Volume 2011, Article ID 840319,25pages doi:10.1155/2011/840319
Research Article
The Shrinking Projection Method for Common
Solutions of Generalized Mixed Equilibrium
Problems and Fixed Point Problems for Strictly
Pseudocontractive Mappings
Thanyarat Jitpeera and Poom Kumam
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand
Correspondence should be addressed to Poom Kumam,[email protected]
Received 21 September 2010; Revised 14 December 2010; Accepted 20 January 2011 Academic Editor: Jewgeni Dshalalow
Copyrightq2011 T. Jitpeera and P. Kumam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We introduce the shrinking hybrid projection method for finding a common element of the set of fixed points of strictly pseudocontractive mappings, the set of common solutions of the variational inequalities with inverse-strongly monotone mappings, and the set of common solutions of generalized mixed equilibrium problems in Hilbert spaces. Furthermore, we prove strong convergence theorems for a new shrinking hybrid projection method under some mild conditions. Finally, we apply our results to Convex Feasibility ProblemsCFP. The results obtained in this paper improve and extend the corresponding results announced by Kim et al.2010 and the previously known results.
1. Introduction
LetHbe a real Hilbert space with inner product·,·and norm · , and letEbe a nonempty closed convex subset of H. Let T:E → E be a mapping. In the sequel, we will useFT
to denote the set offixed pointsof T, that is,FT {x ∈ E : Tx x}. We denote weak convergence and strong convergence by notationsand →, respectively.
LetS:E → Ebe a mapping. ThenSis called
1nonexpansiveif
2strictly pseudocontractivewith the coefficientk∈0,1if
Sx−Sy2≤
x−y2kI−Sx−I−Sy2, ∀x, y∈E, 1.2
3pseudocontractiveif
Sx−Sy2≤
x−y2I−Sx−I−Sy2, ∀x, y∈E. 1.3
The class of strictly pseudocontractive mappings falls into the one between classes of nonexpansive mappings and pseudocontractive mappings. Within the past several decades, many authors have been devoted to the studies on the existence and convergence of fixed points for strictly pseudocontractive mappings. In 2008, Zhou 1 considered a convex combination method to study strictly pseudocontractive mappings. More precisely, take
k∈0,1, and define a mappingSkby
Skxkx 1−kSx, ∀x∈E, 1.4
where S is strictly pseudocontractive mappings. Under appropriate restrictions on k, it is proved that the mappingSk is nonexpansive. Therefore, the techniques of studying
nonex-pansive mappings can be applied to study more general strictly pseudocontractive mappings. Recall that lettingA:E → Hbe a mapping, thenAis called
1monotoneif
Ax−Ay, x−y≥0, ∀x, y∈E, 1.5
2β-inverse-strongly monotoneif there exists a constantβ >0 such that
Ax−Ay, x−y≥βAx−Ay2, ∀x, y∈E. 1.6
Thedomainof the functionϕ:E → Ê∪ {∞}is the set domϕ{x∈E:ϕx<∞}.
Letϕ:E → Ê∪ {∞}be a proper extended real-valued function and letFbe a bifunction of
E×EintoÊsuch thatE∩domϕ /∅, whereÊis the set of real numbers.
There exists thegeneralized mixed equilibrium problemfor findingx∈Esuch that
Fx, yAx, y−xϕy−ϕx≥0, ∀y∈E. 1.7
The set of solutions of1.7is denoted by GMEPF, ϕ, A, that is,
We see thatxis a solution of a problem1.7which implies thatx∈domϕ{x∈E:ϕx< ∞}.
In particular, ifA ≡ 0, then the problem 1.7 is reduced into themixed equilibrium problem2for findingx∈Esuch that
Fx, yϕy−ϕx≥0, ∀y∈E. 1.9
The set of solutions of1.9is denoted by MEPF, ϕ.
IfA ≡ 0 andϕ ≡ 0, then the problem1.7is reduced into theequilibrium problem3
for findingx∈Esuch that
Fx, y≥0, ∀y∈E. 1.10
The set of solutions of1.10is denoted by EPF. This problem contains fixed point problems and includes as special cases numerous problems in physics, optimization, and economics. Some methods have been proposed to solve the equilibrium problem; please consult4,5.
IfF ≡ 0 andϕ ≡ 0, then the problem1.7is reduced into theHartmann-Stampacchia variational inequality6for findingx∈Esuch that
Ax, y−x≥0, ∀y∈E. 1.11
The set of solutions of 1.11 is denoted by VIE, A. The variational inequality has been extensively studied in the literature. See, for example,7–10and the references therein.
Many authors solved the problems GMEPF, ϕ, A, MEPF, ϕ, and EPFbased on iterative methods; see, for instance,4,5,11–25and reference therein.
In 2007, Tada and Takahashi26introduced a hybrid method for finding the common element of the set of fixed point of nonexpansive mapping and the set of solutions of equilibrium problems in Hilbert spaces. Let{xn} and{un} be sequences generated by the
following iterative algorithm:
x1x∈H,
Fun, y
1
rn
y−un, un−xn
≥0, ∀y∈E,
wn 1−αnxnαnSun,
En{z∈H:wn−z ≤ xn−z},
Dn{z∈H:xn−z, x−xn ≥0},
xn1PEn∩Dnx, ∀n≥1.
1.12
Then, they proved that, under certain appropriate conditions imposed on{αn}and{rn}, the
sequence{xn}generated by1.12converges strongly toPFS∩EPFx.
and the set of solutions of variational inequalities with inverse-strongly monotone mappings in Hilbert spaces:
x1∈E,
znPE
xn−μnCxn
,
ynPExn−λnBxn,
xn1nfxn βnxnγn αn1Skxnαn2ynαn3zn
, ∀n≥1.
