Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 308791, 18 pages
doi:10.1155/2012/308791
Research Article
Strong Convergence to Solutions of Generalized
Mixed Equilibrium Problems with Applications
Prasit Cholamjiak,1, 2 Suthep Suantai,2, 3 and Yeol Je Cho4
1
School of Science, University of Phayao, Phayao 56000, Thailand
Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
3
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
4
Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701,
Republic of Korea
2
Correspondence should be addressed to Yeol Je Cho, yjcho@gnu.ac.kr
Received 21 October 2011; Accepted 23 November 2011
Academic Editor: Yonghong Yao
Copyright q 2012 Prasit Cholamjiak et al. 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 a Halpern-type iteration for a generalized mixed equilibrium problem in uniformly
smooth and uniformly convex Banach spaces. Strong convergence theorems are also established
in this paper. As applications, we apply our main result to mixed equilibrium, generalized
equilibrium, and mixed variational inequality problems in Banach spaces. Finally, examples and
numerical results are also given.
1. Introduction
Let E be a real Banach space, C a nonempty, closed, and convex subset of E, and E∗ the dual
space of E. Let T : C → C be a nonlinear mapping. The fixed points set of T is denoted by
FT , that is, FT {x ∈ C : x T x}.
One classical way often used to approximate a fixed point of a nonlinear self-mapping
T on C was firstly introduced by Halpern 1 which is defined by x1 x ∈ C and
xn1 αn x 1 − αn T xn ,
∀n ≥ 1,
1.1
where {αn } is a real sequence in 0, 1. He proved, in a real Hilbert space, a strong convergence
theorem for a nonexpansive mapping T when αn n−a for any a ∈ 0, 1.
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Subsequently, motivated by Halpern 1, many mathematicians devoted time to study
algorithm 1.1 in different styles. Several strong convergence results for nonlinear mappings
were also continuously established in some certain Banach spaces see also 2–9.
Let f : C × C → R be a bifunction, A : C → E∗ a mapping, and ϕ : C → R a
real-valued function. The generalized mixed equilibrium problem is to find x ∈ C such that
f x,
y Ax,
y − x ϕ y ≥ ϕx,
∀y ∈ C.
1.2
The solutions set of 1.2 is denoted by GMEPf, A, ϕ see Peng and Yao 10.
If A ≡ 0, then the generalized mixed equilibrium problem 1.2 reduces to the
following mixed equilibrium problem: finding x ∈ C such that
f x,
y ϕ y ≥ ϕx,
∀y ∈ C.
1.3
The solutions set of 1.3 is denoted by MEPf, ϕ see Ceng and Yao 11.
If f ≡ 0, then the generalized mixed equilibrium problem 1.2 reduces to the following
mixed variational inequality problem: finding x ∈ C such that
Ax,
y − x ϕ y ≥ ϕx,
∀y ∈ C.
1.4
The solutions set of 1.4 is denoted by VIC, A, ϕ see Noor 12.
If ϕ ≡ 0, then the generalized mixed equilibrium problem 1.2 reduces to the following
generalized equilibrium problem: finding x ∈ C such that
f x,
y Ax,
y − x ≥ 0,
∀y ∈ C.
1.5
The solutions set of 1.5 is denoted by GEPf, A see Moudafi 13.
If ϕ ≡ 0, then the mixed equilibrium problem 1.3 reduces to the following equilibrium problem: finding x ∈ C such that
f x,
y ≥ 0,
∀y ∈ C.
1.6
The solutions set of 1.6 is denoted by EPf see Combettes and Hirstoaga 14.
If f ≡ 0, then the mixed equilibrium problem 1.3 reduces to the following convex
minimization problem: finding x ∈ C such that
ϕ y ≥ ϕx,
The solutions set of 1.7 is denoted by CMPϕ.
∀y ∈ C.
1.7
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If ϕ ≡ 0, then the mixed variational inequality problem 1.4 reduces to the following
variational inequality problem: finding x ∈ C such that
Ax,
y − x ≥ 0,
∀y ∈ C.
1.8
The solutions set of 1.8 is denoted by VIC, A see Stampacchia 7.
The problem 1.2 is very general in the sense that it includes, as special cases,
optimization problems, variational inequalities, minimax problems, the Nash equilibrium
problem in noncooperative games, and others. For more details on these topics, see, for
instance, 14–34.
