Parallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappings

0.2 MB·

by  Dang Van Hieu

Noname manuscript No.
(will be inserted by the editor)
Parallel hybrid methods for generalized
equilibrium problems and asymptotically strictly
pseudocontractive mappings
Dang Van Hieu
6
1
Received: date/Accepted: date
0
2

n Abstract Inthispaper,weproposetwonovelparallelhybridmethodsforfinding
a
a common element of the set of solutions of a finite family of generalized equilib-
J
0 riumproblemsformonotonebifunctions{fi}Ni=1andα-inversestronglymonotone
1 operators{Ai}Ni=1 andthesetofcommonfixedpointsofafinitefamilyof(asymp-
totically) κ- strictly pseudocontractive mappings {Sj}Mj=1 in Hilbert spaces. The
] strong convergence theorems are established under the standard assumptions im-
C
posedonequilibriumbifunctionsandoperators.Anumericalexampleispresented
O
to illustratethe efficiency of the proposed parallelmethods.
.
h Keywords Hybrid method · Equilibrium problem · Strictly pseudocontractive
t
a mapping · Parallelcomputation.
m
Mathematics Subject Classification (2010) 65Y05 . 91B50 . 47H09
[

1
v 1 Introduction
8
1
Let H be a real Hilbert space with the inner product h.,.i and the induced norm
2
2 ||.|| and C be a nonempty closed convex subset of H. Let f : C ×C → ℜ be a
0 bifunction and A : C → H be an α - inverse strongly monotone operator. The
. generalized equilibrium problem (GEP) for the bifunction f and the monotone
1
operator A is defined as follows:
0
6
1 Find x∈C such that: f(x,y)+hAx,y−xi≥0, ∀y ∈C. (1)
:
v
Theset ofsolutionsof (1) is denoted by GEP(f,A).The GEP(1) is verygeneral
i
X in the sense that, it includes, as special cases, many mathematical models: opti-
r mization problems, saddle point problems, Nash equilirium point problems, fixed
a
DangVanHieu
DepartmentofMathematics, VietnamNationalUniversity,HaNoi,Vietnam
334-NguyenTraiStreet, HaNoi,VietNam
Tel.:[phone]
E-mail:[email protected]
2 D.V.Hieu
point problems, convex differentiable optimization problems, variational inequal-
ities, complementarity problems, see e.g., [7,20]. In recent years, some methods
have been proposed for finding a point in the solution set GEP(f,A), see [17,18,
28]. We givetwo special cases for the GEP (1):
If f =0 then the GEP (1) becomes the variationalinequalityproblem
Find x∈C such that: hAx,y−xi≥0, ∀y ∈C (2)
If A=0 then the GEP (1) becomes the equilibrium problem
Find x∈C such that: f(x,y)≥0, ∀y ∈C. (3)
The sets of solutions of (2) and (3) are denoted by VI(A,C) and EP(f,C), re-
spectively.
Let S :C →C be a nonexpansive mapping with the set of fixed points F(S).
In 2003,Nakajoand Takahashi[21] introduced thefollowinghybrid algorithmfor
finding a fixed point of the nonexpansivemapping S in Hilbert spaces
x ∈C :=C,
0 0
yn =αnxn+(1−αn)Sxn,

 QCnn=={{zz∈∈CC::|h|xynn−−zz,||x≤0−||xxnn−i≥z|0|}},,
In 2010, Duan [11]proxpno+s1ed=aPChny∩bQrindxa0,lgnor≥it0h.m for finding a common element
of the solution set ∩Mi=1EP(fi,C) of equilibrium problems for the monotone
bifunctions {fi}Mi=1(cid:16)and the fixed po(cid:17)int set ∩Nj=1F(Sj) of the strictly pseudo-
contractive mappings {Sj}Nj=1 in Hilbert sp(cid:16)aces which c(cid:17)ombines three methods
including the proximal method [9], the Mann iteration [19] and the monotone
hybrid (outer approximation)method. Precisely,
x ∈C :=C,
0 0
 un =TrfMM,n...Trf11,nxn,
 Cynn+=1α=nx{zn∈+C(1n−:|α|ynn)(cid:0)−λzn|u|n≤+||(x1n−−λzn||)}S,n(mod)Nun(cid:1), (4)
Clearly,Duan’sxanl+go1r=ithPmCnis+1inxh0e,rnen≥tl0y.sequential.Thusitcanbecostlyonasingle
processor when the numbers of bifunctions M and of strictly pseudocontractive
mappings N are large.