1.13
Then, they proved that, under certain appropriate conditions imposed on{n},{βn},{γn},
{αn1}, {αn2}, and{αn3}, the sequence {xn}generated by 1.13converges strongly to q ∈
FS∩VIE, B∩VIE, C, whereqPFS∩VIE,B∩VIE,Cfq.
In 2010, Kumam and Jaiboon28introduced a new method for finding a common element of the set of fixed point of strictly pseudocontractive mappings, the set of common solutions of variational inequalities with inverse-strongly monotone mappings, and the set of common solutions of a system of generalized mixed equilibrium problems in Hilbert spaces. Then, they proved that, under certain appropriate conditions imposed on {n}, {βn}, and
{αni}, wherei1,2,3,4,5. The sequence{xn}converges strongly toq∈Θ:FS∩VIE, B∩
VIE, C∩GMEPF1, ϕ, A1∩GMEPF2, ϕ, A2, whereqPΘI−Aγfq.
In this paper, motivate, by Tada and Takahashi26, Qin and Kang27, and Kumam and Jaiboon28, we introduce a new shrinking projection method for finding a common element of the set of fixed points of strictly pseudocontractive mappings, the set of common solutions of generalized mixed equilibrium problems, and the set of common solutions of the variational inequalities for inverse-strongly monotone mappings in Hilbert spaces. Finally, we apply our results to Convex Feasibility ProblemsCFP. The results obtained in this paper improve and extend the corresponding results announced by the previously known results.
2. Preliminaries
LetHbe a real Hilbert space, and letEbe a nonempty closed convex subset ofH. In a real Hilbert spaceH, it is well known that
λx 1−λy2
λx2 1−λy2−λ1−λx−y2, 2.1
for allx, y∈Handλ∈0,1.
For anyx∈H, there exists aunique nearest pointinE, denoted byPEx, such that
x−PEx ≤x−y, ∀y∈E. 2.2
The mappingPEis called themetric projectionofHontoE.
It is well known thatPEis a firmly nonexpansive mapping ofHontoE, that is,
x−y, PEx−PEy
Moreover,PExis characterized by the following properties:PEx∈Eand
x−PEx, y−PEx
≤0,
x−y2≥
x−PEx2y−PEx2
2.4
for allx∈H, y∈E.
Lemma 2.1. LetEbe a nonempty closed convex subset of a real Hilbert spaceH. Givenx∈Hand
z∈E, then,
zPEx⇐⇒
x−z, y−z≤0, ∀y∈E. 2.5
Lemma 2.2. LetHbe a Hilbert space, letEbe a nonempty closed convex subset ofH, and letBbe a mapping ofEintoH. Letu∈E. Then, forλ >0,
u∈VIE, B⇐⇒uPEu−λBu, 2.6
wherePEis the metric projection ofHontoE.
Lemma 2.3see1. LetEbe a nonempty closed convex subset of a real Hilbert spaceH, and let
S : E → Ebe ak-strictly pseudocontractive mapping with a fixed point. ThenFSis closed and convex. DefineSk:E → EbySk kx 1−kSxfor eachx∈E. ThenSkis nonexpansive such
thatFSk FS.
Lemma 2.4see29. LetEbe a closed convex subset of a real Hilbert spaceH, and letS:E → E
be a nonexpansive mapping. ThenI−Sis demiclosed at zero; that is,
xn x, xn−Sxn−→0 2.7
impliesxSx.
Lemma 2.5see30. Each Hilbert spaceHsatisfies the Kadec-Klee property, for any sequence{xn}
withxn xandxn → xtogether implyingxn−x → 0.
Lemma 2.6see31. LetEbe a closed convex subset ofH. Let{xn}be a bounded sequence inH.
Assume that
1the weakω-limit setωwxn⊂E,
2for eachz∈E,limn→ ∞xn−zexists.
Then{xn}is weakly convergent to a point inE.
Lemma 2.7see32. LetEbe a closed convex subset ofH. Let{xn}be a sequence inHandu∈H.
LetqPEu. If{xn}isωwxn⊂Eand satisfies the condition
xn−u ≤u−q 2.8
Lemma 2.8see33. LetEbe a nonempty closed convex subset of a strictly convex Banach space
X. Let{Tn:n∈Æ}be a sequence of nonexpansive mappings onE. Suppose
∞
n1FTnis nonempty.
Letδnbe a sequence of positive number with
∞
n1δn1. Then a mappingSonEdefined by
Sx ∞
n1
δnTnx 2.9
forx∈Eis well defined, nonexpansive, andFS ∞n1FTnholds.
For solving the mixed equilibrium problem, let us give the following assumptions for the bifunctionF, the functionA, and the setE:
A1Fx, x 0 for allx∈E
A2Fis monotone, that is,Fx, y Fy, x≤0 for allx, y∈E A3for eachx, y, z∈E, limt→0Ftz 1−tx, y≤Fx, y
A4for eachx∈E,y→Fx, yis convex and lower semicontinuous
A5for eachy∈E,x→Fx, yis weakly upper semicontinuous
B1for eachx ∈Handr >0, there exists a bounded subsetDx ⊆Eandyx ∈ Esuch
that, for anyz∈E\Dx,
Fz, yx
ϕyx
−ϕz 1 r
yx−z, z−x
<0, 2.10
B2Eis a bounded set.
By similar argument as in the proof of Lemma 2.9 in 34, we have the following lemma appearing.
Lemma 2.9. LetEbe a nonempty closed convex subset ofH. LetF : E×E → Ê be a bifunction that satisfies (A1)–(A5), and letϕ:E → Ê∪ {∞}be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. Forr >0andx∈H, define a mappingTF
r :H → E
as follows:
TrFx
z∈E:Fz, yϕy−ϕz 1 r
y−z, z−x≥0, ∀y∈E
, 2.11
for allz∈H. Then, the following hold:
1for eachx∈H,TrFx/∅,
2TF
r is single valued,
3TrFis firmly nonexpansive, that is, for anyx, y∈H,
TrFx−TrFy
2
≤TrFx−TrFy, x−y
, 2.12
4FTF
r MEPF, ϕ,
Lemma 2.10. Let H be a Hilbert space, let E be a nonempty closed convex subset ofH, and let
A:E → Hbeρ-inverse-strongly monotone. If0< r≤2ρ, thenI−ρAis a nonexpansive mapping inH.
