Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 347204, 30 pages
doi:10.1155/2011/347204
Research Article
Approximation of Common Solutions to
System of Mixed Equilibrium Problems,
Variational Inequality Problem, and Strict
Pseudo-Contractive Mappings
Poom Kumam1, 2 and Chaichana Jaiboon2, 3
1
Department of Mathematics, Faculty of Science, King Mongkut’s University of
Technology Thonburi (KMUTT), Bangkok 10140, Thailand
2
Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand
3
Department of Mathematics, Faculty of Liberal Arts, Rajamangala University of
Technology Rattanakosin (RMUTR), Bangkok 10100, Thailand
Correspondence should be addressed to Chaichana Jaiboon, chaichana.j@rmutr.ac.th
Received 3 October 2010; Accepted 5 March 2011
Academic Editor: Jong Kim
Copyright q 2011 P. Kumam and C. Jaiboon. 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 an iterative algorithm for finding a common element of the set of fixed points of strict
pseudocontractions mapping, the set of common solutions of a system of two mixed equilibrium
problems and the set of common solutions of the variational inequalities with inverse strongly
monotone mappings. Strong convergence theorems are established in the framework of Hilbert
spaces. Finally, we apply our results for solving convex feasibility problems in Hilbert spaces. Our
results improve and extend the corresponding results announced by many others recently.
1. Introduction
Throughout this paper, we denote by N and R the sets of positive integers and real numbers,
respectively. Let H be a real Hilbert space with inner product ·, · and norm ·, and let E be a
nonempty closed convex subset of H. We denote weak convergence and strong convergence
by notations and → , respectively. Recall that a mapping f : E → E is an α-contraction on
E if there exists a constant α ∈ 0, 1 such that fx − fy ≤ αx − y for all x, y ∈ E. Let
S : E → E be a mapping. In the sequel, we will use FS to denote the set of fixed points of S;
that is, FS {x ∈ E : Sx x}. In addition, let a mapping S : E → E be called nonexpansive,
if Sx − Sy ≤ x − y, for all x, y ∈ E. It is well known that if E ⊂ H is nonempty, bounded,
closed, and convex and S is a nonexpansive self-mapping on E, then FS is nonempty; see,
2
Fixed Point Theory and Applications
for example, 1. Recall that a mapping S : E → E is called strictly pseudo-contraction if there
exists a constant k ∈ 0, 1 such that
Sx − Sy2 ≤ x − y2 kI − Sx − I − Sy2 ,
∀x, y ∈ E,
1.1
where I denotes the identity operator on E. Note that if k 0, then S is a nonexpansive
mapping. The class of strict pseudo-contractions is one of the most important classes of
mappings among nonlinear mappings. Within the past several decades, many authors have
been devoted to the studies on the existence and convergence of fixed points for strict pseudocontractions. In 1967, Browder and Petryshyn 2 introduced a convex combination method
to study strict pseudo-contractions in Hilbert spaces. On the other hand, Marino and Xu 3
and Zhou 4 developed some iterative scheme for finding a fixed point of a strict pseudocontraction mapping. More precisely, take k ∈ 0, 1 and define a mapping Sk by
Sk x kx 1 − kSx,
∀x ∈ E,
1.2
where S is a strict pseudo-contraction. Under appropriate restrictions on k, it is proved
that the mapping Sk is nonexpansive. Therefore, the techniques of studying nonexpansive
mappings can be applied to study more general strict pseudo-contractions.
Let ϕ : E → R ∪ {∞} be a proper extended real-valued function and let φ be a
bifunction of E × E into R such that E ∩ dom ϕ /
∅, where R is the set of real numbers and
dom ϕ {x ∈ E : ϕx < ∞}. Ceng and Yao 5 considered the following mixed equilibrium
problems for finding x ∈ E such that
φ x, y ϕ y − ϕx ≥ 0,
∀y ∈ E.
1.3
The set of solutions of 1.3 is denoted by MEPφ, ϕ, that is,
MEP φ, ϕ x ∈ E : φ x, y ϕ y − ϕx ≥ 0, ∀y ∈ E .
1.4
We see that x is a solution of a problem 1.3 that implies that x ∈ dom ϕ {x ∈ E :
ϕx < ∞}.
Special Examples
1 If ϕ 0, then the mixed equilibrium problem 1.3 becomes to be the equilibrium problem
which is to find x ∈ E such that
φ x, y ≥ 0,
∀y ∈ E.
1.5
The set of solutions of 1.5 is denoted by EPφ.
2 If ϕ 0 and φx, y Bx, y − x for all x, y ∈ E, where B : E → H is a nonlinear
mapping, then problem 1.5 becomes to be the variational inequality problems which is to
find x ∈ E such that
Bx, y − x ≥ 0,
∀y ∈ E.
1.6
Fixed Point Theory and Applications
3
The set of solutions of 1.6 is denoted by VIE, B. The variational inequality has been
extensively studied in the literature. See, for example, 6–8 and the references therein.
The mixed equilibrium problems include fixed point problems, variational inequality
problems, optimization problems, Nash equilibrium problems, and the equilibrium problem
as special cases. Numerous problems in physics, optimization, and economics reduce to
find a solution of 1.3. Some authors have proposed some useful methods for solving the
MEPφ, ϕ and EPφ; see, for instance 5, 9–27. In 1997, Combettes and Hirstoaga 10
introduced an iterative scheme of finding the best approximation to initial data when EPφ
is nonempty and proved a strong convergence theorem. Next, we recall some definitions.
Definition 1.1. Let B : E → H be nonlinear mappings. Then B is called
1 monotone if
Bx − By, x − y ≥ 0,
∀x, y ∈ E,
1.7
2 ρ-strongly monotone if there exists a constant ρ > 0 such that
2
Bx − By, x − y ≥ ρx − y ,
∀x, y ∈ E,
1.8
3 η-Lipschitz continuous if there exists a constant η > 0 such that
Bx − By ≤ ηx − y,
∀x, y ∈ E,
1.9
4 β-inverse strongly monotone if there exists a constant β > 0 such that
2
Bx − By, x − y ≥ βBx − By ,
∀x, y ∈ E.
1.10
Remark 1.2. It is obvious that any β-inverse strongly monotone mappings B is monotone and
1/β-Lipschitz continuous.
5 A set-valued mapping T : H → 2H is called a monotone if, for all x, y ∈ H, f ∈ T x
and g ∈ T y imply x − y, f − g ≥ 0.
6 A monotone mapping T : H → 2H is a maximal if the graph of GT of T is not
properly contained in the graph of any other monotone mapping. It is known that a
monotone mapping T is maximal if and only if for x, f ∈ H × H, x − y, f − g ≥ 0
for every y, g ∈ GT implies f ∈ T x.
Let B be a monotone map of E into H, η-Lipschitz continuous mapping and let NE ϑ
be the normal cone to E when ϑ ∈ E, that is,
NE ϑ {w ∈ H : u − ϑ, w ≥ 0, ∀u ∈ E}
1.11
4
Fixed Point Theory and Applications
and define a mapping T on E by
Tϑ ⎧
⎨Bϑ NE ϑ, ϑ ∈ E,
⎩∅,
1.12
ϑ∈
/ E.
Then T is the maximal monotone and 0 ∈ T ϑ if and only if ϑ ∈ VIE, B; see 28.