Very recently, Anh and Hieu [4,5] have proposed the followingparallelhybrid
algorithmforfindinga commonfixed pointof afinitefamilyasymptoticallyquasi
φ - nonexpansive mappings {Ti}Ni=1 in uniformly smooth and uniformly convex
Banach spaces
x ∈C, C :=C,
0 0
 yni =J−1(αnJxn+(1−αn)JTinxn),i=1,2,...,N,
 iCnn=+1a:r=gm{vax∈1≤Cin≤N:φn((cid:13)(cid:13)vy,yn¯in−)≤xnφ(cid:13)(cid:13)(ov,,xny¯)n+:=ǫny}ni,n,
 xn+1 =ΠCn+1x0,n≥(cid:13)0, (cid:13)
ParallelMethodsforGEPsandFPPs 3
where ǫn := (kn −1)(ω+||xn||)2, {αn} ⊂ [0,1], lim αn = 0. Note that [1,10]
n→∞
in Hilbert spaces the normalizeduality mapping J is the identity operator I, the
Lyapunov funtional φ(x,y) = ||x−y||2, and the generalized projection Π =
C
P . Arccoding to this algorithm, itermadiate approximations yi can be found
C n
in parallel, among all yni the furthest element from xn, denoted by y¯n, is chosen.
Afterthat,basedony¯n,theclosedconvexsetCn+1isconstructed.Finally,thenext
approximationxn+1 isdefinedastheprojectionofx0 ontoCn+1.Somenumerical
experiments (see [3,4,5]) have implied the efficency of this parallel algorithm.
Moreover, it can be used to solve systems of monotone operator equations in
Hilbert spaces or accretive operator equations in Banach spaces. Other parallel
algorithms for finding a common solution of a finite family of accretive operator
equations in Banach spaces can be found in [2,3,13].
In this paper, motivated and inspired by above results we propose two new
parallelhybrid algorithmsfor finding a commonelement of the set of solutions of
a finite family of GEPs for bifunctions {fi}Ni=1 and operators {Ai}Ni=1, and the
set of common fixed points of finitely many (asymptotically)κ - strictly pseudo-
contractivemappings{Sj}Mj=1 inHilbertspaces.Thestrongconvergencetheorems
are proved under the widely used assumptions of equilibriumbifunctions and op-
erators.
Thispaperisorganizedasfollows:InSection2wecollectsomedefinitionsand
primary results used in the next sections. In Section 3 we propose two parallel
hybrid algorithms and prove their convergence. Finally, Section 4 presents a nu-
mericalexampletoillustratetheefficiencyofparallelcomputationoftheproposed
algorithms.
2 Preliminaries
In this section we recall some definitions and results for further use. Let C be a
nonempty closed and convex subset of a real Hilbert space H.
