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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(8),6087-6098
PublishedOnlineAugust2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.118641
.52ÂMann-HalpernŽ{9ÙA^
NNN___
H“‰ŒÆêÆÆ§H&²
ÂvFϵ2022c726F¶¹^Fϵ2022c819F¶uÙFϵ2022c829F
Á‡
©Ì‡ïÄHilbert˜m¥š*ÜNØÄ:¯K"JÑ˜«.52ÂMann-HalpernŽ
{"3˜½^‡ey²Ž{rÂñ5"òŽ{A^u¦)Fermat-Weber½ ¯K§¿‰Ñ
ꊢ(J"ƒ'®kŽ{§TŽ{3ëêÀþ•ä(¹5"
'…c
š*ÜN§ØÄ:¯K§.5Ž{§2ÂMann-HalpernŽ{
InertialGeneralizedMann-Halpern
AlgorithmandItsApplication
YunxiaXu
SchoolofMathematics,YunnanNormalUniversity,KunmingYunnan
Received:Jul.26
th
,2022;accepted:Aug.19
th
,2022;published:Aug.29
th
,2022
Abstract
We considered the fixed point problem of nonexpansive mapping in Hilbert space.We
proposedaninertialgeneralizedMann-Halpernalgorithm.Givingcertainconditions,
©ÙÚ^:N_..52ÂMann-HalpernŽ{9ÙA^[J].A^êÆ?Ð,2022,11(8):6087-6098.
DOI:10.12677/aam.2022.118641
N_
weprovedthestrongconvergenceofthealgorithm.Thenweappliedthealgorithm
to solve the Fermat-Weber locationproblem,and gaveanumerical experiment.Com-
paredwithalgorithmshadbeenproposedbefore,ouralgorithmhastheflexibilityon
choosingparameters.
Keywords
NonexpansiveMapping,FixedPointProblem,InertialAlgorithm,Generalized
Mann-HalpernAlgorithm
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
©Ì‡ïÄš*ÜNTØÄ:¯K.H“LHilbert˜m,h·,·i,k·k©O“LHþS
È9Ùp‰ê.C⊂H´š˜4à8,T: C→C´š*ÜN.
Mann3©z[1]¥JÑ²;MannŽ{,éux
0
∈C, α∈(0,1) ,ÙS“/ª•:
x
n+1
= αx
n
+(1−α)Tx
n
,n= 0,1,2,···.(1)
Reich3©z[2]¥JÑ˜„/ªMann Ž{,éux
0
∈C, {α
n
}⊂(0,1),ÙS“/ª•:
x
n+1
= α
n
x
n
+(1−α
n
)Tx
n
,n= 0,1,2,···.(2)
3˜½^‡ey²{x
n
}fÂñTØÄ:.•„©z[2].
Halpern3©z[3]¥JÑHalpern Ž{,éux
0
,u∈C,{α
n
}⊂(0,1),ÙS“/ª•:
x
n+1
= α
n
u+(1−α
n
)Tx
n
,n= 0,1,2,···.(3)
y²{x
n
}rÂñTØÄ:7‡^‡´:lim
n→∞
α
n
= 0,
∞
P
n=0
α
n
= ∞.
Wittmann3©z[4]¥y²Ž{(3)¥S{x
n
}rÂñTØÄ:¿©^‡´:
lim
n→∞
α
n
= 0,
∞
P
n=0
α
n
= ∞,…
∞
P
n=0
|α
n+1
−α
n
|<∞.
DOI:10.12677/aam.2022.1186416088A^êÆ?Ð
N_
,˜•¡,.5Ž{•@´dPolyak 3©z[5]¥JÑ,TŽ{3\¯Âñ„Ç•¡åX-
‡Š^.e˜ÚS“dcüÚS“,=
x
n+1
= x
n
+θ
n
(x
n
−x
n−1
),n= 0,1,2,···.(4)
Ù¥x
0
,x
1
∈C, {θ
n
}⊂[0,1).Cc5,é.5Ž{kXŒþïÄ,~X,.5c©Ž{[6],.
5ÝKŽ{[7],.5MannŽ{[8–12].
Mainge3©z[8]¥JÑ.5MannŽ{,éux
0
,x
1
∈C, {α
n
}⊂(0,1),ÙS“/ª•:



w
n
= x
n
+θ
n
(x
n
−x
n−1
),
x
n+1
= α
n
w
n
+(1−α
n
)Tw
n
,n= 0,1,2,···.
(5)
3˜½^‡e,y²Ž{(5)fÂñ5.•„©z[8].
Tan<3©z[12]¥JÑ?.5Mann-HalpernŽ{, éux
0
,x
1
,u∈C,{α
n
},{β
n
}⊂
(0,1),ÙS“/ª•:







w
n
= x
n
+θ
n
(x
n
−x
n−1
),
y
n
= β
n
w
n
+(1−β
n
)Tw
n
,
x
n+1
= α
n
u+(1−α
n
)y
n
,n= 0,1,2,···.
(6)
3˜½^‡e,y²Ž{(6)äkrÂñ5.•„©z[12].
©3©z[12]Ä:þ,JÑ.52ÂMann-Halpern Ž{,S“/ª•:







w
n
= x
n
+θ
n
(x
n
−x
n−1
),
y
n
= s
n
w
n
+t
n
Tw
n
,
x
n+1
= α
n
u+(1−α
n
)y
n
,n= 0,1,2,···.
(7)
Ù¥x
0
,x
1
,u∈C, {θ
n
}⊂[0,1), {s
n
},{t
n
},{α
n
}⊂(0,1).Ž{(7)3ëêÀþ•\(¹:
s
n
+t
n
≡1ž,TŽ{òz¤Ž{(6).3˜½^‡e,·‚òy²Ž{rÂñ5.
2.ý•£
!̇‰Ñ˜½ny²¥I‡^ÚnÚÄVg.
½Â1.NT: C→C,XJé?¿x,y∈C,Ñk
kTx−Tyk≤kx−yk,
K¡T´Cþš*ÜN.XJ•3x∈C,¦x=Tx,K¡x´TØÄ:,PT¤k
ØÄ:¤8Ü•Fix(T).
51.©¥,XÃAÏ`²,þbT´š*ÜN,¿…Fix(T) 6= ∅.
DOI:10.12677/aam.2022.1186416089A^êÆ?Ð
N_
½Â2.HݘmH
∗
:={g:H→R


g•HþëY‚5•¼}.x∈H,S{x
n
}⊂H,
XJé?¿g∈H
∗
Ñklim
n→∞
g(x
n
) = g(x),K¡{x
n
}fÂñux,P¤x
n
*x,(n→∞);XJ
lim
n→∞
kx
n
−xk= 0,
K¡{x
n
}rÂñux,P¤x
n
→x,(n→∞).
Ún1.(©z[13]íØ4.15)C⊂H´š˜4à8,T:C→C´š*ÜN,KFix(T) ´4à
8.
Ún2.(©z[13]Ún2.37){x
n
}´H¥k.S,K{x
n
}•3fÂñf.
Ún3.(©z[14]Ún2.1)é?¿x,y∈H,±e¯¢¤á:
(1)kx+yk
2
≤kxk
2
+2hy,x+yi;
(2)kax+byk
2
= a(a+b)kxk
2
+b(a+b)kyk
2
−abkx−yk
2
,∀a,b∈R.
Ún4.(©z[15]Ún8){b
n
},{η
n
}´šK¢ê,{δ
n
},{ξ
n
}´¢ê,{γ
n
}⊂(0,1),¿…
b
n+1
≤(1−γ
n
)b
n
+γ
n
δ
n
,9b
n+1
≤b
n
−η
n
+ξ
n
,n= 0,1,2,···.
e
(1)
∞
P
n=0
γ
n
= ∞,(2)lim
n→∞
ξ
n
= 0,
(3){η
n
}?¿÷vlim
k→∞
η
n
k
= 0f{η
n
k
}%¹limsup
k→∞
δ
n
k
≤0,
Klim
n→∞
b
n
= 0.
Ún5.(©z[13]íØ4.18)C⊂H´š˜4à8,T:C→C´š*ÜN,{x
n
}⊂C,x∈H.
XJn→∞ž,kx
n
*x,…(x
n
−Tx
n
) →0,K(I−T)x= 0.
Ún6.(©z[16]·K1.78)C⊂H´š˜4à8,¯x∈H,¯y∈C,K¯y´¯x3CþÝK…
=
h¯x−¯y,y−¯yi≤0,∀y∈C.
3.̇(J
Ž{1.b{θ
n
}⊂[0,1),{s
n
},{t
n
},{α
n
}⊂(0,1),÷vs
n
+t
n
≤1.éux
0
,x
1
,u∈C,Ž{S“
/ª•:







w
n
= x
n
+θ
n
(x
n
−x
n−1
),
y
n
= s
n
w
n
+t
n
Tw
n
,
x
n+1
= α
n
u+(1−α
n
)y
n
,n= 0,1,2,···.
(8)
½n1.C⊂H´š˜4à8,T: C→C´š*ÜN,…Fix(T) 6= ∅.éux
0
,x
1
,u∈C,S 
{x
n
}dŽ{1S“),¿…±e^‡¤á
(1)lim
n→∞
α
n
= 0,
∞
P
n=0
α
n
= ∞,
DOI:10.12677/aam.2022.1186416090A^êÆ?Ð
N_
(2)lim
n→∞
θ
n
kx
n
−x
n−1
k
α
n
= 0,
(3)
∞
P
n=0
(1−s
n
−t
n
) <∞,
(4)lim
n→∞
1−s
n
−t
n
α
n
= 0.
K{x
n
}rÂñTØÄ:p,¿…p= P
Fix(T)
u.
yyy²²².1˜Ú:ky²{x
n
}k..p∈Fix(T),K
kx
n+1
−pk= kα
n
u+(1−α
n
)y
n
−pk
= kα
n
(u−p)+(1−α
n
)(y
n
−p)k
≤α
n
ku−pk+(1−α
n
)ky
n
−pk.(9)
þª¥
ky
n
−pk= ks
n
w
n
+t
n
Tw
n
−pk
= ks
n
(w
n
−p)+t
n
(Tw
n
−p)+(s
n
+t
n
−1)pk
≤s
n
kw
n
−pk+t
n
kTw
n
−pk+(1−s
n
−t
n
)kpk
≤s
n
kw
n
−pk+t
n
kw
n
−pk+(1−s
n
−t
n
)kpk
= (s
n
+t
n
)kw
n
−pk+(1−s
n
−t
n
)kpk
≤kw
n
−pk+(1−s
n
−t
n
)kpk.(10)
ò(10)ª‘\(9)ª¥,Œ
kx
n+1
−pk≤(1−α
n
)kw
n
−pk+α
n
ku−pk+(1−α
n
)(1−s
n
−t
n
)kpk.(11)
þª¥
kw
n
−pk= kx
n
+θ
n
(x
n
−x
n−1
)−pk≤kx
n
−pk+θ
n
kx
n
−x
n−1
k.(12)
ò(12)ª‘\(11)ª¥,Œ
kx
n+1
−pk
≤(1−α
n
)kx
n
−pk+α
n
ku−pk+(1−α
n
)θ
n
kx
n
−x
n−1
k+(1−α
n
)(1−s
n
−t
n
)kpk
= (1−α
n
)kx
n
−pk+α
n
h
ku−pk+(1−α
n
)
θ
n
kx
n
−x
n−1
k
α
n
i
+(1−α
n
)(1−s
n
−t
n
)kpk.
DOI:10.12677/aam.2022.1186416091A^êÆ?Ð
N_
3þª¥,-M=2max
n
ku−pk,sup
n≥0
(1 −α
n
)
θ
n
kx
n
−x
n−1
k
α
n
o
,d(1 −α
n
)<1 Ú^‡(2) Œ•
M<∞,K
α
n
h
ku−pk+(1−α
n
)
θ
n
kx
n
−x
n−1
k
α
n
i
≤α
n
M.
¤±
kx
n+1
−pk≤(1−α
n
)kx
n
−pk+α
n
M+(1−α
n
)(1−s
n
−t
n