For finding a common element of the set of fixed points of a nonexpansive mapping
and the set of solution of variational inequalities for β-inverse strongly monotone, Takahashi
and Toyoda 29 first introduced the following iterative scheme:
x0 ∈ E
chosen arbitrary,
xn1 αn xn 1 − αn SPE xn − λn Bxn ,
∀n ≥ 0,
1.13
where B is an β-inverse strongly monotone, {αn } is a sequence in 0, 1, and {λn } is a sequence
in 0, 2β. They showed that if FS ∩ VIE, B is nonempty, then the sequence {xn } generated
by 1.13 converges weakly to some q ∈ FS ∩ VIE, B.
Further, Y. Yao and J.-C. Yao 30 introduced the following iterative scheme:
x1 x ∈ E
xn1
chosen arbitrary,
yn PE xn − λn Bxn ,
αn x βn xn γn SPE yn − λn Byn ,
1.14
∀n ≥ 1,
where B is an β-inverse strongly monotone, {αn }, {βn }, {γn } are three sequences in 0, 1,
and {λn } is a sequence in 0, 2β. They showed that if FS ∩ VIE, B is nonempty, then the
sequence {xn } generated by 1.14 converges strongly to some q ∈ FS ∩ VIE, B.
A map A : H → H is said to be strongly positive if there exists a constant γ > 0 such
that
Ax, x ≥ γx2 ,
∀x ∈ H.
1.15
A typical problem is to minimize a quadratic function over the set of the fixed points of some
nonexpansive mapping on a real Hilbert space H:
1
min Ax, x − x, b,
x∈E 2
1.16
where A is some linear, E is the fixed point set of a nonexpansive mapping S on H and b is a
point in H. Let A be a strongly positive linear bounded map on H with coefficient γ. In 2006,
Marino and Xu 31 studied the following general iterative method:
xn1 n γfxn 1 − n ASxn .
1.17
Fixed Point Theory and Applications
5
They proved that if the sequence n of parameters appropriate conditions, then the sequence
xn generated by 1.17 converges strongly to q PFS I − A γfq. Recently, Plubtieng and
Punpaeng 32 proposed the following iterative algorithm:
1
y − un , un − xn ≥ 0,
φ un , y rn
∀y ∈ H,
1.18
xn1 n γfxn I − n ASun .
They proved that if the sequences {n } and {rn } of parameters satisfy appropriate condition,
then both sequences {xn } and {un } converge to the unique solution q of the variational
inequality
A − γf q, x − q ≥ 0,
∀x ∈ FS ∩ EP φ ,
1.19
which is the optimality condition for the minimization problem
min
1
x∈FS∩EPφ 2
Ax, x − hx,
1.20
where h is a potential function for γf i.e., h x γfx for x ∈ H.
On the other hand, for finding a common element of the set of fixed points of a kstrict pseudo-contraction mapping and the set of solutions of an equilibrium problem in a
real Hilbert space, Liu 33 introduced the following iterative scheme:
1
y − un , un − xn ≥ 0,
φ un , y rn
yn βn un 1 − βn Sun ,
xn1 n γfxn I − n Aun ,
∀y ∈ E,
1.21
∀n ≥ 1,
where S is a k-strict pseudo-contraction mapping and {n }, {βn } are sequences in 0, 1. They
proved that under certain appropriate conditions over {n }, {βn }, and {rn }, the sequences
{xn } and {un } converge strongly to some q ∈ FS ∩ EPφ, which solves some variational
inequality problems.
In 2008, Ceng and Yao 5 introduced an iterative scheme for finding a common fixed
point of a finite family of nonexpansive mappings and the set of solutions of a problem 1.3
in Hilbert spaces and obtained the strong convergence theorem which used the following
condition:
H K : E → R is η-strongly convex with constant σ > 0 and its derivative K is
sequentially continuous from weak topology to strong topology. We note that the condition
H for the function K : E → R is a very strong condition. We also note that the condition
H does not cover the case Kx x2 /2 and ηx, y x − y for each x, y ∈ E × E.
Very recently, R. Wangkeeree and R. Wangkeeree 34 introduced a general iterative method
for finding a common element of the set of solutions of the mixed equilibrium problems,
the set of fixed point of a k-strict pseudo-contraction mapping, and the set of solutions of
6
Fixed Point Theory and Applications
the variational inequality for an inverse strongly monotone mapping in Hilbert spaces. They
obtained a strong convergence theorem except the condition H for the sequences generated
by these processes.
In 2009, Qin et al. 35 introduced a general iterative scheme for finding a common
element of the set of common solution of generalized equilibrium problems, the set of a
common fixed point of a family of infinite nonexpansive mappings in Hilbert spaces. Let
{xn } be the sequence generated iterative by the following algorithm:
x1 ∈ E,
un ∈ E,
vn ∈ E,
1
φ1 un , u Cxn , u − un u − un , un − xn ≥ 0,
r
∀u ∈ E,
1
φ2 vn , v Bxn , v − vn v − vn , vn − xn ≥ 0,
s
∀v ∈ E,
1.22
yn δn un 1 − δn vn ,
xn1 n fxn βn xn γn Wn yn ,
∀n ≥ 1.
They proved that under certain appropriate conditions imposed on {n }, {βn }, {γn } and {δn },
the sequence {xn } generated by 1.22 converges strongly to q ∈ ∩∞
n1 FTn ∩ EPφ1 , C ∩
EPφ2 , B, where q P∩∞n1 FTn ∩EPφ1 ,C∩EPφ2 ,B fq.
In the present paper, motivated and inspired by Qin et al. 35, Plubtieng and
Punpaeng 32, Peng and Yao 17, R. Wangkeeree and R. Wangkeeree 34, and Y. Yao and
J.-C. Yao 30, we introduce a new approximation iterative scheme for finding a common
element of the set of fixed points of strict pseudo-contractions, the set of common solutions
of the system of a mixed equilibrium problem, and the set of common solutions of the
variational inequalities with inverse strongly monotone mappings in Hilbert spaces. We
obtain a strong convergence theorem for the sequences generated by these processes under
some parameter controlling conditions. Moreover, we apply our results for solving convex
feasibility problems in Hilbert spaces. The results in this paper extend and improve some
well-known results in 17, 30, 32, 34, 35.
2. Preliminaries
Let H be a real Hilbert space and E be a closed convex subset of H. In a real Hilbert space H,
it is well known that
λx 1 − λy2 λx2 1 − λy2 − λ1 − λx − y2
2.1
for all x, y ∈ H and λ ∈ 0, 1.
For any x ∈ H, there exists a unique nearest point in E, denoted by PE x, such that
x − PE x ≤ x − y,
∀y ∈ E.
The mapping PE is called the metric projection of H onto E.
2.2
Fixed Point Theory and Applications
7
It is well known that PE is a firmly nonexpansive mapping of H onto E, that is,
2
x − y, PE x − PE y ≥ PE x − PE y ,
∀x, y ∈ H.
2.3
Further, for any x ∈ H and z ∈ E, z PE x if and only if x − z, z − y ≥ 0, for all y ∈ E.
Moreover, PE x is characterized by the following properties: PE x ∈ E and
x − PE x, y − PE x ≤ 0,
x − y2 ≥ x − PE x2 y − PE x2
2.4
2.5
for all x ∈ H, y ∈ E.
