Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 161897, 22 pages
doi:10.1155/2012/161897
Research Article
Iterative Algorithms with Perturbations for Solving
the Systems of Generalized Equilibrium Problems
and the Fixed Point Problems of
Two Quasi-Nonexpansive Mappings
Rabian Wangkeeree and Uraiwan Boonkong
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
Correspondence should be addressed to Rabian Wangkeeree, rabianw@nu.ac.th
Received 3 September 2012; Accepted 1 November 2012
Academic Editor: Xiaolong Qin
Copyright q 2012 R. Wangkeeree and U. Boonkong. 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 new iterative algorithms with perturbations for finding a common element of the
set of solutions of the system of generalized equilibrium problems and the set of common fixed
points of two quasi-nonexpansive mappings in a Hilbert space. Under suitable conditions, strong
convergence theorems are obtained. Furthermore, we also consider the iterative algorithms with
perturbations for finding a common element of the solution set of the systems of generalized
equilibrium problems and the common fixed point set of the super hybrid mappings in Hilbert
spaces.
1. Introduction
Let H be a real Hilbert space with inner product ·, · and norm · and C a nonempty closed
convex subset of H and let T be a mapping of C into H. Then, T : C → H is said to be nonexpansive if T x − T y ≤ x − y for all x, y ∈ C. A mapping T : C → H is said to be quasinonexpansive if T x − y ≤ x − y for all x ∈ C and y ∈ FT : {x ∈ C : T x x}. It is well
known that the set FT of fixed points of a quasi-nonexpansive mapping T is closed and convex; see Itoh and Takahashi 1. A mapping T : C → H is called nonspreading 2 if
2 2
2 2T x − T y ≤ T x − y T y − x ,
1.1
for all x, y ∈ C. We remark that nonlinear every nonspreading mappings are quasi-nonexpansive mappings if the set of fixed points is nonempty.
2
Abstract and Applied Analysis
Recall that a mapping Ψ : C → H is said to be μ-inverse strongly monotone if there
exists a positive real number μ such that
2
Ψx − Ψy, x − y ≥ μΨx − Ψy ,
∀x, y ∈ C.
1.2
If Ψ is a μ-inverse strongly monotone mapping of C into H, then it is obvious that Ψ is 1/μLipschitz continuous.
Let G : C × C → R be a bifuction and Ψ : C → H be μ-inverse strongly monotone
mapping. The generalized equilibrium problem for short, GEP for F and Ψ is to find z ∈ C
such that
G z, y Ψz, y − z ≥ 0,
∀y ∈ C.
1.3
The problem 1.3 was studied by Moudafi 3. The set of solutions for problem 1.3 is
denoted by GEPF, Ψ, that is,
GEPF, Ψ z ∈ C : G z, y Ψz, y − z ≥ 0, ∀y ∈ C .
1.4
If Ψ ≡ 0 in 1.3, then GEP reduces to the classical equilibrium problem and GEPG, 0 is
denoted by EPG, that is,
EPG z ∈ C : G z, y ≥ 0, ∀y ∈ C .
1.5
If G ≡ 0 in 1.3, then GEP reduces to the classical variational inequality and GEP0, Ψ is
denoted by VIΨ, C, that is,
VIΨ, C z ∈ C : Ψz, y − z ≥ 0, ∀y ∈ C .
1.6
The problem 1.3 is very general in the sense that it includes, as special cases, optimization
problems, variational inequalities, Min–Max problems, the Nash equilibrium problems in
noncooperative games, and others; see, for example, Blum and Oettli 4 and Moudafi 3.