Proof. For allx, y∈Eand 0< r≤2ρ, we have
I−rAx−I−rAy2
x−y−rAx−Ay2
x−y2−2rx−y, Ax−Ayr2Ax−Ay2
≤x−y2−2rρAx−Ayr2Ax−Ay2
x−y2rr−2ρAx−Ay2
≤x−y2.
2.13
So,I−ρAis a nonexpansive mapping ofEintoH.
3. Main Results
In this section, we prove a strong convergence theorem of the new shrinking projection method for finding a common element of the set of fixed points of strictly pseudocontractive mappings, the set of common solutions of generalized mixed equilibrium problems and the set of common solutions of the variational inequalities with inverse-strongly monotone mappings in Hilbert spaces.
Theorem 3.1. Let Ebe a nonempty closed convex subset of a real Hilbert spaceH. LetF1 and F2
be two bifunctions fromE×EtoÊsatisfying (A1)–(A5), and letϕ : E → Ê∪ {∞}be a proper lower semicontinuous and convex function with either (B1) or (B2). LetA1,A2,B,Cbe fourρ,ω,
β,ξ-inverse-strongly monotone mappings ofEintoH, respectively. LetS : E → Ebe ak-strictly pseudocontractive mapping with a fixed point. Define a mappingSk:E → EbySkxkx1−kSx,
for allx∈E. Suppose that
Θ:FS∩GMEPF1, ϕ, A1
∩GMEPF2, ϕ, A2
∩VIE, B∩VIE, C/∅. 3.1
Let{xn}be a sequence generated by the following iterative algorithm:
x0∈H, E1E, x1 PE1x0, un∈E, vn∈E,
F1un, u ϕu−ϕun A1xn, u−un 1
rnu−un, un−xn ≥0, ∀u∈E,
F2vn, v ϕv−ϕvn A2xn, v−vn
1
snv−vn, vn−xn ≥0, ∀v∈E,
ynPExn−λnBxn, znPE
xn−μnCxn
tnαn1Skxnαn2ynαn3znαn4unαn5vn,
En1{w∈En:tn−w ≤ xn−w},
xn1PEn1x0, ∀n≥0,
3.2
where{αni}are sequences in0,1, wherei1,2,3,4,5,rn ∈0,2ρ,sn∈0,2ω, and{λn},{μn}
are positive sequences. Assume that the control sequences satisfy the following restrictions:
C15i1αni1,
C2limn→ ∞αniαi∈0,1, wherei1,2,3,4,5,
C3a≤rn≤2ρandb≤sn≤2ω, wherea, bare two positive constants,
C4c≤λn≤2βandd≤μn≤2ξ, wherec, dare two positive constants,
C5limn→ ∞|λn1−λn|limn→ ∞|μn1−μn|0.
Then,{xn}converges strongly toPΘx0.
Proof. Lettingp∈Θand byLemma 2.9, we obtain
pPE
p−λnBp
PE
p−μnCp
TF1
rnI−rnA1pT
F2
snI−snA2p. 3.3
Note thatunTrFn1I−rnA1xn∈domϕandvn T
F2
snI−snA2xn∈domϕ, then we have
un−pTF1
rnI−rnA1xn−T
F1
rnI−rnA1p ≤ xn−p,
vn−pTF2
snI−snA2xn−T
F2
snI−snA2p
≤xn−p.
3.4
Next, we will divide the proof into six steps.
Step 1. We show that{xn}is well defined andEnis closed and convex for anyn≥1.
From the assumption, we see thatE1 E is closed and convex. Suppose thatEk is
closed and convex for somek≥1. Next, we show thatEk1is closed and convex for somek.
For anyp∈Ek, we obtain
tk−p≤xk−p 3.5
is equivalent to
tk−p2
2tk−xk, xk−p
≤0. 3.6
Thus,Ek1 is closed and convex. Then,Enis closed and convex for anyn ≥ 1. This implies
Step 2. We show thatΘ ⊂Enfor eachn≥1. From the assumption, we see thatΘ⊂E E1.
Suppose Θ ⊂ Ek for some k ≥ 1. For anyp ∈ Θ ⊂ Ek, sinceyn PExn −λnBxn and
zn PExn−μnCxn, for eachλn ≤ 2βandμn ≤ 2ξ byLemma 2.10, we haveI−λnBand
I−μnCare nonexpansive. Thus, we obtain
yn−pPExn−λnBxn−PE
p−λnBp
≤xn−λnBxn−
p−λnBp
I−λnBxn−I−λnBp
≤xn−p,
zn−pPE
xn−μnCxn
−PE
p−μnCp
≤xn−μnCxn
−p−μnCp
I−μnC
xn−
I−μnC
p
≤xn−p.
3.7
FromLemma 2.3, we haveSkis nonexpansive withFSk FS. It follows that
tn−pα1
n Skxnαn2ynαn3znαn4unαn5vn−p
≤αn1Skxn−pαn2yn−pαn3zn−pαn4un−pαn5vn−p
≤αn1xn−pαn2xn−pαn3xn−pαn4xn−pαn5xn−p
xn−p.
3.8
It follows thatp∈Ek1. This implies thatΘ⊂Enfor eachn≥1.
Step 3. We claim that limn→ ∞xn1−xn0 and limn→ ∞xn−tn0.
Fromxn PEnx0, we get
x0−xn, xn−y
≥0 3.9
for eachy∈En. UsingΘ⊂En, we have
x0−xn, xn−p
Hence, forp∈Θ, we obtain
0≤x0−xn, xn−p
x0−xn, xn−x0x0−p
−x0−xn, x0−xn
x0−xn, x0−p
≤ −x0−xn2x0−xnx0−p.