It is easy to see that the following is true:
u ∈ VIE, B ⇐⇒ u PE u − λBu,
λ > 0.
2.6
The following lemmas will be useful for proving the convergence result of this paper.
Lemma 2.1 see 36. Let E, ·, · be an inner product space. Then, for all x, y, z ∈ E and α, β, γ ∈
0, 1 with α β γ 1, one has
αx βy γz2 αx2 βy2 γz2 − αβx − y2 − αγx − z2 − βγ y − z2 .
2.7
Lemma 2.2 see 31. Assume that A is a strongly positive linear bounded operator on H with
coefficient γ > 0 and 0 < ρ ≤ A−1 . Then I − ρA ≤ 1 − ργ.
Lemma 2.3 see 4. Let E be a nonempty closed convex subset of a real Hilbert space H and let
S : E → E be a k-strict pseudo-contraction with a fixed point. Then FS is closed and convex.
Define Sk : E → E by Sk kx 1 − kSx for each x ∈ E. Then Sk is nonexpansive such that
FSk FS.
Lemma 2.4 see 37. Let X be a uniformly convex Banach spaces, E be a nonempty closed convex
subset of X and S : E → E be a nonexpansive mapping. Then I − S is demi-closed at zero.
Lemma 2.5 see 38. Let E be a nonempty closed convex subset of strictly convex Banach space X.
Let {Tn : n ∈ N} be a sequence of nonexpansive mappings on E. Suppose ∩∞
n1 FTn is nonempty. Let
∞
δn be a sequence of positive numbers with n1 δn 1. Then a mapping S on E can be defined by
Sx ∞
δn T n x
n1
for x ∈ E is well defined, nonexpansive and FS ∩∞
n1 FTn holds.
2.8
8
Fixed Point Theory and Applications
In order to solve the mixed equilibrium problem, the following assumptions are given
for the bifunction φ, ϕ and the set E:
A1 φx, x 0 for all x ∈ E;
A2 φ is monotone, that is, φx, y φy, x ≤ 0 for all x, y ∈ E;
A3 for each x, y, z ∈ E, limt → 0 φtz 1 − tx, y ≤ φx, y;
A4 for each x ∈ E, y → φx, y is convex and lower semicontinuous;
A5 for each y ∈ E, x → φx, y is weakly upper semicontinuous;
B1 for each x ∈ H and r > 0, there exist abounded subset Dx ⊆ E and yx ∈ E such that
for any z ∈ E \ Dx ,
1
φ z, yx ϕ yx − ϕz yx − z, z − x < 0;
r
2.9
B2 E is a bounded set.
Lemma 2.6 see 39. Let E be a nonempty closed convex subset of H. Let φ : E × E → R be a
bifunction satisfies (A1)–(A5) and let ϕ : E → R ∪ {∞} be a proper lower semicontinuous and
convex function. Assume that either (B1) or (B2) holds. For r > 0 and x ∈ H, define a mapping
φ,ϕ
: H → E as follows:
Tr
φ,ϕ
Tr
x 1
z ∈ E : φ z, y ϕ y − ϕz y − z, z − x ≥ 0, ∀y ∈ E
r
2.10
for all z ∈ H. Then, the following holds:
φ,ϕ
i for each x ∈ H, Tr
x /
∅;
ii
φ,ϕ
Tr
is single-valued;
iii
φ,ϕ
Tr
is firmly nonexpansive, that is, for any x, y ∈ H,
φ,ϕ
Tr
φ,ϕ
iv FTr
φ,ϕ
x − Tr
φ,ϕ
φ,ϕ
y2 ≤ Tr x − Tr y, x − y ;
2.11
MEPφ, ϕ;
v MEPφ, ϕ is closed and convex.
φ,ϕ
Remark 2.7. If ϕ 0, then Tr
φ
is rewritten as Tr .
Remark 2.8. We remark that Lemma 2.6 is not a consequence of Lemma 3.1 in 5, because the
condition of the sequential continuity from the weak topology to the strong topology for the
derivative K of the function K : E → R does not cover the case Kx x2 /2.
Lemma 2.9 see 40. Let {xn } and {ln } be bounded sequences in a Banach space X and let {βn } be
a sequence in 0, 1 with 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1. Suppose xn1 1 − βn ln βn xn
for all integers n ≥ 1 and lim supn → ∞ ln1 − ln − xn1 − xn ≤ 0. Then, limn → ∞ ln − xn 0.
Fixed Point Theory and Applications
9
Lemma 2.10 see 41. Assume that {an } is a sequence of nonnegative real numbers such that
an1 ≤ 1 − n an σn , n ≥ 1,
2.12
where {n } is a sequence in 0, 1 and {σn } is a sequence in R such that
1 ∞
n1 n ∞,
2 lim supn → ∞ σn /n ≤ 0 or ∞
n1 |σn | < ∞.
Then limn → ∞ an 0.
Lemma 2.11. Let H be a real Hilbert space. Then for all x, y ∈ H,
x y2 ≤ x2 2 y, x y .
2.13
3. Main Results
In this section, we will use the new approximation iterative method to prove a strong
convergence theorem for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of the system of a mixed equilibrium problem and
the set of a common solutions of the variational inequalities with inverse strongly monotone
mappings in a real Hilbert space.
Theorem 3.1. Let E be a nonempty closed convex subset of a real Hilbert space H. Let φ1 and φ2 be
two bifunctions from E × E to R satisfying (A1)–(A5) and let ϕ : E → R ∪ {∞} be a proper lower
semicontinuous and convex function. Let C : E → H be an ξ-inverse strongly monotone mapping
and B : E → H be an β-inverse strongly monotone mapping. Let f : E → E be a contraction
mapping with coefficient α 0 < α < 1 and let A be a strongly positive linear bounded operator on H
with coefficient γ > 0 and 0 < γ < γ/α. Let S : E → E be a k-strict pseudo-contraction with a fixed
point. Define a mapping Sk : E → E by Sk x kx 1 − kSx, for all x ∈ E. Assume that
Θ : FS ∩ VIE, C ∩ VIE, B ∩ MEP φ1 , ϕ ∩ MEP φ2 , ϕ / ∅.
3.1
Assume that either (B1) or (B2). Let {xn } be a sequence generated by the following iterative algorithm:
x1 ∈ E,
un ∈ E,
φ1 ,ϕ
un Tr
vn ∈ E,
xn ,
φ ,ϕ
vn Ts 2 xn ,
zn PE un − μn Cun ,
3.2
yn PE vn − λn Bvn ,
xn1
kn an Sk xn bn yn cn zn ,
n γfxn βn xn 1 − βn I − n A kn ,
∀n ≥ 1,
10
Fixed Point Theory and Applications
where {n }, {βn }, {an }, {bn }, and {cn } are sequences in 0, 1 and {λn }, {μn } are positive sequences.
Assume that the control sequences satisfy the following restrictions:
C1 an bn cn 1,
C2 limn → ∞ n 0 and
∞
n1 n
∞,
C3 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1,
C4 limn → ∞ |λn1 − λn | limn → ∞ |μn1 − μn | 0,
C5 d ≤ λn ≤ 2β, e ≤ μn ≤ 2ξ, where d, e are two positive constants,
C6 limn → ∞ an a, limn → ∞ bn b and limn → ∞ cn c, for some a, b, c ∈ 0, 1.