In 2005, Combettes and Hirstoaga 5 introduced an iterative algorithm of finding the
best approximation to the initial data and proved a strong convergence theorem. In 2007,
by using the viscosity approximation method, S. Takahashi and W. Takahashi 6 introduced
another iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a nonexpansive mapping. Subsequently, algorithms constructed for solving the equilibrium problems and fixed point problems have further developed by some authors. In particular, Ceng and Yao 7 introduced an iterative
scheme for finding a common element of the set of solutions of the mixed equilibrium problem and the set of common fixed points of finitely many nonexpansive mappings. Mainge
and Moudafi 8 introduced an iterative algorithm for equilibrium problems and fixed point
problems. Wangkeeree 9 introduced a new iterative scheme for finding the common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an
equilibrium problem, and the set of solutions of the variational inequality. Wangkeeree and
Kamraksa 10 introduced an iterative algorithm for finding a common element of the set
of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of
Abstract and Applied Analysis
3
nonexpansive mappings and the set of solutions of a general system of variational inequalities for a cocoercive mapping in a real Hilbert space. Their results extend and improve many
results in the literature.
In 1967, Wittmann 11 see also 12 proved the strong convergence theorem of
Halpern’s type 13 {xn } defined by, for any x1 x ∈ C,
xn1 αn x 1 − αn T xn ,
∀n ∈ N,
1.7
∞
where {αn } ⊂ 0, 1 satisfies limn → ∞ αn 0, ∞
n1 αn ∞, and
n1 |αn − αn1 | < ∞. In 14,
Kurokawa and Takahashi also studied the following Halpern’s type for nonspreading
mappings in a Hilbert space; see also Hojo and Takahashi 15. Let T be a nonspreading mapping of C into itself. Let u ∈ C and define two sequences {xn } and {zn } in C as follows: x1 x ∈ C and
xn1 αn u 1 − αn zn ,
where zn n−1
1
T k xn
n k0
1.8
for all n 1, 2, . . ., where {αn } ⊂ 0, 1, limn → ∞ αn 0 and ∞
n1 αn ∞. If FT is nonempty, they proved that {xn } and {zn } converge strongly to PFT u, where PFT is the metric
projection of H onto FT . Recently, Yao and Shahzad 16 gave the following iteration process for nonexpansive mappings with perturbation: x1 ∈ C and
xn1 1 − βn xn βn PC αn un 1 − αn T xn ,
∀n ∈ N,
1.9
where {αn } and {βn } are sequences in 0, 1, and the sequence {un } ⊆ H is a small perturbation for the n-step iteration satisfying un → 0 as n → ∞. In fact, there are perturbations
always occurring in the iterative processes because the manipulations are inaccurate.
On the other hand, very recently, Chuang et al. 17 considered the following iteration
process for finding a common element of the set of solutions of the equilibrium problem and
the set of common fixed points for a quasi-nonexpansive mapping T : C → H with perturbation
q1 ∈ H,
xn ∈ C,
1
such that G xn , y y − xn , xn − qn ≥ 0,
rn
yn βn xn 1 − βn T xn ,
qn1 αn un 1 − αn yn ,
∀y ∈ C,
1.10
∀n ∈ N,
where C is a nonempty closed convex subset of H, G : C × C → R is a function, {αn } and {βn }
are real sequences in 0, 1, and {un } ⊂ H is a convergent sequence and {rn } ⊂ a, ∞ for some
a > 0. They obtained a strong convergence theorem for such iterations.
In this paper, inspired and motivated by Yao and Shahzad 16, S. Takahashi and
W. Takahashi 18 and Chuang et al. 17, we introduce a new iterative algorithms with
perturbations for finding a common element of the set of solutions of the system of generalized equilibrium problems and the set of common fixed points of two quasi-nonexpansive
4
Abstract and Applied Analysis
mappings in a Hilbert space. Under suitable conditions, strong convergence theorems are
obtained. Furthermore, we also consider the iterative algorithms with perturbations for finding a common element of the solution set of the system of generalized equilibrium problems
and the common fixed point set of the super hybrid mappings in a Hilbert space.
2. Preliminaries
Let H be a real Hilbert space with inner product ·, · and norm · . We denote the strongly
convergence and the weak convergence of {xn } to x ∈ H by xn → x and xn x, respectively.
In a Hilbert space, it is known that
λx 1 − λy2 λx2 1 − λy2 − λ1 − λx − y2 ,
2.1
for all x, y ∈ H and λ ∈ R; see 19. Furthermore, we have that for any x, y, u, v ∈ H
2
2
2 x − y, u − v x − v2 y − u − x − u2 − y − v .
