/LMAR=50/TMAR=50/RMAR=4095/BMAR=1/PMAR=0/XLINE=0/FONT#0=NGR13/USETI=0000032*TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX**TEX*

␈βαS␈↓ ↓H␈ε_Nonlinear␈α⊃Ergodic␈α⊃Theorems␈↓ ↓I␈ε_Nonlinear␈α⊃Ergodic␈α⊃Theorems
␈βαT␈↓ ↓H␈ε_Nonlinear␈α⊃Ergodic␈α⊃Theorems␈↓ ↓I␈ε_Nonlinear␈α⊃Ergodic␈α⊃Theorems
␈ββ:␈↓ ↓H␈ε∧by␈α⊃H.␈α⊃Br␈↓ αP␈ε∧'␈↓ αP␈ε∧ezis␈α⊃and␈α⊃F.␈α⊃E.␈α⊃Browder
␈β∧ ␈↓ ↓H␈ε∧Department␈α⊃of␈α⊃Mathematics,␈α⊃University␈α⊃of␈α⊃Chicago,␈α⊃Chicago,␈α⊃Illinois␈α⊃60637
␈β¬ε␈↓ ↓H␈ε∧Communicated␈α⊃by␈α⊃Alexandra␈α⊃Bellow
␈β¬[␈↓ ↓H␈ε∧Received␈α⊃May␈α⊃17,␈α⊃1976
␈βε0␈↓ ↓H␈ε∧Keywords:␈α⊃ergodic␈α⊃theory,␈α⊃nonlinear␈α⊃mappings,␈α⊃averaging␈α⊃processes
␈βπ¬␈↓ ↓H␈ε∧AMS␈α⊃(MOS)␈α⊃subject␈α⊃classifications␈α⊃(1970):␈α⊃Primary␈α⊃47A35,␈α⊃47H10;␈α⊃Secondary␈α⊃40G05
␈βπj␈↓ α≥␈ε∧Abstract:␈↓ α≡␈ε∧Abstract:␈↓ εN␈ε∧1␈↓ εO␈ε∧1␈↓ π#␈ε∧2␈↓ π$␈ε∧2
␈βπk␈↓ α≥␈ε∧Abstract:␈α"In␈α⊃two␈α⊃recent␈α⊃notes␈α⊃([␈↓ α≡␈ε∧Abstract:␈↓ εN␈ε∧1],␈α⊃[␈↓ εO␈ε∧1␈↓ π#␈ε∧2]),␈α⊃J-B.␈α⊃Baillon␈α⊃proved␈α⊃the␈α⊃first␈↓ π$␈ε∧2
␈βλ≡␈↓ α≥␈ε∧ergodic␈α⊃theorems␈α⊃for␈α⊃nonlinear␈α⊃mappings␈α⊃in␈α⊃Hilbert␈α⊃space.␈α"We␈α⊃simplify
␈βλQ␈↓ α≥␈ε∧the␈α⊃argument␈α⊃here␈α⊃and␈α⊃obtain␈α⊃an␈α⊃extension␈α⊃of␈α⊃Baillon's␈α⊃theorems␈α⊃from
␈β	∧␈↓ α≥␈ε∧the␈α⊃usual␈α⊃Ces␈↓ βz␈ε∧'␈↓ βz␈ε∧aro␈α⊃means␈α⊃of␈α⊃ergodic␈α⊃theory␈α⊃to␈α⊃general␈α⊃averaging␈α⊃processes
␈β	&␈↓ βX␈ε∧00␈↓ ∧q␈ε∧k
␈β	7␈↓ α.␈ε∧A␈↓ αa␈ε∧=␈↓ ββ␈ε∧"sum"␈↓ ∧≤␈ε∧a␈↓ ∧`␈ε∧T␈↓ ¬$␈ε∧(␈α⊃0␈ε_␈α⊃<␈↓ ε
␈ε∧a␈↓ ε←␈ε∧,␈↓ π#␈ε∧"sum"␈↓ λ<␈ε∧a␈↓ 	⊃␈ε∧=␈α⊃1␈α⊃).
␈β	H␈↓ α?␈ε∧n␈↓ ∧-␈ε∧n,k␈↓ ε≠␈ε∧n,k␈↓ πx␈ε∧k␈ε_>␈ε∧0␈↓ λM␈ε∧n,k
␈β	I␈↓ βX␈ε∧k=0
␈β
>␈↓ ↓H␈ε∧Theorem␈α⊃1.␈↓ ↓I␈ε∧Theorem␈α⊃1.
␈β
?␈↓ ↓H␈ε∧Theorem␈α⊃1.␈ε_␈α"Let␈ε∧␈α"H␈ε_␈α"be␈α⊃a␈α⊃Hilbert␈α⊃space,␈ε∧␈α3C␈ε_␈α"a␈α⊃closed␈α⊃bounded␈α⊃convex␈α⊃subset␈α⊃of␈↓ ↓I␈ε∧Theorem␈α⊃1.
␈ββ␈↓ ↓Y␈ε∧H␈ε_␈α⊃,␈ε∧␈α3T␈ε_␈α"a␈α⊃nonexpansive␈α⊃self␈α⊃map␈α⊃of␈ε∧␈α"C␈ε_␈α⊃.␈α"Suppose␈α⊃that␈α⊃as␈ε∧␈α"n␈↓ 	f␈ε∧->␈↓ 
→␈ε∧00␈↓ 
L␈ε_,␈↓ ⊂␈ε∧a␈↓ e␈ε∧->␈↓ _␈ε∧0
␈β∀␈↓ !␈ε∧n,k
␈β6␈↓ ε≠␈ε∧00
␈β7␈↓ λM␈ε∧+
␈βH␈↓ ↓H␈ε_for␈α⊃each␈ε∧␈α"k␈ε_␈α⊃,␈α⊃and␈↓ ∧␈ε∧"gamma"␈↓ ¬$␈ε∧=␈↓ ¬F␈ε∧"sum"␈↓ εN␈ε∧(␈↓ ε←␈ε∧a␈↓ πV␈ε∧-␈↓ πx␈ε∧a␈↓ λ<␈ε∧)␈↓ λo␈ε∧->␈↓ 	"␈ε∧0␈ε_␈α⊃.␈α"Then␈α⊃for␈α⊃each␈ε∧␈α"x
␈βY␈↓ ¬α␈ε∧n␈↓ εp␈ε∧n,k+1␈↓ λ	␈ε∧n,k
␈βZ␈↓ ε≠␈ε∧k=0
␈β|␈↓ ∧-␈ε∧00
␈β⎇␈↓ ¬F␈ε∧k
␈β∞␈↓ ↓H␈ε_in␈ε∧␈α"C␈ε_␈α⊃,␈↓ αr␈ε∧A␈↓ β∀␈ε∧x␈α⊃=␈↓ βX␈ε∧"sum"␈↓ ∧q␈ε∧a␈↓ ¬5␈ε∧T␈↓ ¬W␈ε∧x␈ε_␈α"converges␈α⊃weakly␈α⊃to␈α⊃a␈α⊃fixed␈α⊃point␈α⊃of␈ε∧␈α"T␈ε_␈α⊃.
␈β∨␈↓ ββ␈ε∧n␈↓ ∧-␈ε∧k=0␈↓ ¬α␈ε∧n,k
␈βs␈↓ ⊂␈ε∧3␈↓ ⊃␈ε∧3
␈βt␈↓ α≥␈ε∧The␈α⊃proof␈α⊃of␈α⊃Theorem␈α⊃1␈α⊃depends␈α⊃upon␈α⊃an␈α⊃extension␈α⊃of␈α⊃Opial's␈α⊃lemma␈α⊃[␈↓ ⊂␈ε∧3].␈↓ ⊃␈ε∧3
␈β
Y␈↓ ↓H␈ε∧Lemma␈α⊃1.␈↓ ↓I␈ε∧Lemma␈α⊃1.
