惯性广义Mann-Halpern算法及其应用 Inertial Generalized Mann-Halpern Algorithm and Its Application

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AdvancesinAppliedMathematics A^êÆ?Ð,2022,11(8),6087-6098

PublishedOnlineAugust2022inHans.//www.abtbus.com/journal/aam

https://doi.org/10.12677/aam.2022.118641

. 52 ÂMann-HalpernŽ{9ÙA^

NNN ___

 H“‰ŒÆêÆÆ § H&²

Â vFÏµ2022c726F¶¹^FÏµ2022c819F¶uÙFÏµ2022c829F

Á ‡

 ©Ì‡ïÄHilbert˜m¥š*ÜNØ Ä:¯K"JÑ˜«.52ÂMann-HalpernŽ

{ "3˜½^‡ey²Ž{rÂñ5"òŽ{A^u¦)Fermat-Weber½¯K§¿‰Ñ

ê Š¢ (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

,2022;accepted:Aug.19

,2022;published:Aug.29

,2022

Abstract

We considered the ﬁxed point problem of nonexpansive mapping in Hilbert space.We

proposedaninertialgeneralizedMann-Halpernalgorithm.Givingcertainconditions,

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

 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Ø Ä:¯K.H“L Hilbert˜m,h·,·i,k·k©O “L HþS

È 9Ùp‰ê.C⊂H´š˜4à8,T:C→C´š*ÜN.

Mann 3©z[1] ¥JÑ²;Mann Ž{, éux

∈ C,α∈ (0,1) ,ÙS“/ª•:

n +1

= αx

+(1 −α) Tx

,n = 0, 1, 2, ···. (1)

Reich 3©z[2] ¥JÑ˜„/ªMann Ž{, éux

∈ C,{ α

}⊂ (0,1),ÙS“/ª•:

n +1

= α

+(1 −α

) Tx

,n = 0, 1, 2, ···. (2)

3 ˜½^‡ey²{x

} fÂñTØÄ:.•„©z[2].

Halpern 3©z[3] ¥JÑHalpern Ž{, éux

,u ∈C ,{α

}⊂ (0,1),ÙS“/ª•:

n +1

= α

u +(1−α

) Tx

,n = 0, 1, 2, ···. (3)

y ²{x

} rÂñTØÄ:7‡^‡´:lim

n →∞

= 0,

∞

n =0

= ∞.

Wittmann 3©z[4] ¥y²Ž{(3) ¥S{x

} rÂñTØÄ:¿©^‡´:

lim

n →∞

= 0,

∞

n =0

= ∞, …

∞

n =0

| α

n +1

− α

| <∞ .

DOI:10.12677/aam.2022.1186416088 A^êÆ?Ð

N _

, ˜•¡,.5Ž{•@´dPolyak3©z[5]¥JÑ,TŽ{3\¯Âñ„Ç•¡åX-

‡ Š^.e˜ ÚS“dcüÚS“,=

n +1

= x

+ θ

( x

− x

n −1

) ,n= 0 ,1 ,2 ,···.(4)

Ù ¥x

∈ C,{ θ

}⊂ [0,1).Cc5,é.5Ž{kXŒþïÄ,~X,.5c©Ž{[6],.

5 ÝKŽ{[7],.5 MannŽ{[8–12].

Mainge 3©z[8] ¥JÑ.5Mann Ž{, éux

∈ C,{ α

}⊂ (0,1),ÙS“/ª•:







= x

+ θ

( x

− x

n −1

) ,

n +1

= α

+(1 −α

) Tw

,n = 0, 1, 2, ···.

(5)

3 ˜½^‡e,y²Ž{(5)fÂñ5.•„©z[8].

Tan <3©z[12] ¥JÑ?.5Mann-Halpern Ž{, éux

,u ∈C ,{α

} ,{ β

}⊂

(0 ,1), ÙS“/ª•:











= x

+ θ

( x

− x

n −1

) ,

= β

+(1 −β

) Tw

n +1

= α

u +(1−α

) y

,n = 0, 1, 2, ···.

(6)

3 ˜½^‡e,y²Ž{(6)äkrÂñ5.•„©z[12].