2.2
Let C be a nonempty closed convex subset of H and x ∈ H. We know that there exists a
unique nearest point z ∈ C such that x−z infy∈C x−y. We denote such a correspondence
by z PC x. The mapping PC is called the metric projection of H onto C. It is known that PC
is nonexpansive and
x − PC x, PC x − u ≥ 0;
2.3
for all x ∈ H and u ∈ C; see 19, 20 for more details.
Let C be a nonempty, closed and convex subset of H and let G : C × C → R be a
bifunction. For solving the generalized equilibrium problem, let us assume that the bifunction
G : C × C → R satisfies the following conditions:
A1 Gx, x 0 for a ll x ∈ C;
A2 G is monotone, that is, Gx, y Gy, x ≤ 0 for any x, y ∈ C;
A3 for each x, y, z ∈ C
lim G tz 1 − tx, y ≤ G x, y ;
t↓0
2.4
A4 for each x ∈ C, Gx, · is convex and lower semicontinuous.
We know the following lemma which appears implicitly in Blum and Oettli 4.
Lemma 2.1 see 4. Let C be a nonempty closed convex subset of H and let G be a bifunction of
C × C into R satisfying A1–A4. Let r > 0 and x ∈ H. Then, there exists a unique z ∈ C such that
1
G z, y y − z, z − x ≥ 0,
r
∀y ∈ C.
The following lemma was also given in Combettes and Hirstoaga 5.
2.5
Abstract and Applied Analysis
5
Lemma 2.2 see 5. Let C be a nonempty closed convex subset of a real Hilbert space H. Let
G : C × C → R be a bifunction which satisfies conditions A1–A4. For r > 0 and x ∈ H, define a
mapping Tr : H → C as follows:
1
Tr x z ∈ C : G z, y y − z, z − x ≥ 0, ∀y ∈ C
2.6
r
for all x ∈ H. Then the following hold:
i Tr is single-valued;
ii Tr is firmly nonexpansive, that is, for any x, y ∈ C,
Tr x − Tr y2 ≤ Tr x − Tr y, x − y ;
2.7
iii EP is a closed convex subset of C;
iv FTr EP G.
Remark 2.3. For any x ∈ H and r > 0, by Lemma 2.2 i, there exists u ∈ H such that
1
G u, y y − u, u − x ≥ 0,
r
∀y ∈ H.
2.8
Replacing x with x − rΨx ∈ H in 2.8, we have
1
G u, y Ψx, y − u y − u, u − x ≥ 0,
r
∀y ∈ H,
2.9
where Ψ : H → H is an inverse strongly monotone mapping.
Lemma 2.4 see 21. Let {Γn } be a sequence of real numbers that does not decrease at infinity in
the sense that there exists a subsequence {Γni } of {Γn } which satisfies Γni < Γni 1 for all i ∈ N. Define
the sequence {τn}n≥n0 of integers as follows:
τn max{k ≤ n : Γk < Γk1 },
2.10
where n0 ∈ N such that {k ≤ n0 : Γk < Γk1 } / ∅. Then, the following hold:
i τ1 ≤ τ2 ≤ · · · and τn → ∞;
ii Γτn ≤ Γτn1 and Γn ≤ Γτn1 , ∀n ∈ N.
Lemma 2.5 see 22. Let {an }n∈N be a sequence of nonnegative real numbers, {αn } a sequence
of real numbers in 0, 1 with ∞
n1 αn ∞, {un } a sequence of nonnegative real numbers with
∞
u
<
∞,
{t
}
a
sequence
of
real
numbers with lim sup tn ≤ 0. Suppose that
n
n
n1
an1 ≤ 1 − αn an αn tn un ,
Then limn → ∞ an 0.
∀n ∈ N.