Then, {xn } converges strongly to a point q ∈ Θ which is the unique solution of the variational
inequality
A − γf q, x − q ≥ 0,
∀x ∈ Θ
3.3
or equivalent q PΘ I − A γfq, where P is a metric projection mapping form H onto Θ.
Proof. Since n → 0, as n → ∞, we may assume, without loss of generality, that n ≤ 1 −
βn A−1 for all n ∈ N. By Lemma 2.2, we know that if 0 ≤ ρ ≤ A−1 , then I − ρA ≤ 1 − ργ.
We will assume that I − A ≤ 1 − γ. Since A is a strongly positive bounded linear operator
on H, we have
A sup{|Ax, x| : x ∈ H, x 1}.
3.4
Observe that
1 − βn I − n A x, x 1 − βn − n Ax, x
≥ 1 − βn − n A
3.5
≥ 0,
so this shows that 1 − βn I − n A is positive. It follows that
1 − βn I − n A sup 1 − βn I − n A x, x : x ∈ H, x 1
sup 1 − βn − n Ax, x : x ∈ H, x 1
3.6
≤ 1 − βn − n γ.
We divide the proof into seven steps.
Step 1. We claim that the mapping PΘ I − A γf where Θ : FS ∩ VIE, C ∩ VIE, B ∩
MEPφ1 , ϕ ∩ MEPφ2 , ϕ has a unique fixed point.
Fixed Point Theory and Applications
11
Since f be a contraction of H into itself with α ∈ 0, 1. Then, we have
PΘ I − A γf x − PΘ I − A γf y ≤ I − A γf x − I − A γf y ≤ I − Ax − y γfx − f y ≤ 1 − γ x − y γαx − y
1 − γ − γα x − y, ∀x, y ∈ H.
3.7
Since 0 < 1−γ −γα < 1, it follows that PΘ I −Aγf is a contraction of H into itself. Therefore
the Banach Contraction Mapping Principle implies that there exists a unique element q ∈ H
such that q PΘ I − A γfq.
Step 2. We claim that I − λn B is nonexpansive.
Indeed, from the β-inverse strongly monotone mapping definition on B and condition
C5, we have
I − λn Bx − I − λn By2 x − y − λn Bx − By2
2
2
x − y − 2λn x − y, Bx − By λ2n Bx − By
2
2
≤ x − y − 2λn βBx − By λ2n Bx − By
2
2
x − y λn λn − 2β Bx − By
2
≤ x − y ,
3.8
where λn ≤ 2β, for all n ∈ N implies that the mapping I − λn B is nonexpansive and so is,
I − μn C.
Step 3. We claim that {xn } is bounded.
Indeed, let p ∈ Θ and Lemma 2.6, we obtain
φ ,ϕ
φ ,ϕ
p PE p − λn Bp PE p − μn Cp Tr 1 p Ts 2 p.
φ1 ,ϕ
Note that un Tr
φ2 ,ϕ
xn ∈ dom ϕ and vn Ts
3.9
xn ∈ dom ϕ, we have
φ1 ,ϕ
φ ,ϕ un − p xn − Tr 1 p ≤ xn − p,
Tr
φ2 ,ϕ
φ ,ϕ vn − p xn − Ts 2 p ≤ xn − p.
Ts
3.10
12
Fixed Point Theory and Applications
Since I − λn B and I − μn C are nonexpansive and from 2.6, we have
zn − p PE un − μn Cun − PE p − μn Cp ≤ un − μn Cun − p − μn Cp I − μ n C un − I − μ n C p ≤ un − p ≤ xn − p,
yn − p PE vn − λn Bvn − PE p − λn Bp ≤ vn − p ≤ xn − p.
3.11
From Lemma 2.3, we have that Sk is nonexpansive with FSk FS. It follows that
kn − p an Sk xn bn yn cn zn − p
≤ an Sk xn − p bn yn − p cn zn − p
≤ an xn − p bn xn − p cn xn − p xn − p.
3.12
It follows that
xn1 − p n γfxn − Ap βn xn − p 1 − βn I − n A kn − p ≤ 1 − βn − n γ kn − p βn xn − p n γfxn − Ap
≤ 1 − βn − n γ xn − p βn xn − p n γfxn − Ap
≤ 1 − n γ xn − p n γ fxn − f p n γf p − Ap
≤ 1 − n γ xn − p n γαxn − p n γf p − Ap
γf p − Ap
1 − γ − αγ n xn − p γ − αγ n
γ − αγ
γf p − Ap
.
≤ max xn − p,
γ − αγ
3.13
By simple induction, we have
γf p − Ap
xn − p ≤ max x1 − p,
,
γ − αγ
∀n ∈ N.
3.14
Hence, {xn } is bounded, so are {un }, {vn }, {zn }, {yn }, {kn }, {fxn }, {Cun }, and {Bvn }.
Fixed Point Theory and Applications
13
Step 4. We claim that limn → ∞ xn1 − xn 0.
φ ,ϕ
φ ,ϕ
Observing that un Tr 1 xn ∈ dom ϕ and un1 Tr 1 xn1 ∈ dom ϕ, by the
φ ,ϕ
nonexpansiveness of Tr 1 , we get
φ ,ϕ
φ ,ϕ
un1 − un Tr 1 xn1 − Tr 1 xn ≤ xn1 − xn .
φ2 ,ϕ
Similarly, let vn Ts
φ2 ,ϕ
xn ∈ dom ϕ and vn1 Ts
3.15
xn1 ∈ dom ϕ, we have
φ ,ϕ
φ ,ϕ
vn1 − vn Ts 2 xn1 − Ts 2 xn ≤ xn1 − xn .
3.16
From zn PE un − μn Cun and yn PE vn − λn Bvn , we compute
zn1 − zn PE un1 − μn1 Cun1 − PE un − μn Cun ≤ un1 − μn1 Cun1 − un − μn Cun un1 − μn1 Cun1 − un − μn1 Cun μn − μn1 Cun ≤ un1 − μn1 Cun1 − un − μn1 Cun μn − μn1 Cun I − μn1 C un1 − I − μn1 C un μn − μn1 Cun ≤ un1 − un μn − μn1 Cun ≤ xn1 − xn μn − μn1 Cun .
3.17
Similarly, we have
yn1 − yn PE vn1 − λn1 Bvn1 − PE vn − λn Bvn ≤ vn1 − vn |λn − λn1 |Bvn 3.18
≤ xn1 − xn |λn − λn1 |Bvn .
Observing that
kn an Sk xn bn yn cn zn ,
kn1 an1 Sk xn1 bn1 yn1 cn1 zn1 ,
3.19
we obtain
kn1 − kn ≤ an1 Sk xn1 − Sk xn |an1 − an |Sk xn bn1 yn1 − yn |bn1 − bn |yn cn1 zn1 − zn |cn1 − cn |zn ≤ an1 xn1 − xn |an1 − an |Sk xn bn1 yn1 − yn |bn1 − bn |yn cn1 zn1 − zn |cn1 − cn |zn .