Definition 1 [8,12,23] A mapping S :C →C is said to be
i. nonexpansive if ||Sx−Sy||≤||x−y|| for all x,y ∈C;
ii. asymptotically nonexpansive if there exists a sequence {kn} ⊂ [1;+∞) with
kn →1 such that
||Snx−Sny||≤kn||x−y||, ∀x,y ∈C,n≥1;
iii. κ - strictly pseudocontractive if there exists a constant κ∈[0;1) such that
||Sx−Sy||2 ≤||x−y||2+κ||(I−S)x−(I−S)y||2, ∀x,y ∈C;
iv. asymptotically κ - strictly pseudocontractive if thereexist a constantκ∈[0;1)
and a sequence {kn}⊂[1;+∞) with kn →1 such that
||Snx−Sny||2 ≤kn||x−y||2+κ||(I−Sn)x−(I−Sn)y||2,∀x,y ∈C,n≥1;
The class of κ - strictly pseudocontractive mappings was introduced by Browder
andPetryshyn[8]in1967.Clearly,eachnonexpansivemappingis0-strictlypseu-
docontractive.Theclassofasymptoticallyκ-strictlypseudocontractivemappings
4 D.V.Hieu
[23] is a generalization of the one of κ - strictly pseudocontractive mappings. A
mentionedexamplein[26]showsthattheclassofasymptoticallyκ-strictlypseu-
docontractivemappingscontainsproperlytheoneofκ-strictlypseudocontractive
mappings. We have the followingresult [8].
Lemma 1 [27] Let H be a real Hilbert space and C be a nonempty closed convex
subset of H. Let S : C → C be an asymptotically κ-strict pseudocontraction with
the sequence {kn}⊂[1;∞),kn →1. Then
i. F(S) is a closed convex subset of H.
ii. I−S is demiclosed, i.e., whenever {xn} is a sequence in C weakly converging
to some x∈C and the sequence {(I−S)xn} strongly converges to some y, it
follows that (I−S)x=y.
iii. S is uniformly L - Lipschitz continuous with the constant
κ+ 1+(1−κ)(kn−1)
L=sup :n≥1 ,
( p 1+κ )
i.e., ||Snx−Sny||≤L||x−y|| for all x,y ∈C and n≥1.
Lemma 2 [25] In a real Hilbert space H, the following equality holds
||ax+(1−a)y||2=a||x||2+(1−a)||y||2−a(1−a)||x−y||2, ∀x,y ∈H,a∈[0,1].
Definition 2 A mapping A:C →H is said to be
i. monotone if hAx−Ay,x−yi≥0, ∀x,y ∈C;
ii. η - strongly monotone if there exists a constantη >0 such that
hAx−Ay,x−yi≥ηkx−yk2, ∀x,y ∈C;
iii. α - inverse strongly monotone if there exists a constantα>0 such that
hAx−Ay,x−yi≥αkAx−Ayk2, ∀x,y ∈C.
Remark 1 AmappingS :C →C isκ-strictlypseudocontractiveiffA=I−S isα
- inversestronglymonotone(0<α<1 and κ=1−2α) and is pseudocontractive
iff A=I−S is monotone.
Remark 2 IfAisη-stronglymonotoneandL-Lipschitzcontinuous,i.e.,kAx−Ayk
≤Lkx−ykforallx,y ∈C thenAisη/L2-inversestronglymonotone.IfT isnon-
expansivethenA=I−T is1/2-inversestronglymonotoneandVI(A,C)=F(T).
Remark 3 IfA:C →H isα-inversestronglymonotonethenAis1/α-Lipschitz
continuous and I−λA is nonexpansive,where λ∈(0,2α).
Indeed, from hAx−Ay,x−yi≥αkAx−Ayk2, we obtain
||Ax−Ay||||x−y||≥αkAx−Ayk2.
This implies that kAx−Ayk ≤ 1/α||x−y||. Therefore A is 1/α - Lipschitz con-
tinuous. Moreover,
||(I−λA)x−(I−λA)y||2 =||x−y||2+λ2||Ax−Ay||2−2λhAx−Ay,x−yi
≤||x−y||2+λ2||Ax−Ay||2−2λα||Ax−Ay||2
=||x−y||2−λ(2α−λ)||Ax−Ay||2
≤||x−y||2.
ParallelMethodsforGEPsandFPPs 5
Hence, I−λA is nonexpansive.
For every x∈H, the element P x is defined by
C
P x=argmin{ky−xk:y ∈C}.