For solving the generalized mixed equilibrium problem, let us assume the following
25:
A1 fx, x 0 for all x ∈ C;
A2 f is monotone, that is, fx, y fy, x ≤ 0 for all x, y ∈ C;
A3 for all x, y, z ∈ C, lim supt↓0 ftz 1 − tx, y ≤ fx, y;
A4 for all x ∈ C, fx, · is convex and lower semicontinuous.
The purpose of this paper is to investigate strong convergence of Halpern-type
iteration for a generalized mixed equilibrium problem in uniformly smooth and uniformly
convex Banach spaces. As applications, our main result can be deduced to mixed equilibrium,
generalized equilibrium, mixed variational inequality problems, and so on. Examples and
numerical results are also given in the last section.
2. Preliminaries and Lemmas
In this section, we need the following preliminaries and lemmas which will be used in our
main theorem.
Let E be a real Banach space and let U {x ∈ E : x
1} be the unit sphere of E. A
Banach space E is said to be strictly convex if, for any x, y ∈ U,
x y
x/
y implies 2 < 1.
2.1
It is also said to be uniformly convex if, for any ε ∈ 0, 2, there exists δ > 0 such that, for any
x, y ∈ U,
x − y ≥ ε implies x y < 1 − δ.
2 2.2
It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a
function δ : 0, 2 → 0, 1 called the modulus of convexity of E as follows:
x y
: x, y ∈ E, x
y 1, x − y ≥ ε .
δε inf 1 − 2 2.3
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Then E is uniformly convex if and only if δε > 0 for all ε ∈ 0, 2. A Banach space E is said
to be smooth if the limit
lim
x ty − x
t→0
2.4
t
exists for all x, y ∈ U. It is also said to be uniformly smooth if the limit 2.4 is attained
∗
uniformly for x, y ∈ U. The normalized duality mapping J : E → 2E is defined by
Jx x∗ ∈ E∗ : x, x∗ x
2 x∗ 2
2.5
for all x ∈ E. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm
continuous on each bounded subset of E see 35.
Let E be a smooth Banach space. The function φ : E × E → R is defined by
2
φ x, y x
2 − 2 x, Jy y ,
∀x, y ∈ E.
2.6
Remark 2.1. We know the following: for any x, y, z ∈ E,
1 x
− y
2 ≤ φx, y ≤ x
y
2 ;
2 φx, y φx, z φz, y 2x − z, Jz − Jy;
3 φx, y x − y
2 in a real Hilbert space.
Lemma 2.2 see 36. Let E be a uniformly convex and smooth Banach space and let {xn } and {yn }
be sequences of E such that {xn } or {yn } is bounded and limn → ∞ φxn , yn 0. Then limn → ∞ xn −
yn 0.
Let E be a reflexive, strictly convex, and smooth Banach space and let C be a nonempty
closed and convex subset of E. The generalized projection mapping, introduced by Alber 37, is
a mapping ΠC : E → C, that assigns to an arbitrary point x ∈ E the minimum point of the
functional φy, x, that is, ΠC x x, where x is the solution to the minimization problem:
φx, x min φ y, x : y ∈ C .
2.7
In fact, we have the following result.
Lemma 2.3 see 37. Let C be a nonempty, closed, and convex subset of a reflexive, strictly convex,
and smooth Banach space E and let x ∈ E. Then there exists a unique element x0 ∈ C such that
φx0 , x min{φz, x : z ∈ C}.
Lemma 2.4 see 36, 37. Let C be a nonempty closed and convex subset of a reflexive, strictly
convex, and smooth Banach space E, x ∈ E, and z ∈ C. Then z ΠC x if and only if
Jx − Jz, y − z ≤ 0,
∀y ∈ C.
2.8
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Lemma 2.5 see 36, 37. Let C be a nonempty closed and convex subset of a reflexive, strictly
convex, and smooth Banach space E and let x ∈ E. Then
φ y, ΠC x φΠC x, x ≤ φ y, x ,
∀y ∈ C.
2.9
Lemma 2.6 see 38. Let E be a uniformly convex and uniformly smooth Banach space and C a
nonempty, closed, and convex subset of E. Then ΠC is uniformly norm-to-norm continuous on every
bounded set.