3.11
It follows that
x0−xn ≤x0−p, ∀p∈Θ, n∈Æ. 3.12
FromxnPEnx0andxn1PEn1x0∈En1 ⊂En, we have
x0−xn, xn−xn1 ≥0. 3.13
Forn∈Æ, we compute
0≤ x0−xn, xn−xn1
x0−xn, xn−x0x0−xn1
−x0−xn, x0−xnx0−xn, x0−xn1
≤ −x0−xn2x0−xn, x0−xn1
≤ −x0−xn2x0−xnx0−xn1,
3.14
and then
x0−xn ≤ x0−xn1, ∀n∈Æ. 3.15
Thus, the sequence{xn−x0}is a bounded and nondecreasing sequence, so limn→ ∞xn−x0
exists; that is, there existsmsuch that
m lim
From3.13, we get
xn−xn12 xn−x0x0−xn12
xn−x022xn−x0, x0−xn1x0−xn12
xn−x022xn−x0, x0−xnxn−xn1x0−xn12
xn−x022xn−x0, x0−xn2xn−x0, xn−xn1x0−xn12
−xn−x022xn−x0, xn−xn1x0−xn12
≤ −xn−x02x0−xn12.
3.17
By3.16, we obtain
lim
n→ ∞xn−xn10. 3.18
Sincexn1 PEn1x0 ∈En1⊂En, we have
xn−tn ≤ xn−xn1xn1−tn ≤2xn−xn1. 3.19
By3.18, we obtain
lim
n→ ∞xn−tn0. 3.20
Step 4. We claim that the following statements hold:
S1limn→ ∞xn−un0,
S2limn→ ∞xn−yn0,
S3limn→ ∞xn−zn0,
S4limn→ ∞xn−vn0.
Forp∈Θ, we note that
zn−p2
PE
xn−μnCxn
−PE
p−μnCp2
≤xn−μnCxn
−p−μnCp2
xn−p
−μn
Cxn−Cp2
≤xn−p2−2μn
xn−p, Cxn−Cp
μ2nCxn−Cp2
≤xn−p2μn
μn−2ξCxn−Cp2
xn−p2−μn
2ξ−μnCxn−Cp2.
Similarly, we also have
yn−p2≤
xn−p2−λn
2β−λnBxn−Bp2. 3.22
We note that
un−p2
TF1
rnI−rnA1xn−T
F1
rnI−rnA1p
2
≤I−rnA1xn−I−rnA1p2
xn−p
−rn
A1xn−A1p2
xn−p2−2rn
xn−p, A1xn−A1p
rn2A1xn−A1p2
≤xn−p2−2rnρA1xn−A1p2rn2A1xn−A1p2
xn−p2rn
rn−2ρA1xn−A1p2
xn−p2−rn
2ρ−rnA1xn−A1p2.
3.23
Similarly, we also have
vn−p2≤
xn−p2−sn2ω−snA2xn−A2p2. 3.24
Observing that
tn−p2≤
αn1Skxn−p2αn2yn−p2αn3zn−p2αn4un−p2αn5vn−p2
≤αn1xn−p2αn2yn−p2αn3zn−p2αn4un−p2αn5vn−p2.
3.25
Substituting3.21,3.22,3.23, and3.24into3.25, we obtain
tn−p2≤
αn1xn−p2αn2
xn−p2−λn
2β−λnBxn−Bp2
αn3
xn−p2−μn
2ξ−μnCxn−Cp2
αn4
xn−p2−rn
2ρ−rnA1xn−A1p2
αn5
xn−p2−sn2ω−snA2xn−A2p2
xn−p2−αn2λn
2β−λnBxn−Bp2−αn3μn
2ξ−μnCxn−Cp2
−αn4rn
2ρ−rnA1xn−A1p2−αn5sn2ω−snA2xn−A2p2.
It follows that
αn3μn
2ξ−μnCxn−Cp2
≤xn−p2−tn−p2−αn2λn
2β−λnBxn−Bp2
−αn4rn
2ρ−rnA1xn−A1p2−αn5sn2ω−snA2xn−A2p2
≤xn−ptn−pxn−tn.
3.27
FromC2,C4, and3.20, we have
lim
n→ ∞Cxn−Cp0. 3.28
Sincesn∈0,2ω, we also have
αn5sn2ω−snA2xn−A2p2
≤xn−p2−tn−p2−αn2λn
2β−λnBxn−Bp2
−αn3μn
2ξ−μnCxn−Cp2−αn4rn
2ρ−rnA1xn−A1p2
≤xn−ptn−pxn−tn.
3.29
FromC2,C3, and3.20, we obtain
lim
n→ ∞A2xn−A2p0. 3.30
Similarly, by3.28and3.30, we can prove that
lim
On the other hand, lettingp∈Θfor eachn≥1, we getpTF1
rnI−rnA1p. SinceT
F1
rn is firmly
nonexpansive, we have
un−p2
TF1
rnI−rnA1xn−T
F1
rnI−rnA1p
2
≤I−rnA1xn−I−rnA1p, un−p
1
2
I−rnA1xn−I−rnA1p2un−p2
−I−rnA1xn−I−rnA1p−
un−p2
≤ 1
2
xn−p2un−p2−xn−un−rn
A1xn−A1p2
≤ 1
2
xn−p2un−p2− xn−un2
2rnxn−unA1xn−A1p−rn2A1xn−A1p2
.