␈β
Z␈↓ ↓H␈ε∧Lemma␈α⊃1.␈ε_␈α"Let␈ε∧␈α"{␈↓ ↓I␈ε∧Lemma␈α⊃1.␈↓ βX␈ε∧x␈↓ βz␈ε∧}␈ε_␈α"and␈ε∧␈α"{␈↓ ¬∪␈ε∧y␈↓ ¬5␈ε∧}␈ε_␈α"be␈α⊃two␈α⊃sequences␈α⊃in␈ε∧␈α"H␈ε_␈α⊃,␈ε∧␈α3F␈ε_␈α"a␈α⊃nonempty␈α⊃subset␈α⊃of
␈β
k␈↓ βi␈ε∧k␈↓ ¬$␈ε∧k
␈β∂<␈↓ εj␈ε∧-1-
␈β⊃ 

␈β↓M␈↓ ↓H␈ε∧Br␈↓ ↓j␈ε∧'␈↓ ↓j␈ε∧ezis␈α⊃and␈α⊃Browder:␈α⊃Nonlinear␈α⊃ergodic␈α⊃theorems
␈βαS␈↓ ↓Y␈ε∧H␈ε_␈α⊃,␈↓ α?␈ε∧C␈↓ ββ␈ε_the␈α⊃convex␈α⊃closure␈α⊃of␈↓ ε
␈ε∧"union"␈↓ π4␈ε∧{␈↓ πE␈ε∧x␈↓ πg␈ε∧}␈ε_␈α⊃.␈α"Suppose␈α⊃that
␈βαd␈↓ αP␈ε∧m␈↓ π↓␈ε∧j␈ε_>␈ε∧m␈↓ πV␈ε∧i
␈ββε␈↓ εp␈ε∧2
␈ββ↔␈↓ α≥␈ε∧(a)␈↓ αa␈ε_For␈α⊃each␈ε∧␈α"f␈ε_␈α"in␈ε∧␈α"F␈ε_␈α⊃,␈ε∧␈α3|␈↓ ¬y␈ε∧x␈↓ ε,␈ε∧-␈α⊃f␈↓ ε←␈ε∧|␈↓ π∩␈ε∧->␈↓ πE␈ε∧p(f)␈α⊃<␈α⊃+␈↓ λM␈ε∧00␈↓ 	␈ε_;
␈ββ(␈↓ ε
␈ε∧j
␈ββ[␈↓ α≥␈ε∧(b)␈↓ αr␈ε∧dist␈↓ β6␈ε∧(␈↓ βG␈ε∧y␈↓ βi␈ε∧,␈↓ ∧␈ε∧C␈↓ ∧-␈ε∧)␈↓ ∧O␈ε∧->␈↓ ¬α␈ε∧0␈ε_␈α"as␈ε∧␈α"k␈↓ ε≠␈ε∧->␈↓ εN␈ε∧00␈↓ π∩␈ε_for␈α⊃each␈ε∧␈α"m␈ε_␈α⊃;
␈ββl␈↓ βX␈ε∧k␈↓ ∧≤␈ε∧m
␈β∧∨␈↓ α≥␈ε∧(c)␈↓ αa␈ε_Any␈α⊃weak␈α⊃limit␈α⊃of␈α⊃an␈α⊃infinite␈α⊃subsequence␈α⊃of␈ε∧␈α"{␈↓ 	␈ε∧y␈↓ 	"␈ε∧}␈ε_␈α"lies␈α⊃in␈ε∧␈α"F␈ε_␈α⊃.
␈β∧0␈↓ 	⊃␈ε∧k
␈β∧⎇␈↓ ↓H␈ε_Then␈↓ α.␈ε∧y␈↓ αr␈ε_converges␈α⊃weakly␈α⊃to␈α⊃a␈α⊃point␈α⊃of␈ε∧␈α"F␈ε_␈α⊃.
␈β¬∞␈↓ α?␈ε∧k
␈β¬c␈↓ α≥␈ε∧Proof␈α⊃of␈α⊃Lemma␈α⊃1.␈α"Since␈α"{␈↓ ¬h␈ε∧y␈↓ ε
␈ε∧}␈α"is␈α⊃bounded,␈α⊃it␈α⊃suffices␈α⊃to␈α⊃show␈α⊃that␈α⊃if␈α"f
␈β¬t␈↓ ¬y␈ε∧k
␈βε'␈↓ ↓H␈ε∧and␈α"g␈α"in␈α"F␈α"are␈α⊃weak␈α⊃limits␈α⊃of␈α⊃infinite␈α⊃subsequences␈α⊃of␈α"{␈↓ 	U␈ε∧y␈↓ 	w␈ε∧}␈α⊃,␈α⊃then␈α"f␈α⊃=␈α⊃g␈α⊃.
␈βε8␈↓ 	f␈ε∧k
␈βεk␈↓ ↓H␈ε∧For␈α⊃each␈α"f␈α⊃,
␈βπ@␈↓ ∧R␈ε∧2␈↓ ε≡␈ε∧2␈↓ πY␈ε∧2
␈βπQ␈↓ βJ␈ε∧|␈↓ β[␈ε∧x␈↓ ∧∞␈ε∧-␈α⊃f␈↓ ∧A␈ε∧|␈↓ ∧t␈ε∧=␈α⊃|␈↓ ¬'␈ε∧x␈↓ ¬Z␈ε∧-␈α⊃g␈↓ ε
␈ε∧|␈↓ ε@␈ε∧+␈α⊃|g␈α⊃-␈α⊃f␈↓ πH␈ε∧|␈↓ π{␈ε∧+␈α⊃2(␈↓ λ?␈ε∧x␈↓ λr␈ε∧-␈α⊃g,␈α⊃g␈α⊃-␈α⊃f).
␈βπb␈↓ βl␈ε∧j␈↓ ¬8␈ε∧j␈↓ λP␈ε∧j
␈βλ7␈↓ ↓H␈ε∧For␈α⊃a␈α⊃given␈↓ β%␈ε∧"epsilon"␈↓ ∧O␈ε∧>␈α⊃0␈α⊃,␈α⊃there␈α⊃exists␈α"m(␈↓ πE␈ε∧"epsilon"␈↓ λ↑␈ε∧)␈α"such␈α⊃that␈α⊃for
␈βλ{␈↓ ↓Y␈ε∧j␈ε_␈α⊃>␈ε∧␈α⊃m(␈↓ α?␈ε∧"epsilon"␈↓ βX␈ε∧)␈α⊃,
␈β	P␈↓ ∧Z␈ε∧2␈↓ 	>␈ε∧2
␈β	a␈↓ αJ␈ε∧|p(g)␈α⊃-␈α⊃|␈↓ βc␈ε∧x␈↓ ∧⊗␈ε∧-␈α⊃g␈↓ ∧I␈ε∧|␈↓ ∧k␈ε∧|␈α⊃<␈↓ ¬/␈ε∧"epsilon"␈↓ εH␈ε∧;␈↓ π.|p(f)␈α⊃-␈α⊃|␈↓ λG␈ε∧x␈↓ λz␈ε∧-␈α⊃f␈↓ 	-␈ε∧|␈↓ 	O␈ε∧|␈α⊃<␈↓ 
∪␈ε∧"epsilon"␈↓ ,␈ε∧.
␈β	r␈↓ βt␈ε∧j␈↓ λX␈ε∧j
␈β
G␈↓ ↓H␈ε∧Let␈↓ α≥␈ε∧K␈↓ βi␈ε∧be␈α⊃the␈α⊃convex␈α⊃set␈α⊃of␈α⊃all␈α"u␈α"such␈α⊃that
␈β
X␈↓ α.␈ε∧"epsilon"
␈β≤␈↓ λP␈ε∧2
␈β-␈↓ β(␈ε∧|2(u␈α⊃-␈α⊃g,␈α⊃g␈α⊃-␈α⊃f)␈α⊃+␈α⊃p(g)␈α⊃-␈α⊃p(f)␈α⊃+␈α⊃|g␈α⊃-␈α⊃f␈↓ λ?␈ε∧|␈↓ λa␈ε∧|␈ε_␈α⊃<␈ε∧␈α⊃2␈↓ 	6␈ε∧"epsilon"␈↓ 
O␈ε∧.