We make use of the following mapping V studied in Alber 37:
V x, x∗ x
2 − 2x, x∗ x∗ 2
2.10
for all x ∈ E and x∗ ∈ E∗ , that is, V x, x∗ φx, J −1 x∗ .
Lemma 2.7 see 39. Let E be a reflexive, strictly convex, smooth Banach space. Then
V x, x∗ 2 J −1 x∗ − x, y∗ ≤ V x, x∗ y∗
2.11
for all x ∈ E and x∗ , y∗ ∈ E∗ .
Lemma 2.8 see 25. Let C be a closed and convex subset of a smooth, strictly convex, and reflexive
Banach space E, let f be a bifunction from C × C to R which satisfies conditions A1–A4, and let
r > 0 and x ∈ E. Then there exists z ∈ C such that
1
f z, y Jz − Jx, y − z ≥ 0,
r
2.12
∀y ∈ C.
Following 25, 40, we know the following lemma.
Lemma 2.9 see 41. Let C be a nonempty closed and convex subset of a smooth, strictly convex,
and reflexive Banach space E. Let A : C → E∗ be a continuous and monotone mapping, let f be a
bifunction from C × C to R satisfying A1–A4, and let ϕ be a lower semicontinuous and convex
function from C to R. For all r > 0 and x ∈ E, there exists z ∈ C such that
1
f z, y Az, y − z ϕ y Jz − Jx, y − z ≥ ϕz,
r
∀y ∈ C.
2.13
Define the mapping Tr : E → 2C as follows:
1
Tr x z ∈ C : f z, y Az, y − z ϕ y Jz − Jx, y − z ≥ ϕz, ∀y ∈ C . 2.14
r
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Then, the followings hold:
1 Tr is single-valued;
2 Tr is firmly nonexpansive-type mapping [42], that is, for all x, y ∈ E,
Tr x − Tr y, JTr x − JTr y ≤ Tr x − Tr y, Jx − Jy ;
2.15
3 FTr GMEPf, A, ϕ;
4 GMEPf, A, ϕ is closed and convex.
Remark 2.10. It is known that T is of firmly nonexpansive type if and only if
φ T x, T y φ T y, T x φT x, x φ T y, y ≤ φ T x, y φ T y, x
2.16
for all x, y ∈ dom T see 42.
The following lemmas give us some nice properties of real sequences.
Lemma 2.11 see 43. Assume that {an } is a sequence of nonnegative real numbers such that
an1 ≤ 1 − αn an bn ,
∀n ≥ 1,
2.17
where {αn } is a sequence in 0, 1 and {bn } is a sequence such that
a ∞
n1 αn ∞;
b lim supn → ∞ bn /αn ≤ 0 or ∞
n1 |bn | < ∞.
Then limn → ∞ an 0.
Lemma 2.12 see 44. Let {γn } be a sequence of real numbers such that there exists a subsequence
{γnj } of {γn } such that γnj < γnj 1 for all j ≥ 1. Then there exists a nondecreasing sequence {mk } of N
such that limk → ∞ mk ∞ and the following properties are satisfied by all (sufficiently large) numbers
k ≥ 1:
γmk ≤ γmk 1 ,
γk ≤ γmk 1 .
2.18
In fact, mk is the largest number n in the set {1, 2, . . . , k} such that the condition γn < γn1 holds.
3. Main Results
In this section, we prove our main theorem in this paper. To this end, we need the following
proposition.
Proposition 3.1. Let C be a nonempty closed and convex subset of a reflexive, strictly convex, and
uniformly smooth Banach space E. Let f be a bifunction from C × C to R satisfying A1–A4,
A : C → E∗ a continuous and monotone mapping, and ϕ a lower semicontinuous and convex function
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from C to R such that GMEPf, A, ϕ /
∅. Let {rn } ⊂ 0, ∞ be such that lim infn → ∞ rn > 0. For
each n ≥ 1, let Trn be defined as in Lemma 2.9. Suppose that x ∈ C and {xn } is a bounded sequence in
C such that limn → ∞ xn − Trn xn 0. Then
lim sup Jx − Jp, xn − p ≤ 0,
n→∞
3.1
where p ΠGMEPf,A,ϕ x and ΠGMEPf,A,ϕ is the generalized projection of C onto GMEPf, A, ϕ.