 ©3©z[12]Ä:þ,JÑ.52ÂMann-HalpernŽ{,S“/ª•:











= x

+ θ

( x

− x

n −1

) ,

= s

+ t

n +1

= α

u +(1−α

) y

,n = 0, 1, 2, ···.

(7)

Ù ¥x

,u ∈C ,{θ

}⊂ [0,1),{ s

} ,{ t

} ,{ α

}⊂ (0,1).Ž{(7)3ëêÀþ•\(¹:

+ t

≡ 1ž,TŽ{òz¤Ž{(6).3˜½^‡e,·‚òy²Ž{rÂñ5.

2. ý•£

 !Ì‡‰Ñ˜½ny²¥I‡^ÚnÚÄ Vg.

½ Â1.NT:C→C,XJé?¿x,y∈C,Ñk

k Tx− Tyk≤k x− yk ,

K ¡T´Cþš*Ü N.XJ•3 x∈C,¦x=Tx,K ¡x´TØÄ:,PT¤k

Ø Ä:¤8Ü•Fix(T).

5 1.©¥,XÃAÏ`²,þbT´š*ÜN,¿…Fix(T)6=∅.

DOI:10.12677/aam.2022.1186416089 A^êÆ?Ð

N _

½ Â2.HÝ˜mH

∗

:= {g: H→R



g •HþëY‚5•¼}.x ∈H,S{x

}⊂H ,

X Jé? ¿g∈H

∗

Ñ klim

n →∞

g (x

) = g( x) ,K¡{x

} fÂñux,P¤x

*x, (n →∞);XJ

lim

n →∞

k x

− xk = 0,

K ¡{x

} rÂñux,P¤x

→ x,(n→∞ ).

Ú n1.(©z[13]íØ4.15)C⊂H´š˜4à8,T:C→C´š*ÜN,KFix(T)´4à

8 .

Ú n2.(©z[13]Ún2.37){x

} ´H ¥k.S,K{ x

} •3fÂñf.

Ú n3.(©z[14]Ún2.1)é?¿x,y∈H,±e¯¢¤á:

(1) kx+ yk

≤k xk

+2 hy,x+ yi;

(2) kax+ byk

= a( a+ b) kxk

+ b( a+ b) kyk

− abk x− yk

, ∀a,b ∈R.

Ú n4.(©z[15]Ún8){b

} ,{ η

} ´šK¢ê,{ δ

} ,{ ξ

} ´¢ê,{ γ

}⊂ (0,1),¿…

n +1

≤ (1− γ

) b

+ γ

, 9b

n +1

≤ b

− η

+ ξ

,n = 0, 1, 2, ···.

(1)

∞

n =0

= ∞,(2)lim

n →∞

= 0,

(3) {η

} ?¿÷vlim

k →∞

= 0 f{η

} %¹limsup

k →∞

≤ 0,

K lim

n →∞

= 0.

Ú n5.(©z[13]íØ4.18)C⊂H´š˜4à8,T:C→C´š*ÜN,{x

}⊂ C,x∈H .

X Jn→∞ž,kx

*x ,…(x

− Tx

) →0, K( I−T) x= 0.

Ú n6.(©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. Ì‡(J

Ž {1.b{θ

}⊂ [0,1),{ s

} ,{ t

} ,{ α

}⊂ (0,1),÷vs

+ t

≤ 1.éux

,u ∈C ,Ž{S“

/ ª•:











= x

+ θ

( x

− x

n −1

) ,

= s

+ t

n +1

= α

u +(1−α

) y

,n = 0, 1, 2, ···.

(8)

½ n1.C⊂H´š˜4à8,T:C→C´š*ÜN,…Fix(T)6=∅.éux

,u ∈C ,S

{ x

} dŽ{1S“),¿…±e^‡¤á

(1)lim

n →∞

= 0 ,

∞

n =0

= ∞,

DOI:10.12677/aam.2022.1186416090 A^êÆ?Ð

N _

(2)lim

n →∞

k x

− x

n −1

= 0,

(3)

∞

n =0

(1 −s

− t

) <∞,

(4)lim

n →∞

1 −s

− t

= 0 .