␈β∪␈↓ ↓H␈ε∧Since␈↓ α?␈ε∧K␈↓ ∧␈ε∧contains␈↓ ¬5␈ε∧"union"␈↓ λ~␈ε∧{␈↓ λ+␈ε∧x␈↓ λM␈ε∧}␈α⊃,␈α⊃it␈α⊃contains
␈β$␈↓ αP␈ε∧"epsilon"␈↓ ε,␈ε∧j␈ε_>␈ε∧m(␈↓ εp␈ε∧"epsilon"␈↓ λ	␈ε∧)␈↓ λ<␈ε∧j
␈βW␈↓ ↓Y␈ε∧C␈↓ βG␈ε∧.␈α"There␈α⊃exists␈↓ ¬h␈ε∧k␈↓ π4␈ε∧such␈α⊃that␈α⊃for␈α"k␈ε_␈α⊃>␈↓ 	w␈ε∧k
␈βh␈↓ ↓j␈ε∧m(␈↓ α␈ε∧"epsilon"␈↓ β%␈ε∧)␈↓ ¬y␈ε∧"epsilon"␈↓ 
λ␈ε∧"epsilon"
␈β
≠␈↓ ↓H␈ε∧we␈α⊃can␈α⊃find␈↓ β%␈ε∧u␈↓ βi␈ε∧in␈↓ ∧-␈ε∧C␈↓ ε,␈ε∧such␈α⊃that␈α"|␈↓ πx␈ε∧y␈↓ λ+␈ε∧-␈↓ λM␈ε∧u␈↓ λo␈ε∧|␈ε_␈α⊃<␈↓ 	3␈ε∧"epsilon"␈↓ 
]␈ε∧.␈α"For
␈β
,␈↓ β6␈ε∧k␈↓ ∧>␈ε∧m(␈↓ ∧`␈ε∧"epsilon"␈↓ ¬y␈ε∧)␈↓ λ	␈ε∧k␈↓ λ↑␈ε∧k
␈β
←␈↓ ↓Y␈ε∧k␈ε_␈α⊃>␈↓ α≥␈ε∧k␈↓ βX␈ε∧,␈α⊃it␈α⊃follows␈α⊃that
␈β
p␈↓ α.␈ε∧"epsilon"
␈β∞4␈↓ π.␈ε∧2
␈β∞E␈↓ ↓u␈ε∧|2(␈↓ α(␈ε∧y␈↓ α[␈ε∧-␈α⊃g,␈α⊃g␈α⊃-␈α⊃f)␈α⊃+␈α⊃p(g)␈α⊃-␈α⊃p(f)␈α⊃+␈α⊃|g␈α⊃-␈α⊃f␈↓ π≥␈ε∧|␈↓ π?␈ε∧|␈ε_␈α⊃<␈ε∧␈α⊃2␈↓ λ∀␈ε∧"epsilon"␈↓ 	>␈ε∧+␈α⊃2␈↓ 	q␈ε∧"epsilon"␈↓ 
␈ε∧|g␈α⊃-␈α⊃f|.
␈β∞V␈↓ α9␈ε∧k
␈β∂<␈↓ εj␈ε∧-2-
␈β⊃ 

␈β↓M␈↓ ↓H␈ε∧Br␈↓ ↓j␈ε∧'␈↓ ↓j␈ε∧ezis␈α⊃and␈α⊃Browder:␈α⊃Nonlinear␈α⊃ergodic␈α⊃theorems
␈βαS␈↓ ↓H␈ε∧Consider␈α⊃an␈α⊃infinite␈α⊃subsequence␈α"{␈↓ ε≠␈ε∧y␈↓ εN␈ε∧}␈α"for␈α⊃which␈α"(␈↓ λM␈ε∧y␈↓ 	⊃␈ε∧-␈α⊃g,␈α⊃g␈α⊃-␈α⊃f)␈↓ 
]␈ε∧->␈↓ ⊂␈ε∧0␈α⊃.␈α"In
␈βαd␈↓ ε,␈ε∧k␈↓ λ↑␈ε∧k
␈βαu␈↓ ε=␈ε∧s␈↓ λo␈ε∧s
␈ββ↔␈↓ ↓H␈ε∧the␈α⊃limit
␈ββl␈↓ ε
␈ε∧2
␈ββ⎇␈↓ β↔␈ε∧|p(g)␈α⊃-␈α⊃p(f)␈α⊃+␈α⊃|g␈α⊃-␈α⊃f␈↓ ¬|␈ε∧|␈↓ ε≡␈ε∧|␈ε_␈α⊃<␈ε∧␈α⊃2␈↓ εs␈ε∧"epsilon"␈↓ λ≥␈ε∧+␈α⊃2␈↓ λP␈ε∧"epsilon"␈↓ 	i␈ε∧|g␈α⊃-␈α⊃f|.
␈β∧R␈↓ 
;␈ε∧2
␈β∧c␈↓ ↓H␈ε∧Since␈↓ α?␈ε∧"epsilon"␈↓ βi␈ε∧>␈α⊃0␈α"is␈α⊃arbitrary,␈α⊃it␈α⊃follows␈α⊃that␈α"p(g)␈α⊃+␈α⊃|g␈α⊃-␈α⊃f␈↓ 
*␈ε∧|␈↓ 
]␈ε∧=␈α⊃p(f)␈α⊃.␈α"By
␈β¬⊗␈↓ ∧q␈ε∧2␈↓ λ<␈ε∧2
␈β¬'␈↓ ↓H␈ε∧symmetry,␈α"p(f)␈α⊃+␈α⊃|g␈α⊃-␈α⊃f␈↓ ∧`␈ε∧|␈↓ ¬∪␈ε∧=␈α⊃p(g)␈α⊃.␈α"Hence,␈α"|f␈α⊃-␈α⊃g␈↓ λ+␈ε∧|␈↓ λ↑␈ε∧=␈α⊃0␈α⊃,␈α3f␈α⊃=␈α⊃g␈α⊃.␈↓ 
␈␈ε∧QED
␈βε∧␈↓ α≥␈ε∧Proof␈α⊃of␈α⊃Theorem␈α⊃1.␈α"We␈α⊃apply␈α⊃Lemma␈α⊃1␈α⊃with␈α"F␈α"the␈α⊃fixed␈α⊃point␈α⊃set␈α⊃of
␈βε7␈↓ ∧≤␈ε∧k␈↓ 
→␈ε∧2
␈βεH␈↓ ↓Y␈ε∧T␈α"in␈α"C␈α⊃,␈↓ β6␈ε∧x␈↓ βi␈ε∧=␈↓ ∧␈ε∧T␈↓ ∧-␈ε∧x␈α⊃,␈↓ ¬α␈ε∧y␈↓ ¬5␈ε∧=␈↓ ¬W␈ε∧"sum"␈↓ εp␈ε∧a␈↓ π4␈ε∧x␈↓ πg␈ε∧.␈α"Since␈α"|␈↓ 	"␈ε∧x␈↓ 	U␈ε∧-␈α⊃f␈↓ 
λ␈ε∧|␈↓ 
L␈ε∧decreases
␈βεY␈↓ βG␈ε∧k␈↓ ¬∪␈ε∧n␈↓ ε,␈ε∧k␈ε_>␈ε∧0␈↓ π↓␈ε∧n,k␈↓ πE␈ε∧k␈↓ 	3␈ε∧j
␈βπ␈↓ ↓H␈ε∧with␈α"j␈α⊃,␈α⊃it␈α⊃converges␈α⊃to␈α"p(f)␈α⊃<␈α⊃+␈↓ ε≠␈ε∧00␈↓ εN␈ε∧.␈α"Since␈↓ πx␈ε∧a␈↓ λM␈ε∧->␈↓ 	␈ε∧0␈α"as␈α"n␈↓ 
→␈ε∧->␈↓ 
L␈ε∧00␈↓ 
␈␈ε∧,
␈βπ≥␈↓ λ	␈ε∧n,k
␈βπP␈↓ ↓Y␈ε∧dist␈↓ α≥␈ε∧(␈↓ α.␈ε∧y␈↓ αP␈ε∧,␈↓ αr␈ε∧C␈↓ β∀␈ε∧)␈↓ β6␈ε∧->␈↓ βi␈ε∧0␈α3(n␈↓ ∧`␈ε∧->␈↓ ¬∪␈ε∧+␈↓ ¬$␈ε∧00␈↓ ¬F␈ε∧)␈α⊃.␈α"To␈α⊃show␈α⊃that␈α⊃(c)␈α⊃holds,␈α⊃it␈α⊃suffices␈α⊃to␈α⊃prove
␈βπa␈↓ α?␈ε∧n␈↓ ββ␈ε∧m
␈βλ∀␈↓ ↓H␈ε∧that␈α"|␈↓ α?␈ε∧y␈↓ αr␈ε∧-␈α⊃T␈↓ β%␈ε∧y␈↓ βG␈ε∧|␈↓ βi␈ε∧->␈↓ ∧≤␈ε∧0␈α"as␈α"n␈↓ ¬5␈ε∧->␈↓ ¬h␈ε∧+␈↓ ¬y␈ε∧00␈↓ ε,␈ε∧.␈α"For␈α⊃any␈α"u␈α"in␈α"H␈α⊃,
␈βλ%␈↓ αP␈ε∧n␈↓ β6␈ε∧n
␈βλi␈↓ β(␈ε∧2␈↓ π∃␈ε∧2
␈βλz␈↓ α ␈ε∧|␈↓ α1␈ε∧y␈↓ αd␈ε∧-␈α⊃u␈↓ β↔␈ε∧|␈↓ βJ␈ε∧=␈↓ β⎇␈ε∧|␈↓ ∧0␈ε∧"sum"␈↓ ¬⊗␈ε∧a␈↓ ¬Z␈ε∧(␈↓ ¬k␈ε∧x␈↓ ε≡␈ε∧-␈α⊃u)␈↓ εs␈ε∧|␈↓ π7␈ε∧=␈↓ πY␈ε∧"sum"␈↓ λ?␈ε∧a␈↓ 	β␈ε∧a␈↓ 	G␈ε∧(␈↓ 	X␈ε∧x␈↓ 
␈ε∧-␈α⊃u,␈↓ 
`␈ε∧x␈↓ ∪␈ε∧-␈α⊃u).