Proof. Let x ∈ C and put p ΠGMEPf,A,ϕ x. Since E is reflexive and {xn } is bounded, there
exists a subsequence {xnk } of {xn } such that xnk v ∈ C and
lim sup Jx − Jp, xn − p Jx − Jp, v − p .
n→∞
3.2
Put yn Trn xn . Since limk → ∞ xnk − ynk 0, we have ynk v. On the other hand, since E is
uniformly smooth, J is uniformly norm-to-norm continuous on bounded subsets of E. So we
have
lim Jxnk − Jynk 0.
k→∞
3.3
Since lim infk → ∞ rnk > 0,
Jxn − Jyn k
k
lim
0.
k→∞
rnk
3.4
By the definition of Trnk , for any y ∈ C, we see that
1 f ynk , y Aynk , y − ynk ϕ y Jynk − Jxnk , y − ynk ≥ ϕ ynk .
rnk
3.5
By A2, for each y ∈ C, we obtain
f y, ynk ϕ ynk ≤ −f ynk , y ϕ ynk
1 ≤ Aynk , y − ynk ϕ y Jynk − Jxnk , y − ynk .
rnk
3.6
For any t ∈ 0, 1 and y ∈ C, we define yt ty 1 − tv. Then yt ∈ C. It follows by the
monotonicity of A that
f yt , ynk ϕ ynk ≤ Aynk − Ayt , yt − ynk Ayt , yt − ynk
1 Jynk − Jxnk , yt − ynk
ϕ yt rnk
1 ≤ Ayt , yt − ynk ϕ yt Jynk − Jxnk , yt − ynk .
rnk
3.7
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By A4, 3.4, and the weakly lower semicontinuity of ϕ, letting k → ∞, we obtain
f yt , v ϕv ≤ Ayt , yt − v ϕ yt .
3.8
By A1, A4, and the convexity of ϕ, we have
0 f yt , yt ϕ yt − ϕ yt
≤ tf yt , y 1 − tf yt , v tϕ y 1 − tϕv − ϕ yt
t f yt , y ϕ y − ϕ yt 1 − t f yt , v ϕv − ϕ yt
≤ t f yt , y ϕ y − ϕ yt 1 − tAyt , yt − v
t f yt , y ϕ y − ϕ yt 1 − ttAyt , y − v.
3.9
f yt , y ϕ y − ϕ yt 1 − t Ayt , y − v ≥ 0.
3.10
It follows that
By A3, the weakly lower semicontinuity of ϕ, and the continuity of A, letting t → 0, we
obtain
f v, y ϕ y − ϕv Av, y − v ≥ 0,
∀y ∈ C.
3.11
This shows that v ∈ GMEPf, A, ϕ. By Lemma 2.4, we have
lim sup Jx − Jp, xn − p Jx − Jp, v − p ≤ 0.
3.12
n→∞
This completes the proof.
Theorem 3.2. Let C be nonempty, closed, and convex subset of a uniformly smooth and uniformly
convex Banach space E. Let f be a bifunction from C × C to R satisfying A1–A4, A : C → E∗ a
continuous and monotone mapping, and ϕ a lower semicontinuous and convex function from C to R
such that GMEPf, A, ϕ /
∅. Define the sequence {xn } as follows: x1 x ∈ C and
1
f yn , y Ayn , y − yn ϕ y Jyn − Jxn , y − yn ≥ ϕ yn ,
rn
xn1 ΠC J −1 αn Jx 1 − αn Jyn , ∀n ≥ 1,
where {αn } ⊂ 0, 1 and {rn } ⊂ 0, ∞ satisfy the following conditions:
a limn → ∞ αn 0;
b ∞
n1 αn ∞;
c lim infn → ∞ rn > 0.
∀y ∈ C,
3.13
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Then {xn } converges strongly to ΠGMEPf,A,ϕ x, where ΠGMEPf,A,ϕ is the generalized projection of C
onto GMEPf, A, ϕ.
Proof. From Lemma 2.94, we know that GMEPf, A, ϕ is closed and convex. Let p ΠGMEPf,A,ϕ x. Put yn Trn xn and zn J −1 αn Jx 1 − αn Jyn for all n ∈ N. So, by Lemma 2.5,
we have
φ p, xn1 ≤ φ p, zn
≤ αn φ p, x 1 − αn φ p, yn
≤ αn φ p, x 1 − αn φ p, xn .