3.32
So, we obtain
un−p2≤
xn−p2− xn−un22rnxn−unA1xn−A1p. 3.33
Observe that
yn−p2
PExn−λnBxn−PE
p−λnBp2
≤I−λnBxn−I−λnBp, yn−p
1
2
I−λnBxn−I−λnBp2yn−p2
−I−λnBxn−I−λnBp−
yn−p2
≤ 1
2
xn−p2yn−p2−xn−yn
−λn
Bxn−Bp2
≤ 1
2
xn−p2yn−p2−xn−yn2−λ2nBxn−Bp2
2λn
xn−yn, Bxn−Bp
,
3.34
and hence
yn−p2 ≤
By using the same argument in3.33and3.35, we can get
vn−p2≤
xn−p2− xn−vn22snxn−vnA2xn−A2p,
zn−p2 ≤
xn−p2− xn−zn22μnxn−znCxn−Cp.
3.36
Substituting3.33,3.35, and3.36into3.25, we obtain
tn−p2≤
αn1xn−p2αn2yn−p2αn3zn−p2
αn4un−p2αn5vn−p2
≤αn1xn−p2αn2
xn−p2−xn−yn22λnxn−ynBxn−Bp
αn3
xn−p2− xn−zn22μnxn−znCxn−Cp
αn4
xn−p2− xn−un22rnxn−unA1xn−A1p
αn5
xn−p2− xn−vn22snxn−vnA2xn−A2p
xn−p2−αn2xn−yn22λnαn2xn−ynBxn−Bp
−αn3xn−zn22μnαn3xn−znCxn−Cp
−αn4xn−un22rnαn4xn−unA1xn−A1p
−αn5xn−vn22snαn5xn−vnA2xn−A2p.
3.37
It follows that
αn4xn−un2≤xn−p2−tn−p2−αn2xn−yn22λnαn2xn−ynBxn−Bp
−αn3xn−zn22μnαn3xn−znCxn−Cp
2rnαn4xn−unA1xn−A1p−αn5xn−vn2
2snαn5xn−vnA2xn−A2p
≤xn−ptn−pxn−tn2λnαn2xn−ynBxn−Bp
2μnαn3xn−znCxn−Cp2rnαn4xn−unA1xn−A1p
2snαn5xn−vnA2xn−A2p.
FromC2,3.20,3.28,3.30, and3.31, we have
lim
n→ ∞xn−un0. 3.39
By using the same argument, we can prove that
lim
n→ ∞xn−ynnlim→ ∞xn−znnlim→ ∞xn−vn0. 3.40
Applying3.20,3.39, and3.40, we can obtain
lim
n→ ∞tn−unnlim→ ∞tn−ynnlim→ ∞tn−znnlim→ ∞tn−vn0. 3.41
Step 5. We show that
z∈FS∩GMEPF1, ϕ, A1
∩GMEPF2, ϕ, A2
∩VIE, B∩VIE, C. 3.42
Assume thatλn → λ∈c,2βandμn → μ∈d,2ξ.
Define a mappingP:E → Eby
Pxα1Skxα2PE1−λBxα3PE
1−μCxα4TF1
r I−rA1x
α5TF2
s I−sA2x, ∀x∈E,
3.43
where limn→ ∞αni αi ∈ 0,1, wheni 1,2,3,4,5. ByC1, then we have
5
i1α
i
n 1.
FromLemma 2.8, we havePis nonexpansive and
FP FSk∩FPE1−λB∩F
PE
1−μC∩FTF1
r I−rA1
∩FTF2
s I−sA2
FSk∩GMEP
F1, ϕ, A1
∩GMEPF2, ϕ, A2
∩VIE, B∩VIE, C.
We note that
Pxn−xn ≤ Pxn−tntn−xn
α1Skxnα2PE1−λBxnα3PE
1−μCxn
α4TF1
r I−rA1xnα5TsF2I−sA2xn
− αn1Skxnαn2PE1−λnBxnαn3PE
1−μnC
xn
αn4TrF1I−rA1xnαn5TsF2I−sA2xntn−xn
≤α1−αn1Skxnα2PEI−λBxn−PEI−λnBxn
α2−αn2PEI−λnBxn
α3PE
I−μCxn−PE
I−μnC
xnα3−αn3PE
I−μnC
xn
α4−αn4TrF1I−rA1xnα5−αn5TsF2I−sA2xntn−xn
≤α1−αn1Skxnα2|λn−λ|Bxnα2−αn2PEI−λnBxn
α3μn−μCxnα3−αn3PE
I−μnC
xn
α4−αn4TrF1I−rA1xnα5−αn5TsF2I−sA2xntn−xn
≤K1
5
i1
αi−αni|λn−λ|μn−μ
tn−xn,
3.45
whereK1is an appropriate constant such that
K1max
sup
n≥1 TF1
r I−rA1xn, sup n≥1
TF2
s I−sA2xn, sup n≥1
PEI−λnBxn,
sup
n≥1
PE
I−μnC
xn, sup n≥1
Bxn, sup n≥1
Cxn, sup n≥1
Skxn
.
3.46
FromC2,C5, and3.20, we obtain
lim
Since{xni}is bounded, there exists a subsequence{xni}of{xn}which converges weakly to
z. Without loss of generality, we may assume that{xni} z. It follows from3.47, that
lim
n→ ∞xni− Pxni0. 3.48
It follows fromLemma 2.4thatz∈FP. By3.44, we havez∈Θ.
Step 6. Finally, we show thatxn → z, wherezPΘx0.
SinceΘis nonempty closed convex subset ofH, there exists a uniquez∈Θsuch that
zPΘx0. Sincez∈Θ⊂EnandxnPEnx0, we have
x0−xnx0−PEnx0 ≤x0−z
3.49
for alln≥1. From3.49,{xn}is bounded, soωwxn/∅. By the weak lower semicontinuity
of the norm, we have
x0−z ≤lim inf
i→ ∞ x0−xni ≤
x0−z.
3.50
Sincez∈ωwxn⊂Θ, we obtain
x0−zx
0−PΘx0 ≤ x0−z. 3.51
Using3.49and3.50, we obtainzz. Thus,ωwxn {z}andxn z. So we have
x0−z≤ x
0−z ≤lim inf
i→ ∞ x0−xn ≤lim supi→ ∞ x0−xn ≤
x0−z.