K {x

} rÂñTØÄ:p,¿…p=P

Fix( T)

u .

yyy ²²².1˜Ú:ky²{x

} k..p∈ Fix(T),K

k x

n +1

− pk =k α

u +(1−α

) y

− pk

= kα

( u−p)+(1 −α

)( y

− p)k

≤ α

k u− pk +(1− α

) ky

− pk .(9)

þ ª¥

k y

− pk =k s

+ t

− pk

= ks

( w

− p)+t

( Tw

− p)+(s

+ t

− 1)pk

≤ s

k w

− pk +t

k Tw

− pk +(1− s

− t

) kpk

≤ s

k w

− pk +t

k w

− pk +(1− s

− t

) kpk

= ( s

+ t

) kw

− pk +(1− s

− t

) kpk

≤k w

− pk +(1− s

− t

) kpk.(10)

ò (10)ª‘\(9)ª¥,Œ

k x

n +1

− pk≤ (1− α

) kw

− pk +α

k u− pk +(1− α

)(1 −s

− t

) kpk.(11)

þ ª¥

k w

− pk =k x

+ θ

( x

− x

n −1

) −pk≤kx

− pk +θ

k x

− x

n −1

k .(12)

ò (12)ª‘\(11)ª¥,Œ

k x

n +1

− pk

≤ (1− α

) kx

− pk +α

k u− pk +(1− α

) θ

k x

− x

n −1

k +(1− α

)(1 −s

− t

) kpk

= (1 −α

) kx

− pk +α

k u− pk +(1− α

)

k x

− x

n −1

+(1 −α

)(1 −s

− t

) kpk.

DOI:10.12677/aam.2022.1186416091 A^êÆ?Ð

N _

3 þª¥,-M=2max

k u− pk ,sup

n ≥0

(1 −α

)

k x

− x

n −1

, d(1 −α

) <1 Ú^‡(2) Œ•

M< ∞,K

k u− pk +(1− α

)

k x

− x

n −1

≤ α

¤ ±

k x

n +1

− pk≤ (1− α

) kx

− pk +α

M +(1−α

)(1 −s

− t

) kpk.

d à|Ü5ŸŒ•

k x

n +1

− pk≤ max

k x

− pk ,M

+(1 −α

)(1 −s

− t

) kpk.

2 d(1−α

) <1, Œ•

k x

n +1

− pk≤ max

k x

− pk ,M

+(1 −s

− t

) kpk.

ò þª'un?18B,Œ

k x

n +1

− pk≤ max

k x

− pk ,M

j =0

(1 −s

− t

) kpk.(13)

( Ü^‡(3)Œ•

j =0

(1 −s

− t

) kpk≤

∞

n =0

(1 −s

− t

) kpk<∞,

 d(13)ªŒ•

k x

n +1

− pk

k .,l{x

} k..

? ˜Ú,Ï•kw

k≤k x

k +θ

k x

− x

n −1

k ,¤±{ w

} k.,dT´š*ÜN•k Tw

− pk≤

k w

− pk ,l{ Tw

} k..

d {y

} ½ÂÚs

+ t

≤ 1•k y

k≤ s

k w

k +t

k Tw

k≤ max

k w

k ,k Tw

, l{y

} k

. ,2dT´š*ÜN•{Ty

} k..

1 Ú:2y²{x

} rÂñp=P

Fix( T)

u .Ï•

k x

n +1

− pk

= kα

u +(1−α

) y

− pk

= k(1 −α

)( y

− p)+α

( u−p) k

≤ (1− α

)

k y

− pk

+2 hα

( u−p) ,x

n +1

− pi

≤ (1− α

) ky

− pk

+2 α

h u− p,x

n +1

− pi .(14)