␈β	␈↓ αB␈ε∧n␈↓ ¬'␈ε∧n,k␈↓ ¬|␈ε∧k␈↓ λP␈ε∧n,j␈↓ 	∀␈ε∧n,k␈↓ 	i␈ε∧j␈↓ 
q␈ε∧k
␈β	"␈↓ ∧A␈ε∧k␈ε_>␈ε∧0␈↓ πY␈ε∧j,k␈ε_>␈ε∧0
␈β	{␈↓ ε≠␈ε∧2␈↓ πg␈ε∧2␈↓ 	D␈ε∧2
␈β
␈↓ ↓H␈ε∧Since␈α"2(␈↓ αa␈ε∧x␈↓ β∀␈ε∧-␈α⊃u,␈↓ βi␈ε∧x␈↓ ∧≤␈ε∧-␈α⊃u)␈α⊃=␈α⊃|␈↓ ¬$␈ε∧x␈↓ ¬W␈ε∧-␈α⊃u␈↓ ε
␈ε∧|␈↓ ε=␈ε∧+␈α⊃|␈↓ εp␈ε∧x␈↓ π#␈ε∧-␈α⊃u␈↓ πV␈ε∧|␈↓ λ	␈ε∧-␈α⊃|␈↓ λ<␈ε∧x␈↓ λo␈ε∧-␈↓ 	⊃␈ε∧x␈↓ 	3␈ε∧|␈↓ 	f␈ε∧,
␈β
≥␈↓ αr␈ε∧j␈↓ βz␈ε∧k␈↓ ¬5␈ε∧j␈↓ π↓␈ε∧k␈↓ λM␈ε∧j␈↓ 	"␈ε∧k
␈β
a␈↓ ¬I␈ε∧2␈↓ λa␈ε∧2
␈β
r␈↓ ∧0␈ε∧2|␈↓ ∧R␈ε∧y␈↓ ¬¬␈ε∧-␈α⊃u␈↓ ¬8␈ε∧|␈↓ ¬k␈ε∧=␈α⊃2␈↓ ε/␈ε∧"sum"␈↓ π∃␈ε∧a␈↓ πY␈ε∧|␈↓ πj␈ε∧x␈↓ λ≥␈ε∧-␈α⊃u␈↓ λP␈ε∧|␈↓ 	β␈ε∧-␈↓ 	%␈ε∧r␈↓ 	G␈ε∧,
␈ββ␈↓ ∧c␈ε∧n␈↓ π&␈ε∧n,k␈↓ π{␈ε∧k␈↓ 	6␈ε∧n
␈β~␈↓ ε@␈ε∧k␈ε_>␈ε∧0
␈βt␈↓ εp␈ε∧2
␈β¬␈↓ ↓H␈ε∧where␈↓ α?␈ε∧r␈↓ αr␈ε∧=␈↓ β∀␈ε∧"sum"␈↓ ∧O␈ε∧a␈↓ ¬∪␈ε∧a␈↓ ¬W␈ε∧|␈↓ ¬h␈ε∧x␈↓ ε≠␈ε∧-␈↓ ε=␈ε∧x␈↓ ε←␈ε∧|␈↓ π∩␈ε∧.␈α"If␈α⊃we␈α⊃choose␈α"u␈α⊃=␈↓ 	w␈ε∧y␈↓ 
*␈ε∧,␈α⊃then
␈β⊗␈↓ αP␈ε∧n␈↓ βi␈ε∧j,k␈ε_>␈ε∧0␈↓ ∧`␈ε∧n,j␈↓ ¬$␈ε∧n,k␈↓ ¬y␈ε∧j␈↓ εN␈ε∧k␈↓ 
λ␈ε∧n
␈βZ␈↓ λP␈ε∧2
␈βk␈↓ ¬⊗␈ε∧r␈↓ ¬I␈ε∧=␈α⊃2␈↓ ε
␈ε∧"sum"␈↓ εs␈ε∧a␈↓ π7␈ε∧|␈↓ πH␈ε∧x␈↓ π{␈ε∧-␈↓ λ≥␈ε∧y␈↓ λ?␈ε∧|␈↓ λa␈ε∧.
␈β|␈↓ ¬'␈ε∧n␈↓ π∧␈ε∧n,k␈↓ πY␈ε∧k␈↓ λ.␈ε∧n
␈β
∪␈↓ ε≡␈ε∧k␈ε_>␈ε∧0
␈β∂<␈↓ εj␈ε∧-3-
␈β⊃ 

␈β↓M␈↓ ↓H␈ε∧Br␈↓ ↓j␈ε∧'␈↓ ↓j␈ε∧ezis␈α⊃and␈α⊃Browder:␈α⊃Nonlinear␈α⊃ergodic␈α⊃theorems
␈βαS␈↓ ↓H␈ε∧If␈α⊃we␈α⊃set␈α"u␈α⊃=␈α⊃T␈↓ βX␈ε∧y␈↓ ∧␈ε∧,␈α⊃we␈α⊃find␈α⊃that
␈βαd␈↓ βi␈ε∧n
␈ββ(␈↓ β∀␈ε∧2␈↓ ¬h␈ε∧2␈↓ 	U␈ε∧2
␈ββ9␈↓ ↓Y␈ε∧2|␈↓ ↓{␈ε∧y␈↓ α.␈ε∧-␈α⊃T␈↓ αa␈ε∧y␈↓ ββ␈ε∧|␈↓ βX␈ε∧=␈α⊃2␈↓ ∧␈ε∧a␈↓ ∧O␈ε∧|x␈α⊃-␈α⊃T␈↓ ¬5␈ε∧y␈↓ ¬W␈ε∧|␈↓ ε
␈ε∧+␈α⊃2␈↓ εN␈ε∧"sum"␈↓ π4␈ε∧a␈↓ πx␈ε∧|T␈↓ λ~␈ε∧x␈↓ λo␈ε∧-␈α⊃T␈↓ 	"␈ε∧y␈↓ 	D␈ε∧|␈↓ 	w␈ε∧-␈↓ 
→␈ε∧r
␈ββJ␈↓ α␈ε∧n␈↓ αr␈ε∧n␈↓ ∧≤␈ε∧n,0␈↓ ¬F␈ε∧n␈↓ πE␈ε∧n,k␈↓ λ+␈ε∧k-1␈↓ 	3␈ε∧n␈↓ 
*␈ε∧n
␈ββa␈↓ ε←␈ε∧k␈ε_>␈ε∧1
␈β∧_␈↓ ¬h␈ε∧2␈↓ 	3␈ε∧2␈↓ \␈ε∧2
␈β∧)␈↓ βX␈ε_<␈ε∧␈α⊃2␈↓ ∧␈ε∧a␈↓ ∧O␈ε∧|x␈α⊃-␈α⊃T␈↓ ¬5␈ε∧y␈↓ ¬W␈ε∧|␈↓ ε
␈ε∧+␈α⊃2␈↓ εN␈ε∧"sum"␈↓ π4␈ε∧a␈↓ πx␈ε∧|␈↓ λ	␈ε∧x␈↓ λ↑␈ε∧-␈↓ 	␈ε∧y␈↓ 	"␈ε∧|␈↓ 	U␈ε∧-␈α⊃2␈↓ 
→␈ε∧"sum"␈↓ 
␈␈ε∧a␈↓ C␈ε∧|␈↓ T␈ε∧x␈↓ π␈ε∧-␈↓ )␈ε∧y␈↓ K␈ε∧|
␈β∧:␈↓ ∧≤␈ε∧n,0␈↓ ¬F␈ε∧n␈↓ πE␈ε∧n,k␈↓ λ~␈ε∧k-1␈↓ 	⊃␈ε∧n␈↓ ⊂␈ε∧n,k␈↓ e␈ε∧k␈↓ :␈ε∧n
␈β∧Q␈↓ ε←␈ε∧k␈ε_>␈ε∧1␈↓ 
*␈ε∧k␈ε_>␈ε∧0
␈β¬	␈↓ ¬h␈ε∧2␈↓ 