2.11
6
Abstract and Applied Analysis
3. Main Results
Let C be a nonempty closed convex subset of a Hilbert space H. For each i 1, 2, . . . , k, let
Gi : C × C → R be a bifunction satisfying A1–A4 and Ψi a μi -inverse strongly monotone
mapping. For each j 1, 2, let Tj : C → H be two mappings. Let {xn } be a sequence
generated in the following manner:
x1 ∈ H,
1
G1 un,1 , y Ψ1 xn , y − un,1 y − un,1 , un,1 − xn ≥ 0,
rn
∀y ∈ C,
1
G2 un,2 , y Ψ2 xn , y − un,2 y − un,2 , un,2 − xn ≥ 0,
rn
∀y ∈ C,
..
.
1
Gk un,k , y Ψk xn , y − un,k y − un,k , un,k − xn ≥ 0,
rn
∀y ∈ C,
3.1
k
1
un,i ,
k i1
yn γn ωn 1 − γn T1 ωn ,
zn βn yn 1 − βn T2 ωn ,
ωn xn1 αn un 1 − αn zn ,
∀n ∈ N,
where {αn }, {βn }, {γn } are sequences in 0, 1 and {un } ⊂ H is a sequence and {rn } ⊂ a, 2μi for some a > 0 and for all i ∈ {1, 2, . . . , k}. Under certain appropriate assumptions imposed
on the sequences {αn }, {βn }, {γn }, the strong convergence theorem of {xn } defined by 3.1 is
studied in the following theorem.
Theorem 3.1. Let C be a nonempty closed convex subset of a Hilbert space H. For each i 1, 2, . . . , k,
let Gi : C × C → R be a bifunction satisfying A1–A4 and Ψi a μi -inverse strongly monotone
mapping. For each j 1, 2, let Tj : C → H be two quasi-nonexpansive mappings such that I − Tj are
∅. Let the sequences {xn },{yn }, and
demiclosed at zero with Ω : FT1 ∩FT2 ∩∩ki1 GEPGi , Ψi /
{zn } be defined by 3.1, where {αn }, {βn }, {γn }, and {un } satisfy the following conditions:
C1 limn → ∞ αn 0 and
∞
n1
αn ∞;
C2 lim infn → ∞ βn 1 − βn > 0;
C3 lim infn → ∞ γn 1 − γn > 0;
C4 limn → ∞ un u for some u ∈ H.
Then {xn } converges strongly to x∗ , where x∗ PΩ u.
Abstract and Applied Analysis
7
Proof. We first have that for all i 1, 2, . . . , k, I − rn Ψi is a nonexpansive mapping. Indeed, for
all x, y ∈ C, we obtain
I − rn Ψi x − I − rn Ψi y2 x − y − rn Ψi x − Ψi y 2
2
2
x − y − 2rn Ψi x − Ψi y, x − y rn2 Ψi x − Ψi y
2
2
≤ x − y − rn 2μi − rn Ψi x − Ψi y
2
≤ x − y .
3.2
Thus I − rn Ψi is nonexpansive for each i ∈ {1, 2, . . . , k}. Now, let w ∈ Ω be arbitrary. By C4,
{un } is a bounded sequence, there exists M ≤ 0 such that
supun − w ≤ M.
n∈N
3.3
For each i 1, 2, . . . , k and n ∈ N, we have from un,i Trn,i xn − rn Ψi xn that
un,i − w Trn,i xn − rn Ψi xn − Trn,i w − rn Ψi w
≤ xn − rn Ψi xn − w − rn Ψi w
3.4
≤ xn − w,
which gives also that
ωn − w ≤
k
1
un,i − w ≤ xn − w ∀w ∈ Ω.
k i1
Since T1 is quasi-nonexpansive, we have
yn − w γn ωn 1 − γn T1 ωn − w
γn ωn − w 1 − γn T1 ωn − w
≤ γn ωn − w 1 − γn T1 ωn − w
3.5
3.6
≤ ωn − w.