)kpk.
dà|Ü5ŸŒ•
kx
n+1
−pk≤max
n
kx
n
−pk,M
o
+(1−α
n
)(1−s
n
−t
n
)kpk.
2d(1−α
n
) <1,Υ
kx
n+1
−pk≤max
n
kx
n
−pk,M
o
+(1−s
n
−t
n
)kpk.
òþª'un?18B,Œ
kx
n+1
−pk≤max
n
kx
0
−pk,M
o
+
n
X
j=0
(1−s
j
−t
j
)kpk.(13)
(Ü^‡(3)Œ•
n
X
j=0
(1−s
j
−t
j
)kpk≤
∞
X
n=0
(1−s
n
−t
n
)kpk<∞,
d(13)ªŒ•
n
kx
n+1
−pk
o
k.,l{x
n
}k..
?˜Ú,Ï•kw
n
k≤kx
n
k+θ
n
kx
n
−x
n−1
k, ¤±{w
n
}k., dT´š*ÜN•kTw
n
−pk≤
kw
n
−pk,l{Tw
n
}k..
d{y
n
}½ÂÚs
n
+t
n
≤1•ky
n
k≤s
n
kw
n
k+t
n
kTw
n
k≤max
n
kw
n
k,kTw
n
k
o
,l{y
n
}k
.,2dT´š*ÜN•{Ty
n
}k..
1Ú:2y²{x
n
}rÂñp= P
Fix(T)
u.Ï•
kx
n+1
−pk
2
= kα
n
u+(1−α
n
)y
n
−pk
2
= k(1−α
n
)(y
n
−p)+α
n
(u−p)k
2
≤(1−α
n
)
2
ky
n
−pk
2
+2hα
n
(u−p),x
n+1
−pi
≤(1−α
n
)ky
n
−pk
2
+2α
n
hu−p,x
n+1
−pi.(14)
DOI:10.12677/aam.2022.1186416092A^êÆ?Ð
N_
(14)ª¥1˜‡ØªdÚn3¯¢(1),…
ky
n
−pk
2
= ks
n
w
n
+t
n
Tw
n
−pk
2
= ks
n
(w
n
−p)+t
n
(Tw
n
−p)+(s
n
+t
n
−1)pk
2
≤ks
n
(w
n
−p)+t
n
(Tw
n
−p)k
2
+2h(s
n
+t
n
−1)p,y
n
−pi
= s
n
(s
n
+t
n
)kw
n
−pk
2
+t
n
(s
n
+t
n
)kTw
n
−pk
2
−s
n
t
n
kw
n
−Tw
n
k
2
+2h(s
n
+t
n
−1)p,y
n
−pi
≤s
n
kw
n
−pk
2
+t
n
kTw
n
−pk
2
−s
n
t
n
kw
n
−Tw
n
k
2
+2(s
n
+t
n
−1)hp,y
n
−pi
≤s
n
kw
n
−pk
2
+t
n
kw
n
−pk
2
−s
n
t
n
kw
n
−Tw
n
k
2
+2(s
n
+t
n
−1)hp,y
n
−pi
= (s
n
+t
n
)kw
n
−pk
2
−s
n
t
n
kw
n
−Tw
n
k
2
+2(s
n
+t
n
−1)hp,y
n
−pi
≤kw
n
−pk
2
−s
n
t
n
kw
n
−Tw
n
k
2
+2(s
n
+t
n
−1)hp,y
n
−pi.(15)
(15)ª¥1˜‡ØªdÚn3¯¢(1),1n‡ªdÚn3¯¢(2).