3.14
By induction, we can show that φp, xn ≤ φp, x for each n ∈ N. Hence {φp, xn } is bounded
and thus {xn } is also bounded.
We next show that if there exists a subsequence {xnk } of {xn } such that
lim φ p, xnk 1 − φ p, xnk 0,
3.15
lim φ p, ynk − φ p, xnk 0.
3.16
lim Jznk − Jynk lim αnk Jx − Jynk 0.
3.17
k→∞
then
k→∞
Since αnk → 0,
k→∞
k→∞
Since J is uniformly norm-to-norm continuous on bounded subsets of E, so is J −1 . It follows
that
lim znk − ynk 0.
k→∞
3.18
Since E is uniformly smooth and uniformly convex, by Lemma 2.6, ΠC is uniformly norm-tonorm continuous on bounded sets. So we obtain
lim xnk 1 − ynk lim ΠC znk − ΠC ynk 0,
3.19
lim Jxnk 1 − Jynk 0.
3.20
k→∞
k→∞
and hence
k→∞
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Furthermore, limk → ∞ φxnk 1 , ynk 0. Indeed, by the definition of φ, we observe that
2
φ xnk 1 , ynk xnk 1 2 − 2 xnk 1 , Jynk ynk xnk 1 , Jxnk 1 − Jynk ynk − xnk 1 , Jynk .
3.21
It follows from 3.19 and 3.20 that limk → ∞ φxnk 1 , ynk 0. On the other hand, from
Remark 2.12, we have
φ p, ynk − φ p, xnk φ p, xnk 1 − φ p, xnk φ p, ynk − φ p, xnk 1
φ p, xnk 1 − φ p, xnk
φ xnk 1 , ynk 2 p − xnk 1 , Jxnk 1 − Jynk .
3.22
φp,ynk −φp,xnk 0.
It follows from 3.20 and 3.21 that klim
→∞
We next consider the following two cases.
Case 1. φp, xn1 ≤ φp, xn for all sufficiently large n. Hence the sequence {φp, xn } is
bounded and nonincreasing. So limn → ∞ φp, xn exists. This shows that limn → ∞ φp, xn1 −
φp, xn 0 and hence
lim φ p, yn − φ p, xn 0.
n→∞
3.23
Since Trn is of firmly nonexpansive type, by Remark 2.10, we have
φ yn , p φ p, yn φ yn , xn φ Trn p, p ≤ φ yn , p φ p, xn ,
3.24
which implies
φ p, yn φ yn , xn ≤ φ p, xn .
3.25
φ yn , xn ≤ φ p, xn − φ p, yn −→ 0
3.26
Hence
as n → ∞. By Lemma 2.2, we obtain
lim xn − yn 0.
3.27
lim sup Jx − Jp, xn − p ≤ 0.
3.28
n→∞
Proposition 3.1 yields that
n→∞
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It also follows that
lim sup Jx − Jp, yn − p ≤ 0.
n→∞
3.29
Finally, we show that xn → p. Using Lemma 2.7, we see that
φ p, xn1 ≤ φ p, zn
V p, αn Jx 1 − αn Jyn
≤ V p, αn Jx 1 − αn Jyn − αn Jx − Jp αn Jx − Jp , zn − p
V p, αn Jp 1 − αn Jyn αn Jx − Jp, zn − p
≤ αn V p, Jp 1 − αn V p, Jyn αn Jx − Jp, zn − p
1 − αn φ p, yn αn Jx − Jp, zn − p
≤ 1 − αn φ p, xn αn Jx − Jp, zn − p
1 − αn φ p, xn αn Jx − Jp, zn − yn Jx − Jp, yn − p .
3.30
Set an φp, xn and bn αn Jx − Jp, zn − yn Jx − Jp, yn − p. We see
∞
that lim supn → ∞ bn /αn ≤ 0. By Lemma 2.11, since
n1 αn ∞, we conclude that
limn → ∞ φp, xn 0. Hence xn → p as n → ∞.
Case 2. There exists a subsequence {φp, xnj } of {φp, xn } such that φp, xnj < φp, xnj 1 for
all j ∈ N. By Lemma 2.12, there exists a strictly increasing sequence {mk } of positive integers
such that the following properties are satisfied by all numbers k ∈ N:
φ p, xmk ≤ φ p, xmk 1 ,
φ p, xk ≤ φ p, xmk 1 .