DOI:10.12677/aam.2022.1186416092 A^êÆ?Ð

N _

(14) ª¥1˜‡ØªdÚn3 ¯¢(1) , …

k y

− pk

= ks

+ t

− pk

= ks

( w

− p)+t

( Tw

− p)+(s

+ t

− 1)pk

≤k s

( w

− p)+t

( Tw

− p)k

+2 h( s

+ t

− 1)p,y

− pi

= s

( s

+ t

) kw

− pk

+ t

( s

+ t

) kTw

− pk

− s

k w

− Tw

+2 h( s

+ t

− 1)p,y

− pi

≤ s

k w

− pk

+ t

k Tw

− pk

− s

k w

− Tw

+2( s

+ t

− 1)h p,y

− pi

≤ s

k w

− pk

+ t

k w

− pk

− s

k w

− Tw

+2( s

+ t

− 1)h p,y

− pi

= ( s

+ t

) kw

− pk

− s

k w

− Tw

+2( s

+ t

− 1)h p,y

− pi

≤k w

− pk

− s

k w

− Tw

+2( s

+ t

− 1)h p,y

− pi .(15)

(15) ª¥1˜‡ØªdÚn3 ¯¢(1) , 1n‡ªdÚn3 ¯¢(2) .

ò (15)ª‘\(14)ª¥,Œ

k x

n +1

− pk

≤ (1− α

) kw

− pk

− (1− α

) s

k w

− Tw

+2(1 −α

)( s

+ t

− 1)h p,y

− pi +2α

h u− p,x

n +1

− pi .(16)

þ ª¥

k w

− pk

= kx

+ θ

( x

− x

n −1

) −pk

= kx

− p+θ

( x

− x

n −1

) k

= kx

− pk

+ θ

k x

− x

n −1

+2 θ

h x

− p,x

− x

n −1

i .(17)

ò (17)ª‘\(16)ª¥,Œ

k x

n +1

− pk

≤ (1− α

) kx

− pk

+(1 −α

) θ

k x

− x

n −1

+2(1 −α

) θ

h x

− p,x

− x

n −1

− (1− α

) s

k w

− Tw

+2(1 −α

)( s

+ t

− 1)h p,y

− pi +2α

h u− p,x

n +1

− pi .

(18)

3 (18)ª¥-b

= kx

− pk

, η

= (1 −α

) s

k w

− Tw

, γ

= α

(1 −α

)

k x

− x

n −1

2(1 −α

)

h x

− p,x

− x

n −1

i +

2(1 −α

)

( s

+ t

− 1)h p,y

− pi +2h u− p,x

n +1

− pi .

- ξ

= α

, K

= (1 −α

) θ

k x

− x

n −1

+2(1 −α

) θ

h x

− p,x

− x

n −1

+2(1 −α

)( s

+ t

− 1)h p,y

− pi +2α

h u− p,x

n +1

− pi .

DOI:10.12677/aam.2022.1186416093 A^êÆ?Ð

N _

l (18)ªŒU¤

n +1

≤ (1− γ

) b

+ γ

, 9b

n +1

≤ b

− η

+ ξ

,n = 0, 1, 2, ···.

∞

n =0

= ∞•

∞

n =0

= ∞.

3 ξ

¥ ,Ï•lim

n →∞

= 0 Ú1˜Ú(Ø{x

} k.,¤±

lim

n →∞

2 α

h u− p,x

n +1

− pi = 0.

d ^‡(2)•lim

n →∞

k x

− x

n −1

k = 0,¤±

lim

n →∞

k x

− x

n −1

= 0 ,lim

n →∞

2(1 −α

) θ

h x

− p,x

− x

n −1

i =0.

d ^‡(3)•lim

n →∞

(1 −s

− t

) = 0, ¤±lim

n →∞

( s

+ t

− 1) = 0,(Ü1˜Ú(Ø{ y

} k.,l

lim

n →∞

2(1 −α

)( s

+ t

− 1)h p,y

− pi = 0.

n ÜŒ•

lim

n →∞

= 0 .

( ÜÚn4,–dTÚn^‡(1),(2)®÷v, e5¦y^‡(3)÷v=Œ.