3.20
14
Fixed Point Theory and Applications
Substituting 3.17 and 3.18 into 3.20, we have
kn1 − kn ≤ an1 xn1 − xn |an1 − an |Sk xn bn1 {xn1 − xn |λn − λn1 |Bvn }
cn1 xn1 − xn μn − μn1 Cun |bn1 − bn |yn |cn1 − cn |zn ≤ xn1 − xn M1 |an1 − an | |bn1 − bn | |cn1 − cn | |λn − λn1 | μn − μn1 ,
3.21
where M1 is an appropriate constant such that M1 max{supn≥1 Sk xn ,yn ,zn , Bvn ,
Cun }.
Putting xn1 1 − βn ln βn xn , for all n ≥ 1, we have
xn1 − βn xn n γfxn 1 − βn I − n A kn
ln .
1 − βn
1 − βn
3.22
Then, we compute
1 − βn1 I − n1 A kn1
ln1 − ln 1 − βn1
n γfxn 1 − βn I − n A kn
−
1 − βn
n1 γfxn1 n1
n
γfxn1 −
γfxn kn1 − kn
1 − βn1
1 − βn
3.23
n
n1
Akn −
Akn1
1 − βn
1 − βn1
n1 n γfxn1 − Akn1 Akn − γfxn 1 − βn1
1 − βn
kn1 − kn .
It follows from 3.21 and 3.23, that
ln1 − ln − xn1 − xn ≤
n1 γfxn1 Akn1 n Akn γfxn 1 − βn1
1 − βn
kn1 − kn − xn1 − xn ≤
n1 γfxn1 Akn1 n Akn γfxn 1 − βn1
1 − βn
M1 |an1 − an | |bn1 − bn | |cn1 − cn | |λn − λn1 | μn − μn1 .
3.24
Fixed Point Theory and Applications
15
This together with C2, C3, C4, and C6 imply that
lim supln1 − ln − xn1 − xn ≤ 0.
n→∞
3.25
Hence, by Lemma 2.9, we obtain ln − xn → 0 as n → ∞. It follows that
lim xn1 − xn lim 1 − βn ln − xn 0.
n→∞
n→∞
3.26
So, we also get
lim un1 − un lim vn1 − vn lim zn1 − zn n→∞
n→∞
n→∞
lim yn1 − yn lim kn1 − kn 0.
n→∞
3.27
n→∞
Observe that
xn1 − xn n γfxn − Axn 1 − βn − n γ kn − xn .
3.28
By condition C2 and 3.26, we have
limn → ∞ kn − xn 0.
3.29
Step 5. We claim that the following statements hold:
s1 limn → ∞ xn − vn 0;
s2 limn → ∞ xn − un 0;
s3 limn → ∞ xn − yn 0;
s4 limn → ∞ xn − zn 0.
Indeed, pick any p ∈ Θ, to obtain
φ1 ,ϕ
φ ,ϕ 2
un − p2 xn − Tr 1 p
Tr
φ ,ϕ
φ ,ϕ
≤ Tr 1 xn − Tr 1 p, xn − p
un − p, xn − p
3.30
1 un − p2 xn − p2 − xn − un 2 .
2
Therefore,
un − p2 ≤ xn − p2 − xn − un 2 .
3.31
16
Fixed Point Theory and Applications
Similarly, we have
vn − p2 ≤ xn − p2 − xn − vn 2 .
3.32
kn − p2 an Sk xn bn yn cn zn − p2
2
2
2
≤ an Sk xn − p bn yn − p cn zn − p
2
≤ an xn − p bn vn − p cn un − p.
3.33
Note that
Substituting 3.31 and 3.32 into 3.33, we obtain
kn − p2 ≤ an xn − p2 bn vn − p cn un − p
2
2
2
≤ an xn − p bn xn − p − xn − vn 2 cn xn − p − xn − un 2 3.34
2
xn − p − bn xn − vn 2 − cn xn − un 2 .
From Lemma 2.1, 3.2 and 3.34, we obtain
xn1 − p2 n γfxn − Ap βn xn − p 1 − βn I − n Akn − p2
2
2 2
≤ n γfxn − Ap βn xn − p 1 − βn − n γ kn − p
2
2
≤ n γfxn − Ap βn xn − p
2
1 − βn − n γ xn − p − bn xn − vn 2 − cn xn − un 2
2 2 n γfxn − Ap 1 − n γ xn − p − 1 − βn − n γ bn xn − vn 2
− 1 − βn − n γ cn xn − un 2
2 2 ≤ n γfxn − Ap xn − p − 1 − βn − n γ bn xn − vn 2
3.35
− 1 − βn − n γ cn xn − un 2 .
It follows that
2 2 2
1 − βn − n γ cn xn − un 2 ≤ n γfxn − Ap xn − p − xn1 − p
2
≤ n γfxn − Ap xn1 − xn xn − p xn1 − p .
3.36
Fixed Point Theory and Applications
17
From C2, C6, and 3.26, we also have
lim xn − un 0.
n→∞
3.37
Similarly, using 3.35 again, we have
2 2 2
1 − βn − n γ bn xn − vn 2 ≤ n γfxn − Ap xn − p − xn1 − p
2
≤ n γfxn − Ap xn1 − xn xn − p xn1 − p .
3.38
From C2, C6, and 3.26, we also have
lim xn − vn 0.
3.39
lim un − vn 0.
3.40
n→∞
From 3.37 and 3.39, we have
n→∞
For p ∈ Θ, we compute
zn − p2 PE un − μn Cun − PE p − μn Cp2
2
≤ un − μn Cun − p − μn Cp
2
un − p − μn Cun − Cp
2
2
≤ un − p − 2μn un − p, Cun − Cp μ2n Cun − Cp
2
2
≤ xn − p μn μn − 2ξ Cun − Cp
2
2
≤ xn − p − μn 2ξ − μn Cun − Cp .
3.41
Similarly, we have
yn − p2 ≤ xn − p2 − λn 2β − λn Bvn − Bp2 .
3.42
18
Fixed Point Theory and Applications
Substituting 3.41 and 3.42 into 3.33, we also have
kn − p2 ≤ an Sk xn − p2 bn yn − p2 cn zn − p2
2
2
2 ≤ an xn − p bn xn − p − λn 2β − λn Bvn − Bp
2
2 cn xn − p − μn 2ξ − μn Cun − Cp
3.43
2
2
2
xn − p − bn λn 2β − λn Bvn − Bp − cn μn 2ξ − μn Cun − Cp .
On the other hand, we note that
xn1 − p2
2
2 2
≤ n γfxn − Ap βn xn − p 1 − βn − n γ kn − p
2
2
≤ n γfxn − Ap βn xn − p
2
2
2 1 − βn − n γ xn − p − bn λn 2β − λn Bvn − Bp − cn μn 2ξ − μn Cun − Cp
2 2 2
n γfxn − Ap 1 − n γ xn − p − 1 − βn − n γ bn λn 2β − λn Bvn − Bp
2
− 1 − βn − n γ cn μn 2ξ − μn Cun − Cp
2 2 2
≤ n γfxn − Ap xn − p − 1 − βn − n γ bn λn 2β − λn Bvn − Bp
2
− 1 − βn − n γ cn μn 2ξ − μn Cun − Cp .
3.44
It follows that
2
1 − βn − n γ cn μn 2ξ − μn Cun − Cp
2 2 2
≤ n γfxn − Ap xn − p − xn1 − p
2
≤ n γfxn − Ap xn1 − xn xn − p xn1 − p .