So, we have from 3.5 and 3.6 and the quasi-nonexpansiveness of T2 that
xn1 − w αn un − w 1 − αn zn − w
≤ αn un − w 1 − αn zn − w
≤ αn un − w 1 − αn βn yn − w 1 − βn T2 ωn − w
≤ αn un − w 1 − αn βn ωn − w 1 − βn ωn − w
≤ αn un − w 1 − αn ωn − w
≤ αn un − w 1 − αn xn − w
≤ max{M, xn − w}.
3.7
8
Abstract and Applied Analysis
By Induction, we have that
xn − w ≤ max{x1 − w, M},
∀n ∈ N.
3.8
Thus we obtain that {xn −w} is bounded, so also {xn }, {yn }, {zn }, {ωn }, {T1 ωn }, and {T2 ωn }
are bounded. Since Ω is closed and convex, we can take x∗ PΩ u. It follows that
yn − x∗ 2 γn ωn − x∗ 1 − γn T1 ωn − x∗ 2
γn ωn − x∗ 2 1 − γn T1 ωn − x∗ 2 − γn 1 − γn ωn − T1 ωn 2
≤ γn ωn − x∗ 2 1 − γn ωn − x∗ 2 − γn 1 − γn ωn − T1 ωn 2
ωn − x∗ 2 − γn 1 − γn ωn − T1 ωn 2
3.9
≤ ωn − x∗ 2 .
From 3.9, we have
2
zn − x∗ 2 βn yn − x∗ 1 − βn T2 ωn − x∗ 2 2
βn yn − x∗ 1 − βn T2 ωn − x∗ 2 − βn 1 − βn yn − T2 ωn 2
≤ βn ωn − x∗ 2 1 − βn ωn − x∗ 2 − βn 1 − βn yn − T2 ωn 2
ωn − x∗ 2 − βn 1 − βn yn − T2 ωn ≤ ωn − x∗ 2 .
Hence we have from 3.5, 3.9, and 3.10 that
ωn1 − x∗ 2 ≤ xn1 − x∗ 2
αn un − x∗ 1 − αn zn − x∗ 2
αn un − x∗ 2 1 − αn zn − x∗ 2 − αn 1 − αn un − zn 2
≤ αn un − x∗ 2 1 − αn zn − x∗ 2
2 αn un − x∗ 2 1 − αn βn yn − x∗ 1 − βn T2 ωn − x∗ 2
2 − βn 1 − βn yn − T2 ωn ≤ αn un − x∗ 2 βn γn ωn − x∗ 2 1 − γn T1 ωn − x∗ 2
− γn 1 − γn ωn − T1 ωn 2 1 − βn T2 ωn − x∗ 2
2
− βn 1 − βn yn − T2 ωn 3.10
Abstract and Applied Analysis
9
≤ αn un − x∗ 2 βn γn ωn − x∗ 2 1 − γn ωn − x∗ 2
− γn 1 − γn ωn − T1 ωn 2 1 − βn ωn − x∗ 2
2
− βn 1 − βn yn − T2 ωn αn un − x∗ 2 ωn − x∗ 2 − γn 1 − γn ωn − T1 ωn 2
2
− βn 1 − βn yn − T2 ωn .
3.11
We also have that
γn 1 − γn ωn − T1 ωn 2 ≤ αn un − x∗ 2 ωn − x∗ 2 − ωn1 − x∗ 2 ,
2
βn 1 − βn yn − T2 ωn ≤ αn un − x∗ 2 ωn − x∗ 2 − ωn1 − x∗ 2 .
3.12
3.13
Furthemore, we have from yn γn ωn 1 − γn T1 ωn that
ωn − T2 ωn ≤ ωn − yn yn − T2 ωn ωn − γn ωn − 1 − γn T1 ωn yn − T2 xn 1 − γn ωn − T1 ωn yn − T2 ωn .
3.14
On the other hand, since xn1 − x∗ αn un − x∗ 1 − αn zn − x∗ , we have
ωn1 − x∗ 2 ≤ xn1 − x∗ 2
≤ 1 − αn zn − x∗ 2 2αn un − x∗ , xn1 − x∗ ≤ 1 − αn ωn − x∗ 2 2αn un − x∗ , xn1 − x∗ 1 − αn ωn − x∗ 2 2αn un − u, xn1 − x∗ 3.15
2αn u − x∗ , xn1 − x∗ 1 − αn ωn − x∗ 2 2αn un − u, xn1 − x∗ 2αn u − x∗ , xn1 − ωn 2αn u − x∗ , ωn − x∗ .