C
Since C is a nonempty closed convex subset of H, P x exists and is unique. The
C
mappingP :H →C is called themetricprojectionof H onto C.It is also well-
C
known that P satisfies the followingproperty
C
hP x−P y,x−yi≥kP x−P yk2, (5)
C C C C
which implies that P is 1 - inverse strongly monotone,and for all x∈C,y ∈H,
C
kx−P yk2+kP y−yk2 ≤kx−yk2. (6)
C C
Moreover, z =P x if and only if
C
hx−z,z−yi≥0, ∀y ∈C. (7)
For solving the GEP (1), we assume that the bifunction f satisfies the following
conditions:
(A1) f(x,x)=0 for all x∈C;
(A2) f is monotone,i.e., f(x,y)+f(y,x)≤0 for all x,y ∈C;
(A3) for all x,y,z ∈C,
limsupf(tz+(1−t)x,y)≤f(x,y);
t→0
(A4) for each x∈C, the function f(x,.) is convex and lower semicontinuos.
The followingresults concern with the befunction f.
Lemma 3 [9] Let C be a closed convex subset of a Hilbert space H, f be a bifunc-
tion from C ×C to ℜ satisfying the conditions (A1)-(A4) and let r > 0, x ∈ H.
Then, there exists z ∈C such that
1
f(z,y)+ hy−z,z−xi≥0, ∀y ∈C.
r
Lemma 4 [9] Let C be a closed convex subset of a Hilbert space H, f be a bi-
function from C ×C to ℜ satisfying the conditions (A1)-(A4). For all r > 0 and
x∈H, define the mapping
1
Tfx={z ∈C :f(z,y)+ hy−z,z−xi≥0, ∀y ∈C}.
r r
Then the following hold:
(B1) Tf is single-valued;
r
(B2) Tf is a firmly nonexpansive, i.e., for all x,y ∈H,
r
||Tfx−Tfy||2 ≤hTfx−Tfy,x−yi;
r r r r
(B3) F(Tf)=EP(f,C);
r
(B4) EP(f,C) is closed and convex.
6 D.V.Hieu
3 Main results
In this section, we propose two parallel hybrid algorithms for finding a common
elementofthesetofsolutionsofafinitefamilyofGEPsformonotonebifunctions
{fi}Ni=1 and α - inversestronglymonotonemappings{Ai}Ni=1 and theset of com-
monfixedpointsofafinitefamilyof(asymptotically)κ-strictlypseudocontractive
mappings {Sj}Mj=1 in Hilbert spaces. We assume that the mappings {Ai}Ni=1 are
inverse strongly monotone with the same constant α and {Sj}Mj=1 are asymptot-
ically κ - strictly pseudocontractive mappings with the same sequence {kn} ⊂
[1;+∞),kn → 1 and constant κ ∈ [0;1). Indeed, if Ai is αi - inverse strongly
monotone,Ai is α - inverse strongly monotonewith α:=min{αi :i=1,...,N}.
Similarly, suppose that Sj is asymptotically κj - strictly pseudocontractive with
the sequence knj ⊂ [1;+∞),knj → 1. Putting kn = max knj :j =1,...,M
and κ:=maxn{κjo:j =1,...,M}. Then, Sj is asymptoticallynκ - strictlypseudoo-
contractivewith the sequence {kn}⊂[1;+∞),kn →1.
Moreover,we also assume that the solutionset
F := ∩Ni=1GEP(fi,Ai) ∩Mj=1F(Sj)
(cid:16) (cid:17)\(cid:16) (cid:17)
is nonempty and bounded, i.e., there exists a positive real number ω such that
F ⊂Ω :={v∈H :||v||≤ω}.
Algorithm 1 Initialization. Choose x ∈C and set n:=0. The control param-
0
eter sequences {α },{β },{r } satisfy the following conditions.
k k k
(a) 0<α <1, lim supα <1;
k k→∞ k
(b) κ≤β ≤b<1 for some b∈(κ;1);
k
(c) 0<d≤r ≤e<2α.
k
Step 1. Find intermediate approximations yi in parallel
n
yni =Trfni(xn−rnAi(xn)), i=1,...,N.
Step 2. Choose the furthest element from xn among all yni, i.e.,
in =argmax{||yni −xn||:i=1,...,N}, y¯n :=ynin.