3.45
From C2, C5, C6, and 3.26, we have
lim Cun − Cp 0.
n→∞
3.46
Thanks to 3.44, we also have
2
1 − βn − n γ bn λn 2β − λn Bvn − Bp
2
≤ n γfxn − Ap xn1 − xn xn − p xn1 − p .
3.47
Fixed Point Theory and Applications
19
From C2, C5, C6, and 3.26, we obtain
lim Bvn − Bp 0.
n→∞
3.48
Observe that
yn − p2
2
PE vn − λn Bvn − PE p − λn Bp
≤ I − λn Bvn − I − λn Bp, yn − p
1 I − λn Bvn − I − λn Bp2 yn − p2 − I − λn Bvn − I − λn Bp − yn − p 2
2
1 vn − p2 yn − p2 − vn − yn − λn Bvn − Bp 2
≤
2
1 xn − p2 yn − p2 − vn − yn 2 − λ2n Bvn − Bp2 2λn vn − yn , Bvn − Bp ,
≤
2
3.49
and hence
yn − p2 ≤ xn − p2 − vn − yn 2 2λn vn − yn Bvn − Bp.
3.50
Similarly, we can obtain that
zn − p2 ≤ xn − p2 − un − zn 2 2μn un − zn Cun − Cp.
3.51
Substituting 3.50 and 3.51 into 3.33, we also have
kn − p2 ≤ an Sk xn − p2 bn yn − p2 cn zn − p2
2
2 2
≤ an xn − p bn xn − p − vn − yn 2λn vn − yn Bvn − Bp
2
cn xn − p − un − zn 2 2μn un − zn Cun − Cp
2
2
xn − p − bn vn − yn 2bn λn vn − yn Bvn − Bp
− cn un − zn 2 2μn un − zn Cun − Cp.
3.52
20
Fixed Point Theory and Applications
On the other hand, we have
xn1 − p2 ≤ n γfxn − Ap2 βn xn − p2 1 − βn − n γ kn − p2
2
2
≤ n γfxn − Ap βn xn − p
2
2
1 − βn − n γ xn − p − bn vn − yn 2bn λn vn − yn Bvn − Bp
−cn un − zn 2 2μn un − zn Cun − Cp
2 2 2
n γfxn − Ap 1 − n γ xn − p − 1 − βn − n γ bn vn − yn 2bn 1 − βn − n γ λn vn − yn Bvn − Bp − 1 − βn − n γ cn un − zn 2
2cn 1 − βn − n γ μn un − zn Cun − Cp
2 2 2
≤ n γfxn − Ap xn − p − 1 − βn − n γ bn vn − yn 2bn 1 − βn − n γ λn vn − yn Bvn − Bp − 1 − βn − n γ cn un − zn 2
2cn 1 − βn − n γ μn un − zn Cun − Cp
3.53
and hence
2 2 2
2
1 − βn − n γ bn vn − yn ≤ n γfxn − Ap xn − p − xn1 − p
2bn 1 − βn − n γ λn vn − yn Bvn − Bp
2cn 1 − βn − n γ μn un − zn Cun − Cp
2
≤ n γfxn − Ap xn1 − xn xn − p xn1 − p
2bn 1 − βn − n γ λn vn − yn Bvn − Bp
2cn 1 − βn − n γ μn un − zn Cun − Cp.
3.54
From C2, C6, 3.26, 3.46, and 3.48, we also have
lim vn − yn 0.
n→∞
3.55
Fixed Point Theory and Applications
21
Similarly, using 3.53 again, we can prove
n→∞
lim un − zn 0.
3.56
lim xn − yn 0.
3.57
lim xn − zn 0.
3.58
From 3.39 and 3.55, we also have
n→∞
From 3.37 and 3.56, we have
n→∞
Step 6. We claim that lim supn → ∞ A − γfq, q − xn ≤ 0, where q PΘ I − A γfq is the
unique solution of the variational inequality A − γfq, x − q ≥ 0, for all x ∈ Θ.
To show this inequality, we choose a subsequence {xni } of {xn } such that
lim sup A − γf q, q − xn lim A − γf q, q − xni .
n→∞
i→∞
3.59
Since {xni } is bounded, there exists a subsequence {xnij } of {xni } which converges weakly to
z ∈ E. Without loss of generality, we can assume that xni z. We claim that z ∈ Θ.
a1 First, we prove that z ∈ FS ∩ VIE, C ∩ VIE, B.
Assume also that λn → λ ∈ d, 2β and μn → μ ∈ e, 2ξ.
Define a mapping Ω : E → E by
Ωx aSk x bPE 1 − μC x cPE 1 − λBx,
∀x ∈ E,
3.60
where limn → ∞ an a, limn → ∞ bn b, and limn → ∞ cn c, for some a, b, c ∈ 0, 1. From
Lemma 2.5, we have that Ω is nonexpansive with
FΩ FSk ∩ F PE 1 − μC ∩ FPE 1 − λB FS ∩ VIE, C ∩ VIE, B.
3.61
22
Fixed Point Theory and Applications
Notice that
Ωxn − xn ≤ Ωxn − kn kn − xn ≤ aSk xn bPE 1 − λBxn cPE 1 − μC xn − an Sk xn bn yn cn zn kn − xn ≤ |a − an |Sk xn bPE I − λBxn − bn PE I − λn Bxn bn PE I − λn Bxn − bn PE I − λn Bvn cPE I − μC xn − cn PE 1 − μn C xn cn PE I − μn B xn − cn PE I − μn C un kn − xn ≤ |a − an |Sk xn |b − bn |xn |bn λn − bλ|Bxn |c − cn |xn cn μn − cμCxn bn xn − vn cn xn − un kn − xn ≤ K1 |a − an | |b − bn | |c − cn | |bλ − bn λn | cμ − cn μn bn xn − vn cn xn − un kn − xn ,
3.62
where K1 is an appropriate constant such that
K1 max supxn , supBxn , supCxn , supSk xn .
n≥1
n≥1
n≥1
3.63
n≥1
From C6, 3.37, 3.39, and 3.29, we obtain
lim xn − Ωxn 0.
3.64
n→∞
By Lemma 2.4, we have z ∈ FΩ, that is, z ∈ FS ∩ VIE, C ∩ VIE, B.
a2 Now, we prove that z ∈ FS ∩ MEPφ1 , ϕ ∩ MEPφ2 , ϕ.
Define a mapping Q : E → E by
φ1 ,ϕ
Qx aSk x bTr
φ2 ,ϕ
x cTs
x,
∀x ∈ E,
3.65
where limn → ∞ an a, limn → ∞ bn b, and limn → ∞ cn c, for some a, b, c ∈ 0, 1. From
Lemma 2.5, we have that Q is nonexpansive with
φ ,ϕ φ ,ϕ ∩ F Ts 2
FS ∩ MEP φ1 , ϕ ∩ φ2 , ϕ .
