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\beginpaper
\title Nonlinear Ergodic Theorems

\shorttitle Nonlinear ergodic theorems

\author H. Br\'ezis and F. E. Browder

\shortauthor Br\'ezis and Browder

\authaddr Department of Mathematics, University of Chicago, Chicago,
Illinois 60637

\commby Alexandra Bellow

\daterecd May 17, 1976

\keywords ergodic theory, nonlinear mappings, averaging processes

\AMSclass Primary 47A35, 47H10; Secondary 40G05

\abstract In two recent notes (\ref1, \ref2), {\nm J-B. Baillon} proved the first
ergodic theorems for nonlinear mappings in Hilbert space. We simplify the
argument here and obtain an extension of {\nm Baillon}'s theorems from the usual
Ces\`aro means of ergodic theory to general averaging processes $A↓n=
\sum↓{k=0}↑\infty a↓{n,k}T↑k$ ($0≤a↓{n,k}$,\xskip $\sum↓{k≥0}a↓{n,k}=1$).

\begintext
\thbegin Theorem 1. Let $H$ be a Hilbert space,\xskip $C$ a closed bounded convex
subset of $H$,\xskip $T$ a nonexpansive self map of $C$. Suppose that as
$n\rtarr\infty$,\xskip $a↓{n,k}\rtarr 0$ for each $k$, and 
$\gamma↓n=\sum↓{k=0}↑\infty (a↓{n,k+1}-a↓{n,k})↑+\rtarr 0$.
Then for each $x$ in $C$,\xskip $A↓n x = 
\sum↓{k=0}↑\infty a↓{n,k}T↑k x$ converges weakly to a fixed point of $T$.

The proof of Theorem 1 depends upon an extension of {\nm Opial}'s lemma \ref3.

\thbegin Lemma 1. Let $\{x↓k\}$ and $\{y↓k\}$ be two sequences in $H$,\xskip $F$ a
nonempty subset of $H$,\xskip $C↓m$ the convex closure of $\union↓{j≥m}\{x↓i\}$.
Suppose that
$$\leave blank 1.5 inches$$
Then $y↓k$ converges weakly to a point of $F$.

\proofbegin Proof of Lemma 1. Since $\{y↓k\}$ is bounded, it suffices to show
that if $f$ and $g$ in $F$ are weak limits of infinite subsequences of $\{y↓k\}$,
then $f=g$. For each $f$,
$$|x↓j-f|↑2 = |x↓j-g|↑2+|g-f|↑2+2(x↓j-g,g-f).$$
For a given $\epsilon>0$, there exists $m(\epsilon)$ such that for $j≥m(\epsilon)$,
$$\bigv p(g)-|x↓j-g|↑2\bigv <\epsilon;\qquad\bigv p(f)-|x↓j-f|↑2\bigv <\epsilon.$$
Let $K↓\epsilon$ be the convex set of all $u$ such that
$$\bigv 2(u-g,g-f)+p(g)-p(f)+|g-f|↑2\bigv ≤2\epsilon.$$
Since $K↓\epsilon$ contains $\union↓{j≥m(\epsilon)}\{x↓j\}$, it contains $C↓{m(
\epsilon)}$. There exists $k↓\epsilon$ such that for $k≥k↓\epsilon$ we can find
$u↓k$ in $C↓{m(\epsilon)}$ such that $|y↓k-u↓k|≤\epsilon$. For $k≥k↓\epsilon$, it
follows that
$$\bigv 2(y↓k-g,g-f)+p(g)-p(f)+|g-f|↑2\bigv ≤2\epsilon+2\epsilon|g-f|.$$
Consider an infinite subsequence $\{y↓{k↓s}\}$ for which $(y↓{k↓s}-g,g-f)\rtarr 0$.
In the limit
$$\bigv p(g)-p(f)+|g-f|↑2\bigv ≤2\epsilon+2\epsilon|g-f|.$$
Since $\epsilon>0$ is arbitrary, it follows that $p(g)+|g-f|↑2=p(f)$. By symmetry,
$p(f)+|g-f|↑2=p(g)$. Hence, $|f-g|↑2=0$,\xskip $f=g$.\QED

\proofbegin Proof of Theorem 1. We apply Lemma 1 with $F$ the fixed point set of
$T$ in $C$,\xskip $x↓k=T↑k x$, $y↓n=\sum↓{k≥0}a↓{n,k}x↓k$. Since $|x↓j-f|↑2$
decreases with $j$, it converges to $p(f)<+\infty$. Since $a↓{n,k}\rtarr 0$ as
$n\rtarr\infty$,\xskip $\roman{dist}(y↓n,C↓m)\rtarr 0$ $(n\rtarr+\infty)$. 
To show that (c) holds, it suffices to prove that $|y↓n-Ty↓n|\rtarr 0$ as
$n\rtarr+\infty$. For any $u$ in $H$,
$$\leave blank 1 inch$$
Since $2(x↓j-u,x↓k-u)=|x↓j-u|↑2+|x↓k-u|↑2-|x↓j-x↓k|↑2$,
$$2|y↓n-u|↑2=2\sum↓{k≥0}a↓{n,k}|x↓k-u|↑2-r↓n,$$
where $r↓n=\sum↓{j,k≥0}a↓{n,j}a↓{n,k}|x↓j-x↓k|↑2$. If we choose $u=y↓n$, then
$$r↓n=2\sum↓{k≥0}a↓{n,k}|x↓k-y↓n|↑2.$$
If we set $u=Ty↓n$, we find that
$$\leave blank 2.5 inches$$

\defbegin Definition 1. The array $\{a↓{n,k}\}$ is said to be {\sl proper} if
for each $l↑\infty$-element $\{\beta(k)\}$ such that $\sum↓k a↓{n,k}\beta(k)\rtarr
\delta$, then $\sum↓{k,l} a↓{n,k} a↓{n,l} \beta(|k-l|)\rtarr\delta$.