Step 3. Find intermediate approximations zj in parallel
n
znj =αnxn+(1−αn) βny¯n+(1−βn)Sjny¯n , j =1,...,M.
Step 4. Choose the furthest elem(cid:0)ent from xn among al(cid:1)l znj, i.e.,
jn =argmax{||znj −xn||:j =1,...,M}, z¯n :=znjn.
Step 5. Construct the closed convex subset Cn+1 of C
Cn+1 ={v∈Cn :||z¯n−v||2 ≤||xn−v||2+ǫn},
where ǫn =(kn−1)(||xn||+ω)2.
Step 6.Thenextapproximationxn+1 isdefinedastheprojectionofx0 ontoCn+1,
i.e.,
xn+1 =PCn+1(x0).
Step 7. Set n:=n+1 and go to Step 1.
ParallelMethodsforGEPsandFPPs 7
Lemma 5 If Algorithm 1reaches totheiteration n≥0then F ⊂Cn+1 andxn+1
is well-defined.
Proof FromLemmas1and4,weseethatGEP(fi,A),i=1,...,N andF(Sj),j =
1,...,M are closed convex subsets. Hence, F is closed and convex. Moreover, we
see that C0 = C is closed and convex. Assume that Cn is closed and convex for
some n≥0. From the definition of Cn+1 we obtain
Cn+1 =Cn∩ v∈H :2hv,xn−z¯ni≤||xn||2−||z¯n||2+ǫn .
n o
Hence,Cn+1isclosedandconvex.Bytheinduction,Cnisclosedandconvexforall
n≥0.Now,weshowthatF ⊂Cn foralln≥0.PuttingSj,βn =βnI+(1−βn)Sjn,
henceznj =αnxn+(1−αn)Sj,βny¯n.FromthenonexpansivenessofTrfnin,theinverse
strongly monotonicityof Ain and the hypothesis of rn, we have, for each u∈F,
||y¯n−u||2 =||Trfnin(xn−rnAinxn)−Trfnin(u−rnAinu)||2
≤||(xn−rnAinxn)−(u−rnAinu)||2
=||xn−u||2+rn2||Ainxn−Ainu||2−2rnhAinxn−Ainu,xn−ui
≤||xn−u||2+rn2||Ainxn−Ainu||2−2rnα||Ainxn−Ain||2
=||xn−u||2−rn(2α−rn)||Ainxn−Ainu||2
≤||xn−u||2. (8)
Therefore, from the convexity of ||.||2 and the asymptoticallyκ - strictly pseudo-
contractivenessof Sjn,
||z¯n−u||2 =||αnxn+(1−αn)Sjn,βny¯n−u||2
≤αn||xn−u||2+(1−αn)||Sjn,βny¯n−u||2
=αn||xn−u||2+(1−αn)||βny¯n+(1−βn)Sjnny¯n−u||2
=αn||xn−u||2
+(1−αn) βn||y¯n−u||2+(1−βn)||Sjnny¯n−Sjnnu||2
−(1−αn)β(cid:16)n(1−βn)||(y¯n−u)−(Sjnny¯n−Sjhnu)||2 (cid:17)
=αn||xn−u||2+(1−αn) βn||y¯n−u||2+(1−βn)kn||y¯n−u||2
+κ(1−αn)(1−βn)||(I−(cid:16)Sjnn)y¯n−(I−Sjnn)u||2 (cid:17)
−(1−αn)βn(1−βn)||(y¯n−u)−(Sjnny¯n−Sjhnu)||2
=αn||xn−u||2+(1−αn) βn||y¯n−u||2+(1−βn)kn||y¯n−u||2
−(βn−κ)(1−αn)(1−β(cid:16)n)||(I−Sjnn)y¯n−(I−Sjnn)u||2 (cid:17)
≤αn||xn−u||2+(1−αn) βn||y¯n−u||2+(1−βn)kn||y¯n−u||2
=αn||xn−u||2+(1−αn)|(cid:16)|y¯n−u||2+(1−βn)(kn−1)||y¯n−u||(cid:17)2
≤αn||xn−u||2+(1−αn)||xn−u||2+(kn−1)||xn−u||2
≤||xn−u||2+(kn−1)(||xn||+||u||)2
≤||xn−u||2+(kn−1)(||xn||+ω)2
=||xn−u||2+ǫn. (9)
8 D.V.Hieu
This implies that u ∈Cn+1 for all u ∈F. Thus, by the induction F ⊂Cn for all
n≥0. Since F is nonempty, so is Cn+1. Hence xn+1 is well-defined. The proof of
Lemma 5 is complete.