3.31
So we have
0 ≤ lim φ p, xmk 1 − φ p, xmk
k→∞
≤ lim sup φ p, xn1 − φ p, xn
n→∞
≤ lim sup φ p, zn − φ p, xn
n→∞
≤ lim sup αn φ p, x 1 − αn φ p, yn − φ p, xn
n→∞
lim sup αn φ p, x − φ p, yn φ p, yn − φ p, xn
n→∞
≤ lim sup αn φ p, x − φ p, yn 0.
n→∞
3.32
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This shows that
lim φ p, xmk 1 − φ p, xmk 0.
3.33
k→∞
Following the proof line in Case 1, we can show that
lim sup Jx − Jp, ymk − p ≤ 0,
k→∞
φ p, xmk 1 ≤ 1 − αmk φ p, xmk αmk
3.34
Jx − Jp, zmk − ymk Jx − Jp, ymk − p .
This implies
αmk φ p, xmk ≤ φ p, xmk − φ p, xmk 1
αmk Jx − Jp, zmk − ymk Jx − Jp, ymk − p
≤ αmk Jx − Jp, zmk − ymk Jx − Jp, ymk − p .
3.35
Hence limk → ∞ φp, xmk 0. Using this and 3.33 together, we conclude that
lim sup φ p, xk ≤ lim φ p, xmk 1 0.
k→∞
k→∞
3.36
This completes the proof.
As a direct consequence of Theorem 3.2, we obtain the following results.
Corollary 3.3. Let C be nonempty closed and convex subset of a uniformly smooth and uniformly
convex Banach space E. Let f be a bifunction from C × C to R satisfying A1–A4 and ϕ a lower
semicontinuous and convex function from C to R such that MEPf, ϕ / ∅. Define the sequence {xn }
as follows: x1 x ∈ C and
1
Jyn − Jxn , y − yn ≥ ϕ yn , ∀y ∈ C,
f yn , y ϕ y rn
xn1 ΠC J −1 αn Jx 1 − αn Jyn , ∀n ≥ 1,
3.37
where {αn } ⊂ 0, 1 and {rn } ⊂ 0, ∞ satisfy the following conditions:
a limn → ∞ αn 0;
b ∞
n1 αn ∞;
c lim infn → ∞ rn > 0.
Then {xn } converges strongly to ΠMEPf,ϕ x, where ΠMEPf,ϕ is the generalized projection of C onto
MEPf, ϕ.
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13
Corollary 3.4. Let C be nonempty, closed, and convex subset of a uniformly smooth and uniformly
convex Banach space E. Let f be a bifunction from C × C to R satisfying A1–A4, and A : C → E∗
a continuous and monotone mapping such that GEPf, A /
∅. Define the sequence {xn } as follows:
x1 x ∈ C and
1
f yn , y Ayn , y − yn Jyn − Jxn , y − yn ≥ 0, ∀y ∈ C,
rn
xn1 ΠC J −1 αn Jx 1 − αn Jyn , ∀n ≥ 1,
3.38
where {αn } ⊂ 0, 1 and {rn } ⊂ 0, ∞ satisfy the following conditions:
a limn → ∞ αn 0;
b ∞
n1 αn ∞;
c lim infn → ∞ rn > 0.
Then {xn } converges strongly to ΠGEPf,A x, where ΠGEPf,A is the generalized projection of C onto
GEPf, A.
Corollary 3.5. Let C be nonempty, closed, and convex subset of a uniformly smooth and uniformly
convex Banach space E. Let A : C → E∗ be a continuous and monotone mapping, and ϕ a lower
semicontinuous and convex function from C to R such that VIC, A, ϕ /
∅. Define the sequence {xn }
as follows: x1 x ∈ C and
1
Ayn , y − yn ϕ y Jyn − Jxn , y − yn ≥ ϕ yn ,
rn
xn1 ΠC J −1 αn Jx 1 − αn Jyn , ∀n ≥ 1,
∀y ∈ C,
3.39
where {αn } ⊂ 0, 1 and {rn } ⊂ 0, ∞ satisfy the following conditions:
a limn → ∞ αn 0;
b ∞
n1 αn ∞;
c lim infn → ∞ rn > 0.