;␈ε∧2
␈β¬~␈↓ βX␈ε_<␈ε∧␈α⊃2␈↓ ∧␈ε∧a␈↓ ∧O␈ε∧|x␈α⊃-␈α⊃T␈↓ ¬5␈ε∧y␈↓ ¬W␈ε∧|␈↓ ε
␈ε∧+␈α⊃2␈↓ εN␈ε∧"sum"␈↓ π#␈ε∧{␈↓ π4␈ε∧a␈↓ λ+␈ε∧-␈↓ λM␈ε∧a␈↓ 	⊃␈ε∧}|␈↓ 	3␈ε∧x␈↓ 	f␈ε∧-␈↓ 
λ␈ε∧y␈↓ 
*␈ε∧|
␈β¬+␈↓ ∧≤␈ε∧n,0␈↓ ¬F␈ε∧n␈↓ πE␈ε∧n,k+1␈↓ λ↑␈ε∧n,k␈↓ 	D␈ε∧k␈↓ 
→␈ε∧n
␈β¬B␈↓ ε←␈ε∧k␈ε_>␈ε∧0
␈β¬y␈↓ πE␈ε∧2
␈βε
␈↓ βX␈ε_<␈ε∧␈α⊃{2␈↓ ∧≤␈ε∧a␈↓ ∧q␈ε∧+␈α⊃2␈↓ ¬$␈ε∧"gamma"␈↓ ε,␈ε∧}␈↓ εN␈ε∧diam␈↓ π∩␈ε∧(C␈↓ π4␈ε∧)␈↓ πg␈ε∧->␈↓ λ~␈ε∧0.␈↓ λ↑␈ε∧QED
␈βε≠␈↓ ∧-␈ε∧n,0␈↓ ε≠␈ε∧n
␈βπ	␈↓ ↓H␈ε∧Definition␈α⊃1.␈↓ ↓I␈ε∧Definition␈α⊃1.
␈βπ
␈↓ ↓H␈ε∧Definition␈α⊃1.␈α"The␈α⊃array␈α"{␈↓ ↓I␈ε∧Definition␈α⊃1.␈↓ ¬∪␈ε∧a␈↓ ¬W␈ε∧}␈α"is␈α⊃said␈α⊃to␈α⊃be␈ε_␈α⊃proper␈ε∧␈α⊃if␈α⊃for␈α⊃each
␈βπ≠␈↓ ¬$␈ε∧n,k
␈βπ=␈↓ ↓j␈ε∧00
␈βπN␈↓ ↓Y␈ε∧l␈↓ α≥␈ε∧-element␈α"{␈↓ βX␈ε∧"beta"␈↓ ∧>␈ε∧(k)}␈α"such␈α⊃that␈↓ ε←␈ε∧"sum"␈↓ πV␈ε∧a␈↓ λ~␈ε∧"beta"␈↓ 	␈ε∧(k)␈↓ 	D␈ε∧->␈↓ 	w␈ε∧"delta"␈↓ 
␈␈ε∧,␈α⊃then
␈βπ←␈↓ π4␈ε∧k␈↓ πg␈ε∧n,k
␈βλ∩␈↓ ↓Y␈ε∧"sum"␈↓ αr␈ε∧a␈↓ β6␈ε∧a␈↓ βz␈ε∧"beta"␈↓ ∧`␈ε∧(|k␈α⊃-␈α⊃l|)␈↓ ε
␈ε∧->␈↓ ε=␈ε∧"delta"␈↓ πE␈ε∧.
␈βλ#␈↓ α.␈ε∧k,l␈↓ ββ␈ε∧n,k␈↓ βG␈ε∧n,l
␈βλV␈↓ α≥␈ε∧Ces␈↓ αP␈ε∧'␈↓ αP␈ε∧aro␈α⊃means␈α⊃are␈α⊃proper␈α⊃in␈α⊃this␈α⊃sense,␈α⊃as␈α⊃we␈α⊃can␈α⊃see␈α⊃from␈α⊃a␈α⊃simple
␈β	~␈↓ ↓H␈ε∧computation,␈α⊃as␈α⊃are␈α⊃other␈α⊃familiar␈α⊃summation␈α⊃methods.
␈β	␈␈↓ ↓H␈ε∧Theorem␈α⊃2.␈↓ ↓I␈ε∧Theorem␈α⊃2.
␈β
␈↓ ↓H␈ε∧Theorem␈α⊃2.␈ε_␈α"Suppose␈α⊃in␈α⊃Theorem␈α⊃1␈α⊃that␈ε∧␈α"0␈↓ ↓I␈ε∧Theorem␈α⊃2.␈↓ π↓␈ε∧"in"␈↓ πV␈ε∧C␈ε_␈α⊃,␈ε∧␈α3T(0)␈α⊃=␈α⊃0␈ε_␈α"and␈α⊃that␈α⊃for␈α⊃some
␈β
D␈↓ ↓Y␈ε∧c␈ε_␈α⊃>␈ε∧␈α⊃0␈ε_␈α⊃,␈ε∧␈α3T␈ε_␈α"satisfies␈α⊃for␈α⊃all␈ε∧␈α"u␈ε_␈α⊃,␈ε∧␈α"v␈ε_␈α"the␈α⊃inequality
␈β→␈↓ ∧I␈ε∧2␈↓ ε∧␈ε∧2␈↓ π≥␈ε∧2␈↓ λ%␈ε∧2␈↓ 	≤␈ε∧2␈↓ 
$␈ε∧2
␈β*␈↓ β0␈ε∧|Tu␈α⊃+␈α⊃Tv␈↓ ∧8␈ε∧|␈↓ ∧k␈ε_<␈ε∧␈α⊃|u␈α⊃+␈α⊃v␈↓ ¬s␈ε∧|␈↓ ε&␈ε∧+␈α⊃c{|u␈↓ π␈ε∧|␈↓ π?␈ε∧-␈α⊃|Tu␈↓ λ∀␈ε∧|␈↓ λG␈ε∧+␈α⊃|v␈↓ 	␈ε∧|␈↓ 	>␈ε∧-␈α⊃|Tv␈↓ 
∪␈ε∧|␈↓ 
5␈ε∧}.␈↓ 
␈ε∧(i)
␈β⊂␈↓ ↓H␈ε_Suppose␈α⊃that␈ε∧␈α"{␈↓ βG␈ε∧a␈↓ ∧␈ε∧}␈ε_␈α"is␈α⊃proper␈α⊃in␈α⊃the␈α⊃sense␈α⊃of␈α⊃Definition␈α⊃1␈α⊃and␈α⊃that
␈β!␈↓ βX␈ε∧n,k
␈βv␈↓ ∧'␈ε∧"sum"␈↓ ¬
␈ε∧|␈↓ ¬≡␈ε∧a␈↓ ε∃␈ε∧-␈↓ ε7␈ε∧a␈↓ ε{␈ε∧|␈↓ π≥␈ε∧->␈↓ πP␈ε∧0␈↓ λ%(n␈↓ λX␈ε∧->␈↓ 	␈ε∧+␈↓ 	≤␈ε∧00␈↓ 	>␈ε∧).