FQ FSk ∩ F Tr 1
3.66
Fixed Point Theory and Applications
23
On the other hand, we have
Qxn − xn ≤ Qxn − kn kn − xn φ ,ϕ
φ ,ϕ
≤ aSk xn bTr 1 xn cTs 2 xn − an Sk xn bn yn cn zn kn − xn φ ,ϕ φ ,ϕ ≤ |a − an |Sk xn |b|Tr 1 xn |c|Ts 2 xn |bn − b|PE I − λn Bvn |cn − c|PE I − μn C un |b|PE I − λn Bvn |c|PE I − μn C un kn − xn ≤ K2 |a − an | |b − bn | |c − cn | 2|b| 2|c| kn − xn ,
3.67
where K2 is an appropriate constant such that
φ ,ϕ φ ,ϕ K2 max supTr 1 xn , supTs 2 xn , supSk xn ,
n≥1
n≥1
n≥1
φ ,ϕ φ ,ϕ 1
2
xn PE I − λn Bvn , sup Ts
xn PE I − μn C un .
sup Tr
n≥1
n≥1
3.68
From C6 and 3.29, we obtain
lim xn − Qxn 0.
n→∞
3.69
Since PΘ I − A γfq is a contraction with the coefficient α ∈ 0, 1, there exists a unique
fixed point. We use q to denote the unique fixed point to the mapping PΘ I − A γfq, that
is, q PΘ I − A γfq. 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 from 3.69, that
lim xni − Qxni 0.
n→∞
3.70
It follows from Lemma 2.4, we obtain that z ∈ FQ. Hence z ∈ Θ, where Θ : FS ∩
VIE, C ∩ VIE, B ∩ MEPφ1 , ϕ ∩ MEPφ2 , ϕ. From 3.59 and 2.4, we arrive at
lim sup A − γf q, q − xn lim sup A − γf q, q − xni
n→∞
n→∞
A − γf q, q − z ≤ 0.
3.71
24
Fixed Point Theory and Applications
On the other hand, we have
A − γf q, q − xn1 A − γf q, xn − xn1 A − γf q, q − xn
≤ A − γf qxn − xn1 A − γf q, q − xn .
3.72
From 3.26 and 3.71, we obtain that
lim sup A − γf q, q − xn1 ≤ 0.
n→∞
Step 7. We claim that limn → ∞ xn − q 0.
Indeed, by 3.2 and using Lemmas 2.2 and 2.11, we observe that
2
xn1 − z2 n γfxn βn xn 1 − βn I − n Akn − q
2
1 − βn I − n A
≤ 1 − βn kn − q βn xn − q 1 − βn
2n γfxn − Aq, xn1 − q
2
2
1 − βn I − n A ≤ 1 − βn kn − q βn xn − q
1 − βn
2n γfxn − f q , xn1 − q 2n γf q − Aq, xn1 − q
2
2
1 − βn I − n A
≤ 1 − βn kn − q
βn xn − q
1 − βn
2n γαxn − qxn1 − q 2n γf q − Aq, xn1 − q
1 − βn I − n A2 kn − q2 βn xn − q2
≤
1 − βn
2 2 n γα xn − q xn1 − q 2n γf q − Aq, xn1 − q
1 − βn I − n A2 xn − q2 βn xn − q2
≤
1 − βn
2 2 n γα xn − q xn1 − q 2n γf q − Aq, xn1 − q
3.73
Fixed Point Theory and Applications
25
≤
2
2
1 − βn − γn
βn n γα xn − q
1 − βn
2
n γαxn1 − q 2n γf q − Aq, xn1 − q
γ 2 n2 xn − q2
≤ 1 − 2γ − αγ n 1 − βn
2
n γαxn1 − q 2n γf q − Aq, xn1 − q
3.74
which implies that
2 γ − αγ n xn − q2
xn1 − z ≤ 1 −
1 − αγn
2
2
γ n2 n
xn − q 2n γf q − Aq, xn1 − q .
1 − αγn 1 − βn
2
3.75
Taking
n
σn 1 − αγn
γ 2 n2 xn − q2 2n γf q − Aq, xn1 − q ,
1 − βn
2 γ − αγ n
n .
1 − αγn
3.76
Then we can rewrite 3.75 as
xn1 − z2 ≤ 1 − n xn − z2 σn .
3.77
We have lim supn → ∞ σn /n ≤ 0. Applying Lemma 2.10 to 3.77, we conclude that {xn }
converges strongly to q in norm. This completes the proof.
If the mapping S is nonexpansive, then Sk S0 S. We can obtain the following result
from Theorem 3.1 immediately.
Corollary 3.2. Let E be a nonempty closed convex subset of a real Hilbert space H. Let φ1 and φ2 be
two bifunction from E × E to R satisfying (A1)–(A4) and let ϕ : E → R ∪ {∞} be a proper lower
semicontinuous and convex function. Let C : E → H be an ξ-inverse strongly monotone mapping
and B : E → H be an β-inverse strongly monotone mapping. Let f : E → E be a contraction
mapping with coefficient α 0 < α < 1 and let A be a strongly positive linear bounded operator on H
with coefficient γ > 0 and 0 < γ < γ/α. Let S : E → E a nonexpansive mapping with a fixed point.
Assume that
Θ : FS ∩ VIE, C ∩ VIE, B ∩ MEP φ1 , ϕ ∩ MEP φ2 , ϕ /
∅.
3.78
26
Fixed Point Theory and Applications
Assume that either B1 or B2. Let {xn } be a sequence generated by the following iterative algorithm:
x1 ∈ E,
un ∈ E,
φ1 ,ϕ
un Tr
vn ∈ E,
xn ,
φ ,ϕ
vn Ts 2 xn ,
zn PE un − μn Cun ,
3.79
yn PE vn − λn Bvn ,
xn1
kn an Sxn bn yn cn zn ,
n γfxn βn xn 1 − βn I − n A kn ,
∀n ≥ 1,
where {n }, {βn }, {an }, {bn }, and {cn } are sequences in 0, 1 and {λn }, {μn } are positive sequences.
Assume that the control sequences satisfy the following restrictions:
C1 an bn cn 1,
C2 limn → ∞ n 0 and
∞
n1 n
∞,
C3 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1,
C4 limn → ∞ |λn1 − λn | limn → ∞ |μn1 − μn | 0,
C5 d ≤ λn ≤ 2β, e ≤ μn ≤ 2ξ, where d, e are two positive constants,
C6 limn → ∞ an a, limn → ∞ bn b and limn → ∞ cn c, for some a, b, c ∈ 0, 1.
Then, {xn } converges strongly to a point q ∈ Θ which is the unique solution of the variational
inequality
A − γf q, x − q ≥ 0,
∀x ∈ Θ
3.80
or equivalent q PΘ I − A γfq, where P is a metric projection mapping form H onto Θ.
Finally, we consider the following convex feasibility problem CFP:
finding an x ∈
N
Ci ,
3.81
i1
where N ≥ 1 is an integer and each Ci is assumed to be the of solutions of equilibrium
problem with the bifunction φi , i 1, 2, 3, . . . , N and the solution set of the variational
inequality problem. There is a considerable investigation on CEP in the setting of Hilbert
spaces which captures applications in various disciplines such as image restoration 42, 43,
computer tomography 44, and radiation therapy treatment planning 45.
The following result can be concluded from Theorem 3.1 easily.
Theorem 3.3. Let E be a nonempty closed convex subset of a real Hilbert space H. Let be a φi
bifunction from E × E to R satisfying (A1)–(A4) and let ϕ : E → R ∪ {∞} be a proper lower
Fixed Point Theory and Applications
27
semicontinuous and convex function. Let Ci : E → H be an ξi -inverse strongly monotone mapping
for each i ∈ {1, 2, 3, . . . , N}. Let f : E → E be a contraction mapping with coefficient α 0 < α < 1
and let A be a strongly positive linear bounded operator on H with coefficient γ > 0 and 0 < γ < γ/α.