Then {xn } converges strongly to ΠVIC,A,ϕ x, where ΠVIC,A,ϕ is the generalized projection of C onto
VIC, A, ϕ.
4. Examples and Numerical Results
In this section, we give examples and numerical results for our main theorem.
Example 4.1. Let E R and C −1, 1. Let fx, y −9x2 xy 8y2 , ϕx 3x2 , and Ax 2x.
Find x ∈ −1, 1 such that
f x,
y Ax,
y − x ϕ y ≥ ϕx,
∀y ∈ −1, 1.
4.1
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Journal of Applied Mathematics
Solution. It is easy to check that f, ϕ, and A satisfy all conditions in Theorem 3.2. For each
r > 0 and x ∈ −1, 1, Lemma 2.9 ensures that there exists z ∈ −1, 1 such that, for any
y ∈ −1, 1,
1
f z, y Az, y − z ϕ y z − x, y − z ≥ ϕz
r
1
⇐⇒ −9z2 yz 8y2 2z y − z 3y2 z − x y − z ≥ 3z2
r
⇐⇒ 11ry2 3rz z − xy − 14rz2 z2 − xz ≥ 0.
4.2
Put Gy 11ry2 3rz z − xy − 14rz2 z2 − xz. Then G is a quadratic function of y
with coefficient a 11r, b 3rz z − x, and c −14rz2 z2 − xz. We next compute the
discriminant Δ of G as follows:
Δ b2 − 4ac
3r 1z − x2 44r 14rz2 z2 − xz
x2 − 23r 1xz 3r 12 z2 616r 2 z2 44rz2 − 44rxz
x2 − 50rxz − 2xz 625r 2 z2 50rz2 z2
x2 − 225rz zx 625r 2 z2 50rz2 z2
4.3
x − 25rz z2 .
We know that Gy ≥ 0 for all y ∈ −1, 1 if it has at most one solution in −1, 1. So Δ ≤ 0 and
hence x 25rz z. Now we have z Tr x x/25r 1.
Let {xn }∞
n1 be the sequence generated by x1 x ∈ −1, 1 and
1
f yn , y Ayn , y − yn ϕ y yn − xn , y − yn ≥ ϕ yn ,
rn
xn1 αn x 1 − αn yn ,
∀y ∈ −1, 1,
4.4
∀n ≥ 1,
and, equivalently,
xn1 αn x 1 − αn Trn xn ,
∀n ≥ 1.
4.5
We next give two numerical results for algorithm 4.5.
Algorithm 4.2. Let αn 1/80n and rn n/n 1. Choose x1 x 1. Then algorithm 4.5
becomes
xn1
1
n1
1
1−
xn ,
80n
80n
26n 1
∀n ≥ 1.
4.6
Journal of Applied Mathematics
15
Table 1
n
1
2
3
4
5
..
.
261
262
xn
1.0000
0.0856
0.0111
0.0047
0.0033
..
.
0.0001
0.0000
Table 2
n
1
2
3
4
5
..
.
217
218
xn
−1.0000
−0.0481
−0.0074
−0.0038
−0.0027
..
.
−0.0001
0.0000
Numerical Result I
See Table 1.
Algorithm 4.3. Let αn 1/100n and rn n 1/2n. Choose x1 x −1. Then algorithm 4.5
becomes
xn1
1
2n
1
1−
xn ,
−
100n
100n
27n 25
∀n ≥ 1.
4.7
Numerical Result II
See Table 2.
5. Conclusion
Tables 1 and 2 show that the sequence {xn } converges to 0 which solves the generalized
mixed equilibrium problem. On the other hand, using Lemma 2.93, we can check that
GMEPf, A, ϕ FTr {0}.
Remark 5.1. In the view of computation, our algorithm is simple in order to get strong
convergence for generalized mixed equilibrium problems.
16
Journal of Applied Mathematics
Acknowledgments
The first and the second authors wish to thank the Thailand Research Fund and the Centre of
Excellence in Mathematics, the Commission on Higher Education, Thailand. The third author
was supported by Basic Science Research Program through the National Research Foundation
of Korea NRF funded by the Ministry of Education, Science and Technology Grant no.
2011-0021821.
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