␈β
π␈↓ ¬/␈ε∧n,k+1␈↓ εH␈ε∧n,k
␈β
≡␈↓ ∧8␈ε∧k␈ε_>␈ε∧0
␈β
x␈↓ ↓H␈ε_Then␈↓ α.␈ε∧A␈↓ αP␈ε∧(x)␈ε_␈α"converges␈α⊃strongly.
␈β∞	␈↓ α?␈ε∧n
␈β∞↑␈↓ α≥␈ε∧Obviously␈α⊃(i)␈α⊃will␈α⊃hold␈α⊃with␈α"c␈α⊃=␈α⊃0␈α"if␈α"C␈α⊃=␈α⊃-C␈α"and␈α"T␈α"is␈α⊃odd.
␈β∂<␈↓ εj␈ε∧-4-
␈β⊃ 

␈β↓M␈↓ ↓H␈ε∧Br␈↓ ↓j␈ε∧'␈↓ ↓j␈ε∧ezis␈α⊃and␈α⊃Browder:␈α⊃Nonlinear␈α⊃ergodic␈α⊃theorems
␈βαS␈↓ ↓H␈ε∧Lemma␈α⊃2.␈↓ ↓I␈ε∧Lemma␈α⊃2.
␈βαT␈↓ ↓H␈ε∧Lemma␈α⊃2.␈ε_␈α"Let␈ε∧␈α"{␈↓ ↓I␈ε∧Lemma␈α⊃2.␈↓ βX␈ε∧x␈↓ βz␈ε∧}␈ε_␈α"be␈α⊃a␈α⊃bounded␈α⊃infinite␈α⊃sequence␈α⊃in␈ε∧␈α"H␈ε_␈α⊃,␈ε∧␈α3|␈↓ 	w␈ε∧x␈↓ 
→␈ε∧|␈ε_␈α⊃<␈↓ 
]␈ε∧d␈↓ ⊂␈ε_,
␈βαe␈↓ βi␈ε∧j␈↓ 
λ␈ε∧j␈↓ 
n␈ε∧0
␈ββ_␈↓ ↓Y␈ε∧y␈↓ α␈ε∧=␈↓ α.␈ε∧"sum"␈↓ βG␈ε∧a␈↓ ∧␈ε∧x␈↓ ∧O␈ε_where␈ε∧␈α"{␈↓ ¬W␈ε∧a␈↓ ε≠␈ε∧}␈ε_␈α"is␈α⊃an␈α⊃array␈α⊃as␈α⊃in␈α⊃the␈α⊃hypothesis␈α⊃of␈α⊃Theorem
␈ββ)␈↓ ↓j␈ε∧n␈↓ ββ␈ε∧k␈ε_>␈ε∧0␈↓ βX␈ε∧n,k␈↓ ∧≤␈ε∧k␈↓ ¬h␈ε∧n,k
␈ββ\␈↓ ↓H␈ε_2.␈α"Suppose␈α⊃that␈↓ βz␈ε∧y␈↓ ∧>␈ε_converges␈α⊃weakly␈α⊃to␈ε∧␈α"y␈ε_␈α"and␈α⊃that␈ε∧␈α"(␈↓ 	⊃␈ε∧x␈↓ 	3␈ε∧,␈↓ 	U␈ε∧x␈↓ 
→␈ε∧)␈ε_␈α"converges␈α⊃to
␈ββm␈↓ ∧␈ε∧k␈↓ 	"␈ε∧j␈↓ 	f␈ε∧j+k
␈β∧ ␈↓ ↓Y␈ε∧q(k)␈ε_␈α"as␈ε∧␈α"j␈↓ β%␈ε∧->␈↓ βX␈ε∧+␈↓ βi␈ε∧00␈↓ ∧≤␈ε_,␈α⊃uniformly␈α⊃in␈ε∧␈α"k␈ε_␈α⊃.␈α"Then␈↓ πg␈ε∧y␈↓ λ+␈ε_converges␈α⊃strongly␈α⊃to␈ε∧␈α"y␈ε_␈α⊃.
␈β∧1␈↓ πx␈ε∧n
␈β∧u␈↓ 
L␈ε∧2
␈β¬ε␈↓ α≥␈ε∧Proof␈α⊃of␈α⊃Lemma␈α⊃2.␈α"We␈α⊃first␈α⊃show␈α⊃that␈↓ π4␈ε∧"sum"␈↓ λM␈ε∧a␈↓ 	⊃␈ε∧q(k)␈↓ 	f␈ε∧->␈↓ 
→␈ε∧|y␈↓ 
;␈ε∧|␈↓ 
n␈ε∧.␈α"Let
␈β¬↔␈↓ λ	␈ε∧k␈ε_>␈ε∧0␈↓ λ↑␈ε∧n,k
␈β¬J␈↓ ↓Y␈ε∧"epsilon"␈↓ ββ␈ε∧>␈α⊃0␈α"be␈α⊃given.␈α"We␈α⊃may␈α⊃find␈α"j(␈↓ π∩␈ε∧"epsilon"␈↓ λ+␈ε∧)␈α"such␈α⊃that␈α⊃for
␈βε∞␈↓ ↓Y␈ε∧j␈ε_␈α⊃>␈ε∧␈α⊃j(␈↓ α?␈ε∧"epsilon"␈↓ βX␈ε∧)␈α"and␈α⊃all␈α"k␈α⊃,␈α3|(␈↓ ε,␈ε∧x␈↓ εN␈ε∧,␈↓ εp␈ε∧x␈↓ π4␈ε∧)␈α⊃-␈α⊃q(k)|␈α⊃<␈↓ 	␈ε∧"epsilon"␈↓ 
*␈ε∧.␈α"We␈α⊃note␈α⊃that
␈βε∨␈↓ ε=␈ε∧j␈↓ π↓␈ε∧j+k
␈βεx␈↓ λ↑␈ε∧j-1
␈βπ
␈↓ 
λ␈ε∧2
␈βπ≡␈↓ α␈ε∧|␈↓ α.␈ε∧(␈↓ α?␈ε∧x␈↓ αa␈ε∧,␈α⊃y)␈α⊃-␈↓ βX␈ε∧"sum"␈↓ ∧>␈ε∧a␈↓ ¬α␈ε∧q(k)␈↓ ¬W␈ε∧|␈↓ ε
␈ε_<␈ε∧␈α⊃|(␈↓ εN␈ε∧x␈↓ εp␈ε∧,␈α⊃y␈α⊃-␈↓ πV␈ε∧y␈↓ πx␈ε∧)|␈α⊃+␈↓ λM␈ε∧"sum"␈↓ 	3␈ε∧a␈↓ 	w␈ε∧d
␈βπ/␈↓ αP␈ε∧j␈↓ ∧O␈ε∧n,k␈↓ ε←␈ε∧j␈↓ πg␈ε∧n␈↓ 	D␈ε∧n,k
␈βπ0␈↓ 
λ␈ε∧0
␈βπF␈↓ βi␈ε∧k␈ε_>␈ε∧0␈↓ λ↑␈ε∧k=0
␈βπ}␈↓ αx␈ε∧+␈↓ β~␈ε∧"sum"␈↓ ∧␈ε∧|␈↓ ∧⊃␈ε∧a␈↓ ¬λ␈ε∧-␈↓ ¬*␈ε∧a␈↓ ¬n␈ε∧|␈α⊃|(␈↓ ε2␈ε∧x␈↓ εT␈ε∧,␈↓ εv␈ε∧x␈↓ π:␈ε∧)|␈α⊃+␈↓ λ∂␈ε∧"sum"␈↓ λu␈ε∧a␈↓ 	9␈ε∧|q(k)␈α⊃-␈α⊃(␈↓ 
R␈ε∧x␈↓ 
t␈ε∧,␈↓ ⊗␈ε∧x␈↓ Z␈ε∧)|.