Ces\`aro means are proper in this sense, as we can see from a simple
computation, as are other familiar summation methods.

\thbegin Theorem 2. Suppose in Theorem 1 that $0\in C$,\xskip $T(0)=0$ and that for
some $c≥0$,\xskip $T$ satisfies for all $u$, $v$ the inequality
$$|Tu+Tv|↑2≤|u+v|↑2+c\{|u|↑2-|Tu|↑2+|v|↑2-|Tv|↑2\}.\eqno(\roman i)$$
Suppose that $\{a↓{n,k}\}$ is proper in the sense of Definition 1 and that
$$\sum↓{k≥0}|a↓{n,k+1}-a↓{n,k}|\rtarr 0\qquad(n\rtarr+\infty).$$
Then $A↓n(x)$ converges strongly.

Obviously (i) will hold with $c=0$ if $C=-C$ and $T$ is odd.

\thbegin Lemma 2. Let $\{x↓j\}$ be a bounded infinite sequence in $H$,\xskip $|x↓j|≤
d↓0$,\xskip $y↓n=\sum↓{k≥0}a↓{n,k}x↓k$ where $\{a↓{n,k}\}$ is an array as in the
hypothesis of Theorem 2. Suppose that $y↓k$ converges weakly to $y$ and that
$(x↓j,x↓{j+k})$ converges to $q(k)$ as $j\rtarr+\infty$, 
uniformly in $k$. Then $y↓n$ converges strongly to $y$.

\proofbegin Proof of Lemma 2. We first show that $\sum↓{k≥0}a↓{n,k}q(k)\rtarr|y|↑2$.
Let $\epsilon>0$ be given. We may find $j(\epsilon)$ such that for $j≥j(\epsilon)$
and all $k$,\xskip $|(x↓j,x↓{j+k})-q(k)|<\epsilon$. We note that
$$\leave blank 1 inch$$
If we choose $j≥j(\epsilon)$ and then $n≥n(\epsilon,j)$, it follows that we can
make $|(x↓j,y)-\sum↓{k≥0}a↓{n,k}q(k)|<2\epsilon$.
Hence for $j, j↓1 ≥j(\epsilon)$,\xskip
$|(x↓j,y)-(x↓{j↓1},y)<4\epsilon$. Thus $(x↓j,y)$ converges as $j\rtarr\infty$, 
and since $\sum↓{j≥0}a↓{n,j}(x↓j,y)=(y↓n,y)$ must converge to the same limit,
$(x↓j,y)\rtarr|y|↑2$. Thus for a large choice of $j$ and $n≥n(\epsilon,j)$,
$$\leave blank 1 in$$
i.e., $\sum↓{k≥0}a↓{n,k}q(k)\rtarr|y|↑2$.

To prove the lemma, it suffices to show that $\limsup|y↓n|↑2≤|y|↑2$. We have
$|y↓n|↑2=\sum↓{k,l≥0}a↓{n,k}a↓{n,l}\,(x↓k,x↓l)$. From the assumption of Lemma 2,
we have $\bigv(x↓k,x↓l)-q(|k-l|)\bigv<\epsilon↓{\min(k,l)}$ where $\epsilon↓p
\rtarr 0$ as $p\rtarr\infty$. Therefore
$$\eqalign{|y↓n|↑2⊗≤\sum↓{k,l≥0}a↓{n,k}a↓{n,l}\,q(|k-l|)+
\sum↓{k,l≥0}a↓{n,k}a↓{n,l}\,\epsilon↓{\min(k,l)}\cr
⊗≤\sum↓{k,l≥0}a↓{n,k}a↓{n,l}\,q(|k-l|)+2\sum↓{k≥0}a↓{n,k}\epsilon↓k.\cr}$$
Since $a↓{n,k}$ is proper we conclude that $\limsup|y↓n|↑2≤|y|↑2$.\QED

\proofbegin Proof of Theorem 2. It follows from the inequality (i) that $|(x↓
{j+s},x↓{j+k+s})-(x↓j,x↓{j+k})|≤(c+1)\{|x↓j|↑2-q(0)\}\rtarr 0$ as $j\rtarr+\infty$.
Hence we may apply Lemma 2 to obtain Theorem 2.\QED

\references

\refno 1. \au{J-B. Baillon} \paper{Un th\'eor\`eme de type ergodique pour les
contractions non lin\'eaires dans un espace de Hilbert} \jour{C. R. Acad.\
Sci.\ Paris S\'er. A-B} \vol{280} \year{1975} \issue{no. 22, Aii}
\pages{A1511--A1514} \MR{51#11205}

\refno 2. \au{\sameauthor} \paper{Quelques propri\'etes de convergence
asymptotique pour les contractions impaires} \jour{C. R. Acad.\ Sci.\
Paris} \toappear

\refno 3. \au{Z. Opial} \paper{Weak convergence of the sequence of the successive
approximations for nonexpansive mappings in Banach spaces} \jour{Bull.\
Amer.\ Math.\ Soc.} \vol{73} \year{1967} \pages{591--597} \MR{35#2183}

\refno 4. \au{F. Hausdorff} \book{Set Theory} \publ{Chelsea} \publaddr{New
York} \year{1962}

\endpaper