Lemma 6 Suppose that {xn}, yni and znj are the sequences generated by Al-
gorithm 1. Then, {xn} is a Caunchyosequennce aond there hold the following relations
lim ||xn−yni||= lim ||xn−znj||= lim ||xn−Sjxn||=0
n→∞ n→∞ n→∞
for all i=1,...,N and j =1,...,M.
Proof FromthedefinitionofCn,wehaveCn+1 ⊂Cn.Moreover,xn+1 =PCn+1(x0)∈
Cn+1. Thus xn+1 ∈Cn. From xn =PCnx0 and the definition of PCn, we obtain
||xn−x0||≤||xn+1−x0||.
This implies that the sequence {||xn−x0||} is nondecreasing. From xn =PCnx0,
we also have ||xn −x0|| ≤ ||u−x0|| for each u ∈ F ⊂ Cn. Thus, the sequence
{||xn−x0||}isbounded.Hence,thereexiststhelimitofthesequence{||xn−x0||}.
For allm≥n≥0, wehavexm ∈Cn. By xn =PCnx0 and the property(6) of the
metric projection,we get
||xm−xn||2 ≤||xm−x0||2−||xn−x0||2.
Letting m,n→∞ in the last inequality,we obtain
lim ||xm−xn||=0. (10)
m,n→∞
Therefore, {xn} is a Cauchy sequence. From xn+1 ∈ Cn+1 and the definition of
Cn+1 we have
||z¯n−xn+1||2 ≤||xn−xn+1||2+ǫn. (11)
From the boundedness of {xn} and kn →1, one has
ǫn =(kn−1)(||xn||+ω)2 →0 (12)
as n→∞. Combining(10),(11),(12) we get
lim ||z¯n−xn+1||=0. (13)
n→∞
From (10),(13) and ||z¯n−xn||≤||z¯n−xn+1||+||xn+1−xn|| we get
lim ||z¯n−xn||=0. (14)
n→∞
From the definition of jn, we see that
lim ||znj −xn||=0 (15)
n→∞
for all j =1,...,M. From znj =αnxn+(1−αn)Sj,βny¯n we obtain
||znj −xn||=(1−αn)||Sj,βny¯n−xn||.
ParallelMethodsforGEPsandFPPs 9
This equality together (15) and limn→∞supαn <1 implies that
nl→im∞||Sj,βny¯n−xn||=0. (16)
For each u∈F, from the firmly nonexpansiveness of Trfnin, we have
2||y¯n−u||2 =2||Trfnin(xn−rnAinxn)−Trfnin(u−rnAinu)||2
≤2h(xn−rnAinxn)−(u−rnAinu),y¯n−ui
=||(xn−rnAinxn)−(u−rnAinu)||2+||y¯n−u||2
−||(xn−rnAinxn)−(u−rnAinu)−(y¯n−u)||2
≤||xn−u||2+||y¯n−u||2−||(xn−y¯n)−rn(Ainxn−Ainu)||2
≤||xn−u||2+||y¯n−u||2−||xn−y¯n||2−rn2||Ainxn−Ainu||2
+2rnhAinxn−Ainu,xn−y¯ni.