ò(15)ª‘\(14)ª¥,Œ
kx
n+1
−pk
2
≤(1−α
n
)kw
n
−pk
2
−(1−α
n
)s
n
t
n
kw
n
−Tw
n
k
2
+2(1−α
n
)(s
n
+t
n
−1)hp,y
n
−pi+2α
n
hu−p,x
n+1
−pi.(16)
þª¥
kw
n
−pk
2
= kx
n
+θ
n
(x
n
−x
n−1
)−pk
2
= kx
n
−p+θ
n
(x
n
−x
n−1
)k
2
= kx
n
−pk
2
+θ
2
n
kx
n
−x
n−1
k
2
+2θ
n
hx
n
−p,x
n
−x
n−1
i.(17)
ò(17)ª‘\(16)ª¥,Œ
kx
n+1
−pk
2
≤(1−α
n
)kx
n
−pk
2
+(1−α
n
)θ
2
n
kx
n
−x
n−1
k
2
+2(1−α
n
)θ
n
hx
n
−p,x
n
−x
n−1
i
−(1−α
n
)s
n
t
n
kw
n
−Tw
n
k
2
+2(1−α
n
)(s
n
+t
n
−1)hp,y
n
−pi+2α
n
hu−p,x
n+1
−pi.
(18)
3(18)ª¥-b
n
= kx
n
−pk
2
,η
n
= (1−α
n
)s
n
t
n
kw
n
−Tw
n
k
2
,γ
n
= α
n
,
δ
n
=
(1−α
n
)
α
n
θ
2
n
kx
n
−x
n−1
k
2
+
2(1−α
n
)
α
n
θ
n
hx
n
−p,x
n
−x
n−1
i+
2(1−α
n
)
α
n
(s
n
+t
n
−1)hp,y
n
−pi+2hu−p,x
n+1
−pi.
-ξ
n
= α
n
δ
n
,K
ξ
n
= (1−α
n
)θ
2
n
kx
n
−x
n−1
k
2
+2(1−α
n
)θ
n
hx
n
−p,x
n
−x
n−1
i
+2(1−α
n
)(s
n
+t
n
−1)hp,y
n
−pi+2α
n
hu−p,x
n+1
−pi.
DOI:10.12677/aam.2022.1186416093A^êÆ?Ð
N_
l(18)ªŒU¤
b
n+1
≤(1−γ
n
)b
n
+γ
n
δ
n
,9b
n+1
≤b
n
−η
n
+ξ
n
,n= 0,1,2,···.
d
∞
P
n=0
α
n
= ∞•
∞
X
n=0
γ
n
= ∞.
3ξ
n
¥,Ï•lim
n→∞
α
n
= 0Ú1˜Ú(Ø{x
n
}k.,¤±
lim
n→∞
2α
n
hu−p,x
n+1
−pi= 0.
d^‡(2)•lim
n→∞
θ
n
kx
n
−x
n−1
k= 0,¤±
lim
n→∞
θ
2
n
kx
n
−x
n−1
k
2
= 0,lim
n→∞
2(1−α
n
)θ
n
hx
n
−p,x
n
−x
n−1
i=0.
d^‡(3)•lim
n→∞
(1−s
n
−t
n
) = 0,¤±lim
n→∞
(s
n
+t
n
−1) = 0,(Ü1˜Ú(Ø{y
n
}k.,l
lim
n→∞
2(1−α
n
)(s
n
+t
n
−1)hp,y
n
−pi= 0.
nÜŒ•
lim
n→∞
ξ
n
= 0.
(ÜÚn4,–dTÚn^‡(1),(2)®÷v,e5¦y^‡(3)÷v=Œ.