3.52
Thus,
x0−z lim
i→ ∞x0−xnx0−z
.
3.53
Fromxn z, we obtainx0−xnx0−z. UsingLemma 2.5, we obtain that
xn−zxn−x0−z−x0 −→0 3.54
asn → ∞and hencexn → zin norm. This completes the proof.
If the mappingSis nonexpansive, thenSkS0S. We can obtain the following result
fromTheorem 3.1immediately.
Corollary 3.2. LetEbe a nonempty closed convex subset of a real Hilbert spaceH. LetF1and F2
ξ-inverse-strongly monotone mappings ofEintoH, respectively. LetS:E → Ebe a nonexpansive mapping with a fixed point. Suppose that
Θ:FS∩GMEPF1, ϕ, A1
∩GMEPF2, ϕ, A2
∩VIE, B∩VIE, C/∅. 3.55
Let{xn}be a sequence generated by the following iterative algorithm3.1, where{αni}are sequences
in 0,1, wherei 1,2,3,4,5,rn ∈ 0,2ρ,sn ∈ 0,2ω, and{λn},{μn}are positive sequences.
Assume that the control sequences satisfy (C1)–(C5) inTheorem 3.1. Then,{xn}converges strongly
toPΘx0.
Ifϕ 0 and A1 A2 0 inTheorem 3.1, then we can obtain the following result
immediately.
Corollary 3.3. LetEbe a nonempty closed convex subset of a real Hilbert spaceH. LetF1andF2be
two bifunctions fromE×EtoÊsatisfying (A1)–(A5), and letϕ:E → Ê∪ {∞}be a proper lower semicontinuous and convex function with either (B1) or (B2). LetB, Cbe twoβ,ξ-inverse-strongly monotone mappings ofEintoH, respectively. LetS:E → Ebe a nonexpansive mapping with a fixed point. Suppose that
Θ:FS∩EPF1∩EPF2∩VIE, B∩VIE, C/∅. 3.56
Let{xn}be a sequence generated by the following iterative algorithm:
x0∈H, E1 E, x1PE1x0, un∈E, vn∈E,
F1un, u
1
rnu−un, un−xn ≥
0, ∀u∈E,
F2vn, v
1
snv−vn, vn−xn ≥0, ∀v∈E,
znPE
xn−μnCxn
, ynPExn−λnBxn,
tnαn1Sxnαn2ynαn3znαn4unαn5vn,
En1{w∈En:tn−w ≤ xn−w},
xn1PEn1x0, ∀n≥1,
3.57
where{αni}are sequences in (0,1), wherei1,2,3,4,5,rn∈0,∞,sn∈0,∞and{λn},{μn}are
positive sequences. Assume that the control sequences satisfy the condition (C1)–(C5) inTheorem 3.1. Then,{xn}converges strongly toPΘx0.
IfB0,C0, andF1un, u F1vn, v 0 inCorollary 3.3, thenPE Iand we get
unynxnandvnznxn; hence, we can obtain the following result immediately.
Corollary 3.4. LetEbe a nonempty closed convex subset of a real Hilbert spaceH. LetS:E → Ebe ak-strictly pseudocontractive mapping with a fixed point. Define a mappingSk:E → EbySkx
iterative algorithm:
x0∈H, E1E, x1 PE1x0,
tnαnSkxn 1−αnxn,
En1{w∈En:tn−w ≤ xn−w},
xn1 PEn1x0, ∀n≥1,
3.58
where {αn} are sequences in 0,1. Assume that the control sequences satisfy the condition
limn→ ∞αnα∈0,1inTheorem 3.1. Then,{xn}converges strongly to a pointPFSx0.
4. Convex Feasibility Problem
Finally, we consider the followingConvex Feasibility ProblemCFP: finding anx ∈ Mj1Cj,
whereM≥1 is an integer and eachCiis assumed to be the solutions of equilibrium problem
with the bifunctionFj, j 1,2,3, . . . , Mand the solution set of the variational inequality
problem. There is a considerable investigation on CFP in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration 35, 36, computer tomography37, and radiation therapy treatment planning38.
The following result can be obtained fromTheorem 3.1. We, therefore, omit the proof.
Theorem 4.1. LetEbe a nonempty closed convex subset of a real Hilbert spaceH. Let{Fj}Mj1be a
family of bifunction fromE×EtoÊsatisfying (A1)–(A5), and letϕ :E → Ê∪ {∞}be a proper lower semicontinuous and convex function with either (B1) or (B2). LetAj :E → Hbeρj
-inverse-strongly monotone mapping for eachj ∈ {1,2,3, . . . , M}. LetBi : E → H beβi-inverse-strongly
monotone mapping for eachi ∈ {1,2,3, . . . , N}. LetS : E → Ebe ak-strictly pseudocontractive mapping with a fixed point. Define a mappingSk:E → EbySkxkx 1−kSx, for allx∈E.
Suppose that
Θ:FS∩
⎛
⎝M
j1
GMEPFj, ϕ, Aj
⎞⎠
∩
N
i1
VIE, Bi
/
∅. 4.1
Let{xn}be a sequence generated by the following iterative algorithm:
x0 ∈H, E1E, x1PE1x0, v1, v2, . . . , vM∈E,
F1vn,1, v1 ϕv1−ϕvn,1 A1xn, v1−vn,1
1
r1v1−vn,1, vn,1−xn ≥
F2vn,2, v2 ϕv2−ϕvn,2 A2xn, v2−vn,2
1
r2v2−vn,2, vn,2−xn ≥0, ∀v2∈E,
.. .