␈βλ∂␈↓ ∧"␈ε∧n,k+j␈↓ ¬;␈ε∧n,k␈↓ εC␈ε∧j␈↓ ππ␈ε∧j+k␈↓ 	ε␈ε∧n,k␈↓ 
c␈ε∧j␈↓ '␈ε∧j+k
␈βλ&␈↓ β+␈ε∧k␈ε_>␈ε∧0␈↓ λ ␈ε∧k␈ε_>␈ε∧0
␈βλ␈␈↓ ↓H␈ε∧If␈α⊃we␈α⊃choose␈α"j␈ε_␈α⊃>␈ε∧␈α⊃j(␈↓ ∧≤␈ε∧"epsilon"␈↓ ¬5␈ε∧)␈α"and␈α⊃then␈α"n␈ε_␈α⊃>␈ε∧␈α⊃n(␈↓ πx␈ε∧"epsilon"␈↓ 	⊃␈ε∧,␈α⊃j)␈α⊃,␈α⊃it␈α⊃follows
␈β	C␈↓ ↓H␈ε∧that␈α⊃we␈α⊃can␈α⊃make␈α"|(␈↓ ∧≤␈ε∧x␈↓ ∧>␈ε∧,␈α⊃y)␈α⊃-␈↓ ¬5␈ε∧"sum"␈↓ εN␈ε∧a␈↓ π∩␈ε∧q(k)|␈α⊃<␈α⊃2␈↓ λ+␈ε∧"epsilon"␈↓ 	U␈ε∧.␈α"Hence␈α⊃for
␈β	T␈↓ ∧-␈ε∧j␈↓ ε
␈ε∧k␈ε_>␈ε∧0␈↓ ε←␈ε∧n,k
␈β
π␈↓ ↓Y␈ε∧j,␈↓ α␈ε∧j␈↓ α?␈ε_>␈ε∧␈α⊃j(␈↓ ββ␈ε∧"epsilon"␈↓ ∧≤␈ε∧)␈α⊃,␈α3|(␈↓ ¬$␈ε∧x␈↓ ¬F␈ε∧,␈α⊃y)␈α⊃-␈α⊃(␈↓ εN␈ε∧x␈↓ π↓␈ε∧,␈α⊃y)␈α⊃<␈α⊃4␈↓ λ	␈ε∧"epsilon"␈↓ 	3␈ε∧.␈α"Thus␈α"(␈↓ 
]␈ε∧x␈↓ 
␈␈ε∧,␈α⊃y)
␈β
_␈↓ α≥␈ε∧1␈↓ ¬5␈ε∧j␈↓ ε←␈ε∧j␈↓ 
n␈ε∧j
␈β
)␈↓ εp␈ε∧1
␈β
K␈↓ ↓H␈ε∧converges␈α⊃as␈α"j␈↓ βX␈ε∧->␈↓ ∧␈ε∧00␈↓ ∧>␈ε∧,␈α⊃and␈α⊃since␈↓ ε≠␈ε∧"sum"␈↓ π4␈ε∧a␈↓ πx␈ε∧(␈↓ λ	␈ε∧x␈↓ λ+␈ε∧,␈α⊃y)␈α⊃=␈α⊃(␈↓ 	3␈ε∧y␈↓ 	U␈ε∧,␈α⊃y)␈α"must␈α⊃converge
␈β
\␈↓ εp␈ε∧j␈ε_>␈ε∧0␈↓ πE␈ε∧n,j␈↓ λ~␈ε∧j␈↓ 	D␈ε∧n
␈β
}␈↓ ε
␈ε∧2
␈β∂␈↓ ↓H␈ε∧to␈α⊃the␈α⊃same␈α⊃limit,␈α"(␈↓ ∧-␈ε∧x␈↓ ∧O␈ε∧,␈α⊃y)␈↓ ¬$␈ε∧->␈↓ ¬W␈ε∧|y␈↓ ¬y␈ε∧|␈↓ ε,␈ε∧.␈α"Thus␈α⊃for␈α⊃a␈α⊃large␈α⊃choice␈α⊃of␈α"j␈α"and
␈β ␈↓ ∧>␈ε∧j
␈βS␈↓ ↓Y␈ε∧n␈ε_␈α⊃>␈ε∧␈α⊃n(␈↓ α?␈ε∧"epsilon"␈↓ βX␈ε∧,␈α⊃j)␈α⊃,
␈β(␈↓ π7␈ε∧2
␈β9␈↓ ∧0␈ε∧|␈↓ ∧c␈ε∧"sum"␈↓ ¬I␈ε∧a␈↓ ε
␈ε∧q(k)␈α⊃-␈α⊃|y␈↓ π&␈ε∧|␈↓ πY␈ε∧|␈↓ λ␈ε∧<␈α⊃4␈↓ λ?␈ε∧"epsilon"␈↓ 	X␈ε∧,
␈βJ␈↓ ¬Z␈ε∧n,k
␈βa␈↓ ∧t␈ε∧k␈ε_>␈ε∧0
␈β
;␈↓ ¬W␈ε∧2
␈β
L␈↓ ↓H␈ε∧i.e.,␈↓ α?␈ε∧"sum"␈↓ βX␈ε∧a␈↓ ∧≤␈ε∧q(k)␈↓ ∧q␈ε∧->␈↓ ¬$␈ε∧|y␈↓ ¬F␈ε∧|␈↓ ¬y␈ε∧.
␈β
]␈↓ β∀␈ε∧k␈ε_>␈ε∧0␈↓ βi␈ε∧n,k
␈β
x␈↓ λ+␈ε∧___
␈β
␈␈↓ 	3␈ε∧2␈↓ 
*␈ε∧2
␈β∞⊂␈↓ α≥␈ε∧To␈α⊃prove␈α⊃the␈α⊃lemma,␈α⊃it␈α⊃suffices␈α⊃to␈α⊃show␈α⊃that␈↓ λ+␈ε∧lim␈↓ λo␈ε∧|␈↓ 	␈ε∧y␈↓ 	"␈ε∧|␈↓ 	U␈ε_<␈ε∧␈α⊃|y␈↓ 
→␈ε∧|␈↓ 
L␈ε∧.␈α"We
␈β∞!␈↓ 	⊃␈ε∧n
␈β∞C␈↓ αr␈ε∧2
␈β∞T␈↓ ↓H␈ε∧have␈α"|␈↓ α?␈ε∧y␈↓ αa␈ε∧|␈↓ β∀␈ε∧=␈↓ β6␈ε∧"sum"␈↓ ∧q␈ε∧a␈↓ ¬5␈ε∧a␈↓ ε
␈ε∧(␈↓ ε≠␈ε∧x␈↓ ε=␈ε∧,␈↓ ε←␈ε∧x␈↓ π↓␈ε∧)␈α⊃.␈α"From␈α⊃the␈α⊃assumption␈α⊃of␈α⊃Lemma␈α⊃2,
␈β∞e␈↓ αP␈ε∧n␈↓ ∧␈ε∧k,l␈ε_>␈ε∧0␈↓ ¬α␈ε∧n,k␈↓ ¬F␈ε∧n,l␈↓ ε,␈ε∧k␈↓ εp␈ε∧l
␈β∂<␈↓ εj␈ε∧-5-
␈β⊃ 

␈β↓M␈↓ ↓H␈ε∧Br␈↓ ↓j␈ε∧'␈↓ ↓j␈ε∧ezis␈α⊃and␈α⊃Browder:␈α⊃Nonlinear␈α⊃ergodic␈α⊃theorems
␈βαS␈↓ ↓H␈ε∧we␈α⊃have␈α"|(␈↓ ββ␈ε∧x␈↓ β%␈ε∧,␈↓ βG␈ε∧x␈↓ βi␈ε∧)␈α⊃-␈α⊃q(|k␈α⊃-␈α⊃l|)|␈α⊃<␈↓ ε≠␈ε∧"epsilon"␈↓ λ↑␈ε∧where␈↓ 	U␈ε∧"epsilon"␈↓ ⊂␈ε∧->␈↓ C␈ε∧0␈α"as
␈βαd␈↓ β∀␈ε∧k␈↓ βX␈ε∧l␈↓ π4␈ε∧min␈↓ πg␈ε∧(k,l)␈↓ 
n␈ε∧p
␈ββ↔␈↓ ↓Y␈ε∧p␈↓ ↓{␈ε∧->␈↓ α.␈ε∧00␈↓ αa␈ε∧.␈α"Therefore
␈ββ[␈↓ αu␈ε∧2
␈ββl␈↓ α1␈ε∧|␈↓ αB␈ε∧y␈↓ αd␈ε∧|␈↓ β9␈ε_<␈↓ β[␈ε∧"sum"␈↓ ∧A␈ε∧a␈↓ ¬¬␈ε∧a␈↓ ¬Z␈ε∧q(|k␈α⊃-␈α⊃l|)␈α⊃+␈↓ π7␈ε∧"sum"␈↓ λ≥␈ε∧a␈↓ λa␈ε∧a␈↓ 	6␈ε∧"epsilon"
␈ββ⎇␈↓ αS␈ε∧n␈↓ ∧R␈ε∧n,k␈↓ ¬⊗␈ε∧n,l␈↓ λ.␈ε∧n,k␈↓ λr␈ε∧n,l␈↓ 
O␈ε∧min␈↓ α␈ε∧(k,l)
␈β∧∀␈↓ β[␈ε∧k,l␈ε_>␈ε∧0␈↓ π7␈ε∧k,l␈ε_>␈ε∧0
␈β∧K␈↓ β9␈ε_<␈↓ β[␈ε∧"sum"␈↓ ∧A␈ε∧a␈↓ ¬¬␈ε∧a␈↓ ¬Z␈ε∧q(|k␈α⊃-␈α⊃l|)␈α⊃+␈α⊃2␈↓ πY␈ε∧"sum"␈↓ λ?␈ε∧a␈↓ 	β␈ε∧"epsilon"␈↓ 
-␈ε∧.