¯¢þ,?{η
n
}f{η
n
k
},¦lim
k→∞
η
n
k
=0.d{α
n
},{s
n
},{t
n
}⊂(0,1)Úη
n
½Â,Œ
•
lim
k→∞
kw
n
k
−Tw
n
k
k= 0.(19)
dlim
n→∞
θ
n
kx
n
−x
n−1
k= 0,Υ
kw
n
k
−x
n
k
k= kx
n
k
+θ
n
k
(x
n
k
−x
n
k−1
)−x
n
k
k= θ
n
k
kx
n
k
−x
n
k−1
k→0,(k→∞).(20)
Ï•{x
n
}k.,¤±{x
n
k
}k.,¤±dÚn2••3{x
n
k
}f{x
n
k
j
}÷v
x
n
k
j
*¯x,(j→∞) 9limsup
k→∞
hu−p,x
n
k
−pi=lim
j→∞
hu−p,x
n
k
j
−pi.
d(20)ªŒ•,w
n
k
j
*¯x,(j→∞),¤±d(19)ªÚÚn5•¯x= T¯x,=¯x∈Fix(T).
Ï•p= P
Fix(T)
u,¤±(ÜÚn1ÚÚn6Œ•hu−p,¯x−pi≤0.¤±
limsup
k→∞
hu−p,x
n
k
−pi=lim
j→∞
hu−p,x
n
k
j
−pi= hu−p,¯x−pi≤0.
DOI:10.12677/aam.2022.1186416094A^êÆ?Ð
N_
e5,y²limsup
k→∞
hu−p,x
n
k+1
−pi≤0.Ï•
ky
n
k
−w
n
k
k= ks
n
k
w
n
k
+t
n
k
Tw
n
k
−w
n
k
k= kt
n
k
(Tw
n
k
−w
n
k
)+(s
n
k
+t
n
k
−1)w
n
k
k
≤t
n
k
kTw
n
k
−w
n
k
k+(1−s
n
k
−t
n
k
)kw
n
k
k.
¤±d(19)ª,t
n
k
⊂(0,1),lim
n→∞
(1−s
n
−t
n
) = 0±9{w
n
k
}k.Υ
lim
k→∞
ky
n
k
−w
n
k
k= 0.(21)
Ï•ky
n
k
−x
n
k
k≤ky
n
k
−w
n
k
k+kw
n
k
−x
n
k
k,¤±d(20)ªÚ(21)ªŒ•
lim
k→∞
ky
n
k
−x
n
k
k= 0.(22)
qϕ
kx
n
k+1
−x
n
k
k= kα
n
k
u+(1−α
n
k
)y
n
k
−x
n
k
k≤α
n
k
ku−x
n
k
k+(1−α
n
k
)ky
n
k
−x
n
k
k.
¤±dlim
n→∞
α
n
= 0,{x
n
k
}k.Ú(22)ªŒ•
lim
k→∞
kx
n
k+1
−x
n
k
k= 0.
¤±
limsup
k→∞
hu−p,x
n
k+1
−pi≤0.
2(ܽn^‡(2),^‡(4)Úδ
n
½Â
Υ
limsup
k→∞
δ
n
k
≤0.
–dyÚn4^‡(3)÷v.
nþ,dÚn4Œ,lim
n→∞
b
n
= 0,=
lim
n→∞
x
n
= p.
–d,¤y².
52.e¡‰Ñ÷v½n^‡ëêŠ«~:
α
n
=
1
n
,s
n
= t
n
=
1
2
−
1
n
2
,0 ≤θ
n
≤
¯
θ
n
,Ù¥
¯
θ
n
=