FMvn,M, vM ϕvM−ϕvn,M AMxn, vM−vn,M
1
rMvM−vn,M, vn,M−xn ≥
0, ∀vM∈E,
yn,1PExn−λn,1B1xn,
yn,2PExn−λn,2B2xn,
.. .
yn,NPExn−λn,NBNxn,
tnαn,0Skxn N
i1
αn,iyn,i M
j1
αn,jvn,j,
En1{w∈En:tn−w ≤ xn−w},
xn1PEn1x0, ∀n≥1,
4.2
whereαn,0, αn,1, αn,2, . . . , αn,N andαn, 1, αn,2, . . . , αn,M ∈ 0,1such that
N
i0αn,i
M
j1αn,j 1,
{λn,i} are positive sequences in 0,1. Assume that the control sequences satisfy the following
restrictions:
C1limn→ ∞αniαi∈0,1, for each0≤i≤N,
C2limn→ ∞αnjαj∈0,1, for each1≤j≤M,
C3aj≤rj≤2ρj, whereajis some positive constants for each1≤j ≤M,
C4ci≤λn,i≤2βi, whereciis some positive constants for each1≤i≤N,
C5limn→ ∞|λn1,i−λn,i|0, for each1≤i≤N.
Then,{xn}converges strongly toPΘx0.
IfAj 0, for each 1≤ j ≤ MandFivn,i, vi 0, for each 1 ≤ i≤ NinTheorem 4.1,
thenvn,ixn; hence, we can obtain the following result immediately.
Theorem 4.2. LetEbe a nonempty closed convex subset of a real Hilbert spaceH. Letϕ:E → Ê∪
{∞}be a proper lower semicontinuous and convex function with either (B1) or (B2). LetBi:E → H
beβi-inverse-strongly monotone mapping for eachi∈ {1,2,3, . . . , N}. LetS:E → Ebe ak-strictly
pseudocontractive mapping with a fixed point. Define a mappingSk:E → EbySkxkx1−kSx,
for allx∈E. Suppose that
Θ:FS∩
N
i1
VIE, Bi
/
Let{xn}be a sequence generated by the following iterative algorithm:
x0∈H, E1E, x1 PE1x0,
yn,1PExn−λn,1B1xn,
yn,2PExn−λn,2B2xn,
.. .
yn,NPExn−λn,NBNxn,
tnαn,0Skxn N
i1
αn,iyn,i,
En1{w∈En:tn−w ≤ xn−w},
xn1PEn1x0, ∀n≥1,
4.4
whereαn,0, αn,1, αn,2, . . . , αn,N∈0,1such that
N
i0αn,i1,{λn,i}are positive sequences in0,1.
Assume that the control sequences satisfy the following restrictions:
C1limn→ ∞αniαi∈0,1, for each0≤i≤N,
C2ci≤λn,i≤2βi, whereciis some positive constants for each1≤i≤N,
C3limn→ ∞|λn1,i−λn,i|0, for each1≤i≤N.
Then,{xn}converges strongly toPΘx0.
IfBi0, for each 1≤i≤NinTheorem 4.1, then we getyn,ixn. Hence, we can obtain
the following result immediately.
Theorem 4.3. LetEbe a nonempty closed convex subset of a real Hilbert spaceH. Let be a{Fj}Mj1be a
family of bifunction fromE×EtoÊsatisfying (A1)–(A5), and letϕ:E → Ê∪{∞}be a proper lower semicontinuous and convex function with either (B1) or (B2). LetAj:E → Hbeρj-inverse-strongly
monotone mapping for eachj ∈ {1,2,3, . . . , M}. LetS:E → Ebe a k-strictly pseudocontractive mapping with a fixed point. Define a mappingSk:E → EbySkxkx 1−kSx, for allx∈E.
Suppose that
Θ:FS∩
⎛
⎝M
j1
GMEPFj, ϕ, Aj
⎞⎠
/
Let{xn}be a sequence generated by the following iterative algorithm:
x0∈H, E1E, x1PE1x0, v1, v2, . . . , vM∈E,
F1vn,1, v1 ϕv1−ϕvn,1 A1xn, v1−vn,1
1
r1v1−vn,1, vn,1−xn ≥
0, ∀v1∈E,
F2vn,2, v2 ϕv2−ϕvn,2 A2xn, v2−vn,2 1
r2v2−vn,2, vn,2−xn ≥0, ∀v2∈E,
.. .
FMvn,M, vM ϕvM−ϕvn,M AMxn, vM−vn,M
1
rMvM−vn,M, vn,M−xn ≥
0, ∀vM∈E,
tnαn,0Skxn M
j1
αn,jvn,j,
En1{w∈En:tn−w ≤ xn−w},
xn1PEn1x0, ∀n≥1,
4.6
whereαn,0andαn,1, αn,2, . . . , αn,M ∈0,1such thatαn,0jM1αn,j 1. Assume that the control
sequences satisfy the following restrictions:
C1limn→ ∞αn0α0∈0,1,
C2limn→ ∞αnjαj∈0,1, for each1≤j≤M,
C3aj≤rj≤2ρj, whereajis some positive constants for each1≤j ≤M.
Then,{xn}converges strongly toPΘx0.
Acknowledgments
The authors would like to thank the anonymous referees for helpful comments to improve this paper, and the second author was supported by the Commission on Higher Education and the Thailand Research Fund under Grant MRG5380044. Moreover, they also would like to thank the National Research University Project of Thailand’s Office of the Higher Education Commission for financial support under NRU-CSEC Project no. 54000267.
References
1 H. Zhou, “Convergence theorems of fixed points forκ-strict pseudo-contractions in Hilbert spaces,”
Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 2, pp. 456–462, 2008.
2 L.-C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems,”Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008. 3 E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,”The
4 S. D. Fl˚am and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,” Mathemat-ical Programming, vol. 78, pp. 29–41, 1997.
5 S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,”Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 506–515, 2007.
6 P. Hartman and G. Stampacchia, “On some non-linear elliptic differential-functional equations,”Acta Mathematica, vol. 115, pp. 271–310, 1966.