¯¢ þ,?{η

} f{ η

} ,¦lim

k →∞

=0. d{α

} ,{ s

} ,{ t

}⊂ (0,1)Úη

½ Â,Œ

•

lim

k →∞

k w

− Tw

k = 0.(19)

d lim

n →∞

k x

− x

n −1

k = 0,Œ•

k w

− x

k =k x

+ θ

( x

− x

k −1

) −x

k =θ

k x

− x

k −1

k→ 0,(k→∞ ).(20)

Ï •{x

} k.,¤±{ x

} k.,¤±dÚn2••3{ x

} f{ x

} ÷v

* ¯x,(j→∞)9limsup

k →∞

h u− p,x

− pi =lim

j →∞

h u− p,x

− pi .

d (20)ªŒ•,w

* ¯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 →∞

h u− p,x

− pi =lim

j →∞

h u− p,x

− pi =h u− p,¯x−pi≤0.

DOI:10.12677/aam.2022.1186416094 A^êÆ?Ð

N _

 e5,y²limsup

k →∞

h u− p,x

k +1

− pi≤ 0.Ï•

k y

− w

k =k s

+ t

− w

k =k t

( Tw

− w

)+( s

+ t

− 1)w

≤ t

k Tw

− w

k +(1− s

− t

) kw

k .

¤ ±d(19)ª,t

⊂ (0,1),lim

n →∞

(1 −s

− t

) = 0 ±9{w

} k.Œ•

lim

k →∞

k y

− w

k = 0.(21)

Ï •ky

− x

k≤k y

− w

k +k w

− x

k ,¤±d(20)ªÚ(21)ªŒ•

lim

k →∞

k y

− x

k = 0.(22)

q Ï•

k x

k +1

− x

k =k α

u +(1−α

) y

− x

k≤ α

k u− x

k +(1− α

) ky

− x

k .

¤ ±dlim

n →∞

= 0, {x

} k.Ú(22)ªŒ•

lim

k →∞

k x

k +1

− x

k = 0.

¤ ±

limsup

k →∞

h u− p,x

k +1

− pi≤ 0.

2 (Ü½n^‡(2),^‡(4)Úδ

½ Â

Œ •

limsup

k →∞

≤ 0.

– dy Ún4^‡(3)÷v.

n þ,dÚn4Œ,lim

n →∞

= 0, =

lim

n →∞

= p.

– d,¤y ².

5 2.e¡‰Ñ÷v½n^‡ëêŠ«~:

 α

, s

= t

−

, 0≤θ

≤

, Ù¥











n −1

n +2

= x

n −1

;

min



n −1

n +2

( n+1)

k x

− x

n −1



6 =x

n −1

DOI:10.12677/aam.2022.1186416095 A^êÆ?Ð

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4. Ž{3Fermat-Weber ½¯K¥A^

 !Ì‡•ÄH¥XeFermat-Weber½¯K:=Ïéx∈R

, ¦)

min

f (x ) =

i =1

k x− a

, (FW)

Ù ¥,ω

> 0,i = 1, 2, ···,m ´-,a

∈ R

´ ‰½e:.





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

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T (·) =

i =1

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i =1

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n +1

= Tx

,n = 0, 1, 2, ···.

(23)

Š •A~,·‚•ÄR

˜ m¥,m= 8œ/,-A={a

, ···,a

} ,

A =







010010010010

001010001010

000010101010







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≡ 1,i=1,2,··· ,8.dž¯K(FW)£ã´R

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• :‹ I,dáNAÛ•£´•,• `)´x

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,α

,θ

 ëêŠÓ52,š*ÜNT

Ó (23)ª,Ð©Šx

d Matlab¼ê10∗rand(3,1)‘Å),u=0.5(x

+ x

), •ŒS“g

ê 1000g,Ø½OK•kx

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, u= 0 .9 x

,0 ≤θ

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n +3

= x

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n −1

n +3

( n+1)

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− x

n −1

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6 =x

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∗

(5 ,5 ,5)

. lã2 Œ±wÑ,A1 S“gê'A2 , ùNyA1 ƒ'ƒcŽ{äk˜½`³.

DOI:10.12677/aam.2022.1186416096 A^êÆ?Ð

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Figure1. Iterationprocess

ã 1.S“L§ã

Figure2. Errorcomparison

ã 2.Øé'ã

5. o(†Ð"

é ',NyTŽ{äk˜½ `³,l3¢SA^¥äk•r (¹5.3™5 ïÄ¥,·‚

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ë •©z

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