Let S : E → E be a k-strict pseudo-contraction with a fixed point. Define a mapping Sk : E → E by
Sk x kx 1 − kSx, for all x ∈ E. Assume that
F : FS ∩
N
∩
VIE, Ci i1
N
MEP φi , ϕ
∅.
/
3.82
i1
Assume that either B1 or B2. Let {xn } be a sequence generated by the following iterative algorithm:
x1 ∈ E,
un,i ∈ E,
1
ui − un,i , un,i − xn ≥ 0,
ri
φi un,i , ui ϕui − ϕun,i kn αn,0 Sk xn N
i1
xn1 n γfxn βn xn ∀ui ∈ E, ∀i ∈ {1, 2, 3, . . . , N},
αn,i PE un,i − μn,i Ci un,i ,
1 − βn I − n A kn ,
∀n ≥ 1,
3.83
where αn,0 , αn,1 , αn,2 , αn,3 , . . . , αn,N ∈ 0, 1 such that N
i0 αn,i 1, {μn,i } are positive sequences and
{n }, {βn } are sequences in 0, 1. Assume that the control sequences satisfy the following restrictions:
C1 limn → ∞ n 0 and
∞
n1 n
∞,
C2 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1,
C3 limn → ∞ |μn1,i − μn,i | 0, for each 1 ≤ i ≤ N,
C4 ei ≤ μn,i ≤ 2ξi , where ei is some positive constant for each 1 ≤ i ≤ N,
C5 limn → ∞ αn,i αi ∈ 0, 1, for each 1 ≤ i ≤ N.
Then, {xn } converges strongly to a point q ∈ F which is the unique solution of the variational
inequality
A − γf q, x − q ≥ 0,
∀x ∈ F
3.84
or equivalent q PF I − A γfq, where P is a metric projection mapping form H onto F.
Acknowledgments
The authors would like to thank the referees for their valuable suggestions to improve this
paper. This work was supported by the Center of Excellence in Mathematics, the Commission
on Higher Education, Thailand.
28
Fixed Point Theory and Applications
References
1 W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.
2 F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert
space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.
3 G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in
Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007.
4 H. Zhou, “Convergence theorems of fixed points for κ-strict pseudo-contractions in Hilbert spaces,”
Nonlinear Analysis. Theory, Methods & Applications. Series A, vol. 69, no. 2, pp. 456–462, 2008.
5 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.
6 E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The
Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.
7 J. C. Yao and O. Chadli, “Pseudomonotone complementarity problems and variational inequalities,”
in Handbook of Generalized Convexity and Monotonicity, J. P. Crouzeix, N. Haddjissas, and S. Schaible,
Eds., pp. 501–558, 2005.
8 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.
9 K. Aoyama, Y. Kimura, and W. Takahashi, “Maximal monotone operators and maximal monotone
functions for equilibrium problems,” Journal of Convex Analysis, vol. 15, no. 2, pp. 395–409, 2008.
10 P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming using proximal-like algorithms,”
Mathematical Programming, vol. 78, no. 1, pp. 29–41, 1997.
11 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.
12 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.
13 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.
14 C. Jaiboon and P. Kumam, “Strong convergence theorems for solving equilibrium problems and fixed
point problems of ξ-strict pseudo-contraction mappings by two hybrid projection methods,” Journal
of Computational and Applied Mathematics, vol. 234, no. 3, pp. 722–732, 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. Second Series, vol. 32, no. 2, pp. 173–185, 2009.
16 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.
17 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.
18 X. Qin, Y. J. Cho, and S. M. Kang, “Convergence analysis on hybrid projection algorithms for
equilibrium problems and variational inequality problems,” Mathematical Modelling and Analysis, vol.
14, no. 3, pp. 335–351, 2009.
19 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.
20 S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem
and a nonexpansive mapping in a Hilbert space,” Nonlinear Analysis. Theory, Methods & Applications,
vol. 69, no. 3, pp. 1025–1033, 2008.
21 W. Takahashi and K. Zembayashi, “Strong convergence theorem by a new hybrid method for
equilibrium problems and relatively nonexpansive mappings,” Fixed Point Theory and Applications,
vol. 2008, Article ID 528476, 11 pages, 2008.
Fixed Point Theory and Applications
29
22 W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems
and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis. Theory, Methods &
Applications, vol. 70, no. 1, pp. 45–57, 2009.
23 Y. Yao, Y.-C. Liou, and J.-C. Yao, “A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems,” Fixed Point Theory and Applications,
vol. 2007, Article ID 417089, 15 pages, 2008.
24 Y. Yao, Y.-C. Liou, and Y.-J. Wu, “An extragradient method for mixed equilibrium problems and fixed
point problems,” Fixed Point Theory and Applications, vol. 2009, Article ID 632819, 15 pages, 2009.
25 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.
26 W.-Y. Zeng, N.-J. Huang, and C.-W. Zhao, “Viscosity approximation methods for generalized mixed
equilibrium problems and fixed points of a sequence of nonexpansive mappings,” Fixed Point Theory
and Applications, vol. 2008, Article ID 714939, 15 pages, 2008.
27 Y. Yao, Y.-C. Liou, and J.-C. Yao, “A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems,” Fixed Point Theory and Applications,
vol. 2008, Article ID 417089, 15 pages, 2008.
28 R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the
American Mathematical Society, vol. 149, pp. 75–88, 1970.
29 W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and
monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428,
2003.
30 Y. Yao and J.-C. Yao, “On modified iterative method for nonexpansive mappings and monotone
mappings,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 1551–1558, 2007.
31 G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”
Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.
32 S. Plubtieng and R. Punpaeng, “A general iterative method for equilibrium problems and fixed point
problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 455–
469, 2007.
33 Y. Liu, “A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert
spaces,” Nonlinear Analysis. Theory, Methods & Applications. Series A, vol. 71, no. 10, pp. 4852–4861,
2009.
34 R. Wangkeeree and R. Wangkeeree, “A general iterative method for variational inequality problems,
mixed equilibrium problems, and fixed point problems of strictly pseudocontractive mappings in
Hilbert spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 519065, 32 pages, 2009.
35 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.
36 M. O. Osilike and D. I. Igbokwe, “Weak and strong convergence theorems for fixed points of
pseudocontractions and solutions of monotone type operator equations,” Computers & Mathematics
with Applications, vol. 40, no. 4-5, pp. 559–567, 2000.
37 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.
38 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.
39 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 Mathematics, vol. 12, no. 6, pp. 1401–1432, 2008.
40 T. Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications,
vol. 305, no. 1, pp. 227–239, 2005.
41 H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical
Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
42 P. L. Combettes, “The convex feasibility problemml: in image recovery,” in Advances in Imaging and
Electron Physics, P. Hawkes, Ed., vol. 95, pp. 155–270, Academic Press, Orlando, Fla, USA, 1996.
30
Fixed Point Theory and Applications
43 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.
44 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.
45 Y. Censor and S. A. Zenios, Parallel Optimization, Numerical Mathematics and Scientific Computation,
Oxford University Press, New York, NY, USA, 1997.