␈β∧\␈↓ ∧R␈ε∧n,k␈↓ ¬⊗␈ε∧n,l␈↓ λP␈ε∧n,k␈↓ 
≤␈ε∧k
␈β∧s␈↓ β[␈ε∧k,l␈ε_>␈ε∧0␈↓ πj␈ε∧k␈ε_>␈ε∧0
␈β¬F␈↓ π↓␈ε∧___
␈β¬M␈↓ λ	␈ε∧2␈↓ 	␈ε∧2
␈β¬↑␈↓ ↓H␈ε∧Since␈↓ α?␈ε∧a␈↓ β%␈ε∧is␈α⊃proper␈α⊃we␈α⊃conclude␈α⊃that␈↓ π↓␈ε∧lim␈↓ πE␈ε∧|␈↓ πV␈ε∧y␈↓ πx␈ε∧|␈↓ λ+␈ε_<␈ε∧␈α⊃|y␈↓ λo␈ε∧|␈↓ 	"␈ε∧.␈↓ 	U␈ε∧QED
␈β¬o␈↓ αP␈ε∧n,k␈↓ πg␈ε∧n
␈βε;␈↓ α≥␈ε∧Proof␈α⊃of␈α⊃Theorem␈α⊃2.␈α"It␈α⊃follows␈α⊃from␈α⊃the␈α⊃inequality␈α⊃(i)␈α⊃that
␈βεn␈↓ πE␈ε∧2
␈βε␈␈↓ ↓Y␈ε∧|(␈↓ ↓{␈ε∧x␈↓ α?␈ε∧,␈↓ αa␈ε∧x␈↓ βG␈ε∧)␈α⊃-␈α⊃(␈↓ ∧≤␈ε∧x␈↓ ∧>␈ε∧,␈↓ ∧`␈ε∧x␈↓ ¬$␈ε∧)|␈ε_␈α⊃<␈ε∧␈α⊃(c␈α⊃+␈α⊃1){|␈↓ π∩␈ε∧x␈↓ π4␈ε∧|␈↓ πg␈ε∧-␈α⊃q(0)}␈↓ λo␈ε∧->␈↓ 	"␈ε∧0␈α"as␈α"j␈↓ 
;␈ε∧->␈↓ 
n␈ε∧+␈↓ 
␈␈ε∧00␈↓ 2␈ε∧.␈α"Hence
␈βπ⊂␈↓ α␈ε∧j+s␈↓ αr␈ε∧j+k+s␈↓ ∧-␈ε∧j␈↓ ∧q␈ε∧j+k␈↓ π#␈ε∧j
␈βπC␈↓ ↓H␈ε∧we␈α⊃may␈α⊃apply␈α⊃Lemma␈α⊃2␈α⊃to␈α⊃obtain␈α⊃Theorem␈α⊃2.␈↓ π#␈ε∧QED
␈βλB␈↓ ↓H␈ε_References␈↓ ↓I␈ε_References
␈βλC␈↓ ↓H␈ε_References␈↓ ↓I␈ε_References
␈β	J␈↓ ↓T␈ε∧1.␈↓ ↓U␈ε∧1.
␈β	K␈↓ ↓T␈ε∧1.␈↓ ↓U␈ε∧1.␈↓ α≥␈ε∧J-B.␈α⊃Baillon,␈α⊃"Un␈α⊃th␈↓ ∧q␈ε∧'␈↓ ∧q␈ε∧eor␈↓ ¬$␈ε∧'␈↓ ¬$␈ε∧eme␈α⊃de␈α⊃type␈α⊃ergodique␈α⊃pour␈α⊃les␈α⊃contractions␈α⊃non
␈β
∞␈↓ T␈ε∧280␈↓ U␈ε∧280
␈β
∂␈↓ α≥␈ε∧lin␈↓ αP␈ε∧'␈↓ αP␈ε∧eaires␈α⊃dans␈α⊃un␈α⊃espace␈α⊃de␈α⊃Hilbert,"␈ε_␈α⊃C.␈α⊃R.␈α⊃Acad.␈α⊃Sci.␈α⊃Paris␈α⊃S␈↓ 
;␈ε_'␈↓ 
;␈ε_er.␈α"A-B␈↓ T␈ε∧280,␈↓ U␈ε∧280
␈β
S␈↓ α≥␈ε∧no.␈α"22,␈α⊃Aii␈α⊃(1975),␈α⊃A1511--A1514.␈ε_␈α"MR␈α⊃51␈ε∧␈α⊃#11205.
␈β⊗␈↓ ↓T␈ε∧2.␈↓ ↓U␈ε∧2.
␈β↔␈↓ ↓T␈ε∧2.␈↓ ↓U␈ε∧2.␈↓ α≥␈ε∧----,␈α⊃"Quelques␈α⊃propri␈↓ ¬∪␈ε∧'␈↓ ¬∪␈ε∧etes␈α⊃de␈α⊃convergence␈α⊃asymptotique␈α⊃pour␈α⊃les␈α⊃contractions
␈β[␈↓ α≥␈ε∧impaires,"␈ε_␈α⊃C.␈α⊃R.␈α⊃Acad.␈α⊃Sci.␈α⊃Paris␈ε∧,␈α⊃to␈α⊃appear.
␈β≡␈↓ ↓T␈ε∧3.␈↓ ↓U␈ε∧3.
␈β∨␈↓ ↓T␈ε∧3.␈↓ ↓U␈ε∧3.␈↓ α≥␈ε∧Z.␈α⊃Opial,␈α⊃"Weak␈α⊃convergence␈α⊃of␈α⊃the␈α⊃sequence␈α⊃of␈α⊃the␈α⊃successive␈α⊃approximations
␈βb␈↓ !␈ε∧73␈↓ "␈ε∧73
␈βc␈↓ α≥␈ε∧for␈α⊃nonexpansive␈α⊃mappings␈α⊃in␈α⊃Banach␈α⊃spaces,"␈ε_␈α⊃Bull.␈α⊃Amer.␈α⊃Math.␈α⊃Soc.␈↓ !␈ε∧73␈↓ "␈ε∧73
␈β
'␈↓ α≥␈ε∧(1967),␈α⊃591--597.␈ε_␈α"MR␈α⊃35␈ε∧␈α⊃#2183.
␈β
j␈↓ ↓T␈ε∧4.␈↓ ↓U␈ε∧4.
␈β
k␈↓ ↓T␈ε∧4.␈↓ ↓U␈ε∧4.␈↓ α≥␈ε∧F.␈α⊃Hausdorff,␈ε_␈α⊃Set␈α⊃Theory␈ε∧,␈α⊃Chelsea,␈α⊃New␈α⊃York,␈α⊃1962.
␈β∂<␈↓ εj␈ε∧-6-
␈β⊃ /FONT#4=delgat[fnt,dek]=_"#'()+,-.0123456789:;<=>ABCDEFGHIJKLMNOPQRSTUWYZ[]abcdefghijklmnopqrstuvwxyz{|}}/FONT#24=unddel[fnt,dek]=',-.1235;<>ABCDEFHLMNPRSTabcdefghiklmnopqrstuvwxyy