7 J.C. Yao and O. Chadli, “Pseudomonotone complementarity problems and variational inequalities,” inHandbook of Generalized Convexity and Generalized Monotonicity, J. P. Crouzeix, N. Haddjissas, and S. Schaible, Eds., vol. 76, pp. 501–558, Springer, New York, NY, USA, 2005.
8 H. Zegeye, E. U. Ofoedu, and N. Shahzad, “Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings,”
Applied Mathematics and Computation, vol. 216, no. 12, pp. 3439–3449, 2010.
9 H. Zegeye and N. Shahzad, “A hybrid approximation method for equilibrium, variational inequality and fixed point problems,”Nonlinear Analysis. Hybrid Systems, vol. 4, no. 4, pp. 619–630, 2010. 10 L. C. Zeng, S. Schaible, and J. C. Yao, “Iterative algorithm for generalized set-valued strongly
nonlinear mixed variational-like inequalities,”Journal of Optimization Theory and Applications, vol. 124, no. 3, pp. 725–738, 2005.
11 X. Qin, Y. J. Cho, and S. M. Kang, “Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 1, pp. 99–112, 2010.
12 X. Gao and Y. Guo, “Strong convergence of a modified iterative algorithm for mixed-equilibrium problems in Hilbert spaces,”Journal of Inequalities and Applications, vol. 2008, Article ID 454181, 23 pages, 2008.
13 C. Jaiboon and P. Kumam, “A hybrid extragradient viscosity approximation method for solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,”Fixed Point Theory and Applications, vol. 2009, Article ID 374815, 32 pages, 2009.
14 C. Jaiboon and P. Kumam, “Strong convergence for generalized equilibrium problems, fixed point problems and relaxed cocoercive variational inequalities,”Journal of Inequalities and Applications, vol. 2010, Article ID 728028, 43 pages, 2010.
15 C. Jaiboon, P. Kumam, and U. W. Humphries, “Weak convergence theorem by an extragradient method for variational inequality, equilibrium and fixed point problems,”Bulletin of the Malaysian Mathematical Sciences Society, vol. 32, no. 2, pp. 173–185, 2009.
16 J. S. Jung, “Strong convergence of composite iterative methods for equilibrium problems and fixed point problems,”Applied Mathematics and Computation, vol. 213, no. 2, pp. 498–505, 2009.
17 J. K. Kim, S. Y. Cho, and X. Qin, “Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings,”Journal of Inequalities and Applications, vol. 2010, Article ID 312602, 18 pages, 2010.
18 P. Kumam and C. Jaiboon, “A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems,”Nonlinear Analysis. Hybrid Systems, vol. 3, no. 4, pp. 510–530, 2009.
19 S.-Y. Matsushita, K. Nakajo, and W. Takahashi, “Strong convergence theorems obtained by a generalized projections hybrid method for families of mappings in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 6, pp. 1466–1480, 2010.
20 J.-W. Peng and J.-C. Yao, “Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems,” Mathematical and Computer Modelling, vol. 49, no. 9-10, pp. 1816–1828, 2009.
21 P. Katchang and P. Kumam, “A general iterative method of fixed points for mixed equilibrium problems and variational inclusion problems,”Journal of Inequalities and Applications, vol. 2010, Article ID 370197, 25 pages, 2010.
22 W. Kumam, C. Jaiboon, P. Kumam, and A. Singta, “A shrinking projection method for gener-alized mixed equilibrium problems, variational inclusion problems and a finite family of quasi-nonexpansive mappings,”Journal of Inequalities and Applications, vol. 2010, Article ID 458247, 25 pages, 2010.
24 H. Zegeye and N. Shahzad, “A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems,”Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 1, pp. 263– 272, 2011.
25 H. Zegeye and N. Shahzad, “Approximating common solution of variationalinequality problems for two monotone mappings in Banach spaces,”Optimization Letters. In press.
26 A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem,”Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359–370, 2007.
27 X. Qin and S. M. Kang, “Convergence theorems on an iterative method for variational inequality problems and fixed point problems,”Bulletin of the Malaysian Mathematical Sciences Society, vol. 33, no. 1, pp. 155–167, 2010.
28 P. Kumam and C. Jaiboon, “A system of generalized mixed equilibrium problems and fixed point problems for pseudocontractive mappings in Hilbert spaces,”Fixed Point Theory and Applications, vol. 2010, Article ID 361512, 33 pages, 2010.
29 F. E. Browder, “Nonlinear operators and nonlinear equations of evolution in Banach spaces,”
Proceedings of Symposia in Pure Mathematics, vol. 18, pp. 78–81, 1976.
30 W. Takahashi,Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.
31 G. L. Acedo and H.-K. Xu, “Iterative methods for strict pseudo-contractions in Hilbert spaces,”
Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 7, pp. 2258–2271, 2007.
32 C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,”Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 11, pp. 2400–2411, 2006. 33 R. E. Bruck Jr., “Properties of fixed-point sets of nonexpansive mappings in Banach spaces,”
Transactions of the American Mathematical Society, vol. 179, pp. 251–262, 1973.
34 J.-W. Peng and J.-C. Yao, “A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems,”Taiwanese Journal of Mathemat-ics, vol. 12, no. 6, pp. 1401–1432, 2008.
35 P. L. Combettes, “The convex feasibility problem: in image recovery,” in Advances in Imaging and Electron Physics, P. Hawkes, Ed., pp. 155–270, Academic Press, Orlando, Fla, USA, 1996.
36 T. Kotzer, N. Cohen, and J. Shamir, “Images to ration by a novel method of parallel projection onto constraint sets,”Optics Letters, vol. 20, pp. 1172–1174, 1995.
37 M. I. Sezan and H. Stark, “Application of convex projection theory to image recovery in tomograph and related areas,” in Image Recovery: Theory and Application, H. Stark, Ed., pp. 155–270, Academic Press, Orlando, Fla, USA, 1987.
