\int _Qk(t,s)f(s,u(s))ds+g(t)\eqno (4.1)
$$
where $g$ is a measurable function. There is a vast literature on
the solvability of (4.1) 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 (4.1) can be
written in the form $u-KFu=g$.
\begin{theorem} \label{thm4.1}
Let $K:H\to H$ be continuous, $\lambda \notin \Sigma (K)$, and let
$d^{-1}=\mathop{\rm dist}(\lambda ,\Sigma (K))$. Let $\delta =\max\|P_n\|$
and for some $k\in (0,d/k)$,
$$
|f(s,u)-\lambda u-(f(s,v)-\lambda v)|\le k|u-v|\quad\mbox{for }
s\in Q,\;u,v\in R.
$$
Then (4.1) is uniquely solvable for each $g\in L_2$.
\end{theorem}
For the proof of this theorem, it is easy to see that the mappings $K$
and $F$ satisfy all conditions of Theorem 3.1. Hence, its conclusions
follow from it.
\begin{theorem} \label{thm4.2}
Let $K:H\to H$ be selfadjoint and for some $\alpha, \beta \in R$,
$$
\alpha |u-v|^2\le (f(s,u)-f(s,v))(u-v)\le \beta |u-v|^2\quad
\mbox{for } s\in Q,\;u,v\in R
$$
(i) If $-c<\alpha \le \beta <-c+d_c^+$ for some $c\in \Sigma (K)
\cap (-\infty ,\mu ^*)$ or $-c-d_c^+<\alpha \le \beta <-c$ for
some $c\in \Sigma (K)\cap (\mu ^*,\infty)$, then (5.1) is
uniquely solvable for each $g\in L_2$.
\vskip 2mm
\noindent
(ii) If $<$ is replaced by $\le $ in (i) and if, for some $a<\lambda $
with $\lambda =c-d_c^-/2$ if $c\le \mu ^*$ and $\lambda =c+d_c^+/2$
if $c>\mu^*$, and some $b\in L_2$, we assume
$$
|f(s,u)-\lambda u|\le a|u|+b(s)\quad\mbox{for }s\in Q,\;u\in R
$$
then (4.1) is solvable for each $g\in L_2$.
\end{theorem}
For the proof of this theorem, it is easy to see that the mappings $K$ and
$F$ satisfy all conditions of Theorem 3.4. Hence, its conclusions
follow from it.
Part (i) of this theorem extends a result of Dolph \cite{Do}. Assuming
compactness of $K$, we can relax the condition on $F$.
\begin{theorem} \label{thm4.3}
Let $K:H\to H$ be selfadjoint and compact and for some
$\beta >0$,
$$
(f(s,u)-f(s,v))(u-v)\le \beta |u-v|^2\quad \mbox{for }
s\in Q,\;u,v\in R
$$
If $\beta$ is sufficiently small, then (4.1) is
uniquely solvable for each $g\in L_2$.
\end{theorem}
To prove this theorem, it is easy to see that the mappings $K$ and $F$ satisfy
all conditions of Theorem 3.5. Hence, its conclusions follow from it.
\begin{thebibliography}{00} \frenchspacing
\bibitem{APZ}
J. A, Appell, E. de Pascale, P.P. Zabrejko, (2000) On the unique
solvability of Hammerstein integral equations with non-symmetric
kernels, in Progress in Nonlinear Differential Equations and Their
Applications, vol. 40, pp. 27-34.
\bibitem{BM}
A.K. Ben-Naoum and J. Mawhin, (1992) The periodic-Dirichlet problem
for some semilinear wave equations, J. Differential Equations,
96, 340-354.
\bibitem{Do}
C. Dolph, (1949) Nonlinear integral equations of Hammerstein type.
Trans. Amer. Math. Soc., 66, 289-307.
\bibitem{Ka}
T. Kato, (1966) Perturbation Theory for Linear Operators, Springer
Verlag, NY.
\bibitem{Kr}
M. A. Krasnosel'skii, (1964) Topological Methods in the Theory of Nonlinear
Integral Equations, MacMillan, N.Y.
\bibitem{K-Z}
M. A. Krasnoselskii and P. P. Zabreiko, (1984) Geometrical
methods of Nonlinear Analysis, Springer-Verlag.
\bibitem{Ma}
J. Mawhin, (1981) Semilinear equations of gradient type in
Hilbert spaces and applications to differential equations, in
Nonlinear Differential Equations: Invariance, Stability and Bifurcation,
Acad. Press, NY, 269-282.
\bibitem{Mi-1}
P.S. Milojevi\'c, (1977) A generalization of the Leray-Schauder theorem
and surjectivity results for multivalued $A$-proper and pseudo $A$-proper
mappings, Nonlinear Anal., TMA, 1(3), 263-276.
\bibitem{Mi-2}
P.S. Milojevi\'c, (1989) Solvability of semilinear operator equations and
applications to semilinear hyperbolic equations, in Nonlinear
Functoinal Analysis, Marcel Dekker (Ed. P.S.Milojevi\'c), vol.121,
95-178.
\bibitem{Mi-3}
P. S. Milojevi\'c, (1994) Approximation-solvability of nonlinear equations
and applications, in Fourier Analysis (Ed.'s W. Bray, P. S.Milojevi\'c,
C. V. Stanojevi\'c), Lecture Notes in Pure and Applied Mathematics,
vol. 157, pp. 311-373, Marcel Dekker Inc., N.Y.
\bibitem{Mi-4}
P. S. Milojevi\'c, (1995) Approximation solvability of Hammerstein
equations, Publication de l'Institute de Mathematique, 58, 71-84.
\bibitem{Mi-5}
P. S. Milojevi\'c, (1996) Approximation-Solvability of Semilinear
Equations and Applications, in Theory and Applications of Nonlinear
Operators of Accretive and Monotone Type (Ed. A.G. Kartsatos),
Lecture Notes in Pure and Applied Mathematics, vol.178, pp. 149-208,
Marcel Dekker Inc., NY.
\bibitem{Mi-6}
P.S. Milojevi\'c, (1997) Implicit Function Theorems, Approximate
Solvability of Nonlinear Equations, and Error Estimates, J. Math.
Anal. Appl., 211, 424-459.
\bibitem{Mi-7}
P.S. Milojevi\'c, (2002) Continuation Theory for A-proper
Mappings and their Uniform Limits and Nonlinear Perturbations
of Fredholm Mappings, SIMMA 4, 311-359.
\bibitem{P}
W. V. Petryshyn, (1975) On the approximation - solvability of
equations involving A-proper and pseudo A-proper mappings, Bull. Amer.
Math. Soc., 81, 223-312.
\bibitem{V}
M. M. Vainberg, (1973) Variational Method and Method of Monotone operators
in the theory of nonlinear equations, Holsted Press, NY.
\end{thebibliography}
\noindent\textsc{Petronije S. Milojevi\'c}\\
Department of Mathematical Sciences and CAMS, \\
New Jersey Institute of Technology, \\
Newark, NJ, 07102 USA\\
e-mail address: pemilo@m.njit.edu
\end{document}