n−1
n+2
,x
n
= x
n−1
;
min

n−1
n+2
,
10
(n+1)
2
kx
n
−x
n−1
k

,x
n
6= x
n−1
.
DOI:10.12677/aam.2022.1186416095A^êÆ?Ð
N_
4.Ž{3Fermat-Weber½ ¯K¥A^
!̇•ÄH¥XeFermat-Weber ½ ¯K:=Ïéx∈R
d
,¦)
min
x
n
f(x) =
m
X
i=1
ω
i
kx−a
i
k
o
,(FW)
Ù¥,ω
i
>0,i= 1,2,···,m´-,a
i
∈R
d
´‰½e:.
¦)¯K(FW)~^Ž{´Weiszfeld Ž{,TŽ{•@dWeiszfeld 3©z[17]¥JÑ,3`
zÚ½ +•kXŒþïÄ,–8E3?ØÚ¦^,äNŒë•©z[17,18].3©z[18]¥,Šö
EØÄ:S“Ž{¿?1Âñ5©Û,'uš*ÜNTäNŽ{Xe:









T(·) =
m
X
i=1
ω
i
a
i
k·−a
i
k
.
m
X
i=1
a
i
k·−a
i
k
,
x
n+1
= Tx
n
,n= 0,1,2,···.
(23)
Š•A~,·‚•ÄR
3
˜m¥,m= 8 œ/,-A= {a
1
,a
2
,···,a
8
},
A=








010010010010
001010001010
000010101010








,
-ω
i
≡1,i=1,2,···,8.dž¯K(FW)£ã´R
3
˜m¥,¦•N8‡º:ålÚ
•:‹I,dáNAÛ•£´•,•`)´x
∗
= (5,5,5)
T
.
e5,|^Ž{1(PŠA1)¦)T¯K.s
n
,t
n
,α
n
,
¯
θ
n
,θ
n
ëêŠÓ52,š*ÜNT
Ó(23)ª,ЩŠx
0
,x
1
dMatlab ¼ê10∗rand(3,1) ‘Å),u=0.5(x
0
+x
1
),•ŒS“g
ê1000 g,Ø½OK•kx
n+1
−x
n
k≤10
−3
.3MatlabR2020b‚¸e,A1†©z[12]Ž
{(PŠA2)êŠé'(J„ã1§ã2.
Ù¥,A2ëêÀ•,β
n
=
1
100(n+1)
2
,α
n
=
1
n+1
,u= 0.9x
0
,0 ≤θ
n
≤
¯
θ
n
,
¯
θ
n
=





n−1
n+3
,x
n
= x
n−1
,
min

n−1
n+3
,
10
(n+1)
2
kx
n
−x
n−1
k

,x
n
6= x
n−1
,
Ù{ÓA1.
3Щ:ƒÓ,Ø½OKƒÓcJe,lã1Œ±wÑ,A1,A2ÑÂñu•`)x
∗
=
(5,5,5)
T
.lã2Œ±wÑ,A1S“gê'A2,ùNyA1ƒ'ƒcŽ{äk˜½`³.
DOI:10.12677/aam.2022.1186416096A^êÆ?Ð
N_
Figure1.Iterationprocess
ã1.S“L§ã
Figure2.Errorcomparison
ã2.Øé'ã
5.o(†Ð"
©JÑHilbert ˜m¥.52ÂMann-Halpern Ž{,3˜½^‡ey²TŽ{r
Âñ5.òŽ{A^u¦)äN¯K,|^ꊢ?1`²,¿†©z[12]¥Ž{?1êŠ
é',NyTŽ{äk˜½`³,l3¢SA^¥äk•r(¹5.3™5ïÄ¥,·‚
„òïÄTŽ{Âñ„Ç.
ë•©z
[1]Mann,W.R.(1953)MeanValueMethodsinIteration.ProceedingsoftheAmericanMathe-
maticalSociety,4,993-993.https://doi.org/10.2307/2031845
[2]Reich,S.(1979)WeakConvergenceTheoremsforNonexpansiveMappingsinBanachSpaces.
JournalofMathematicalAnalysisandApplications,67,274-276.
https://doi.org/10.1016/0022-247X(79)90024-6
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[4]Wittmann,R.(1992)ApproximationofFixedPointsofNonexpansiveMappings.Archivder
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[7]Tan, B.,Xu, S.and Li,S.(2020) ModifiedInertialHybridand ShrinkingProjectionAlgorithms
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DOI:10.12677/aam.2022.1186416098A^êÆ?Ð

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