Therefore
||y¯n−u||2 ≤||xn−u||2−||xn−y¯n||2−rn2||Ainxn−Ainu||2
+2rnhAinxn−Ainu,xn−y¯ni
≤||xn−u||2−||xn−y¯n||2−rn2||Ainxn−Ainu||2
+2rn||Ainxn−Ainu||||xn−y¯n||
≤||xn−u||2−||xn−y¯n||2+2rn||Ainxn−Ainu||||xn−y¯n||.(17)
By arguing similarlyas in (9), we obtain
||z¯n−u||2 ≤αn||xn−u||2+ǫn+(1−αn)||y¯n−u||2. (18)
The last inequalitytogether with the relation(8) one has
||z¯n−u||2 ≤αn||xn−u||2+ǫn+(1−αn)||y¯n−u||2
≤αn||xn−u||2+ǫn+(1−αn)||xn−u||2
−(1−αn)rn(2α−rn)||Ainxn−Ainu||2
=||xn−u||2+ǫn−(1−αn)rn(2α−rn)||Ainxn−Ainu||2.
Therefore
(1−αn)rn(2α−rn)||Ainxn−Ainu||2 ≤||xn−u||2−||z¯n−u||2+ǫn.
This implies that
(1−αn)rn(2α−rn)||Ainxn−Ainu||2 ≤||xn−z¯n||(||xn−u||+||z¯n−u||)+ǫn.
(19)
From (12),(14),(19) and the boundedness of {xn},{z¯n} we obtain
nl→im∞||Ainxn−Ainu||=0. (20)
From (17) and (18), we get
(1−αn)||xn−y¯n||2 ≤||xn−u||2−||z¯n−u||2+ǫn
+(1−αn)2rn||Ainxn−Ainu||||xn−y¯n||
≤||xn−z¯n||(||xn−u||+||z¯n−u||)+ǫn
+(1−αn)2rn||Ainxn−Ainu||||xn−y¯n||. (21)
10 D.V.Hieu
Combining(12),(14),(20),(21), we have
lim ||xn−y¯n||=0. (22)
n→∞
From the definition of in, we conclude that
lim ||xn−yni||=0, i=1,...,N,
n→∞
From (16) and (22), one has
nl→im∞||Sj,βny¯n−y¯n||=0
or
lim (1−βn)||Sjny¯n−y¯n||=0.
n→∞
Since βn ≤b<1,
lim ||Sjny¯n−y¯n||=0. (23)
n→∞
Thus, it followsfrom the uniformly L - Lipschitz continuityof Sj that
||Sjnxn−xn||≤||Sjnxn−Sjny¯n||+||Sjny¯n−y¯n||+||y¯n−xn||
≤L||xn−y¯n||+||Sjny¯n−y¯n||+||y¯n−xn||
=(L+1)||xn−y¯n||+||Sjny¯n−y¯n||,
where L is defined as in Lemma 1. This together with (22) and (23) implies that
lim ||Sjnxn−xn||=0. (24)
n→∞
From the triangleinequalityand the uniformly L - Lipschitz continuityof Sj,
||Sjxn−xn||≤||Sjxn−Sjn+1xn||+||Sjn+1xn−Sjn+1xn+1||
+||Sjn+1xn+1−xn+1||+||xn+1−xn||
=L||xn−Sjnxn||+(L+1)||xn−xn+1||+||Sjn+1xn+1−xn+1||,
which, from the relations(10) and (24), implies that
lim ||Sjxn−xn||=0. (25)
n→∞
The proof of Lemma 6 is complete.
Lemma 7 Suppose that p is a limit point of {xn} then p∈F.
Proof By Lemma6, {xn} is a Cauchysequencein C.SinceC is closed,xn →p∈
C.FromLemma1weseethatSj isLipschitzcontinuousandsocontinuous.Thus,
the equality(25) gives Sjp=p. Hence p∈F(Sj) for all j =1,...,M.
Next, we show that p ∈∩Ni=1GEP(fi,Ai). Note that limn→∞ yni −xn =0.
This together with rn ≥d>0 implies that (cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
yni −xn
lim =0. (26)
n→∞(cid:13)(cid:13) rn (cid:13)(cid:13)
(cid:13) (cid:13)