We also have that
xn1 − ωn ≤ xn1 − yn yn − ωn αn un − yn 1 − αn zn − yn 1 − γn ωn − T1 ωn ≤ αn un − yn 1 − αn βn yn 1 − βn T2 ωn − yn 1 − γn ωn − T1 ωn αn un − yn 1 − αn 1 − βn yn − T2 ωn 1 − γn ωn − T1 ωn .
3.16
10
Abstract and Applied Analysis
Moreover, for any i ∈ {1, 2, . . . , k}, we have from un,i Trn,i xn − rn Ψi xn that
un,i − x∗ 2 ≤ xn − x∗ − rn Ψi xn − Ψi x∗ 2
xn − x∗ 2 − 2rn xn − x∗ , Ψi xn − Ψi x∗ rn2 Ψi xn − Ψi x∗ 2
≤ xn − x∗ 2 − rn 2μi − rn Ψi xn − Ψi x∗ 2 .
3.17
It follows that
2
k
1
∗ un,i − x ωn − x i1 k
∗ 2
≤
k
1
un,i − x∗ 2
k i1
≤ xn − x∗ 2 −
3.18
k
1
rn 2μi − rn Ψi xn − Ψi x∗ 2 .
k i1
This implies that
xn1 − x∗ 2 αn un − x∗ 1 − αn zn − x∗ 2
≤ αn un − x∗ 2 1 − αn ωn − x∗ 2
3.19
≤ αn un − x∗ 2 1 − αn xn − x∗ 2
− 1 − αn k
1
rn 2μi − rn Ψi xn − Ψi x∗ 2 ,
k i1
and hence
1 − αn k
1
rn 2μi − rn Ψi xn − Ψi x∗ 2 ≤ αn un − x∗ 2 xn − x∗ 2 − xn1 − x∗ 2 .
k i1
3.20
Furthermore, we have from Lemma 2.2 that for any i ∈ 1, 2, . . . , k, we have
un,i − x∗ 2 ≤ xn − rn Ψi xn − x∗ − rn Ψi x∗ , un,i − x∗ 1
xn − rn Ψi xn − x∗ − rn Ψi x∗ 2 un,i − x∗ 2
2
− xn − rn Ψi xn − x∗ − rn Ψi x∗ − un,i − x∗ 2
1
xn − x∗ 2 un,i − x∗ 2 − xn − un,i − rn Ψi xn − Ψi x∗ 2
2
1
xn − x∗ 2 un,i − x∗ 2 − xn − un,i 2 − rn2 Ψi xn − Ψi x∗ 2
2
2rn xn − un,i , Ψi xn − Ψi x∗ .
≤
3.21
Abstract and Applied Analysis
11
This implies that
un,i − x∗ 2 ≤ xn − x∗ 2 − xn − un,i 2 2rn xn − un,i Ψi xn − Ψi x∗ .
3.22
Then we have from 3.22 that
ωn − x∗ 2 ≤
k
1
un,i − x∗ 2
k i1
≤ xn − x∗ 2 −
k
1
un,i − xn 2
k i1
3.23
k
1
2rn xn − un,i Ψi xn − Ψi x∗ .
k i1
Hence we have from 3.23 that
xn1 − x∗ 2 ≤ αn un − x∗ 2 1 − αn ωn − x∗ 2
k
1
≤ αn un − x 1 − αn xn − x −
un,i − xn 2
k i1
k
1
∗
2rn xn − un,i Ψi xn − Ψi x .
1 − αn k i1
∗ 2
∗ 2
3.24
It follows that
1 − αn k
1
un,i − xn 2 ≤ αn un − x∗ 2 xn − x∗ 2 − xn1 − x∗ 2
k i1
k
1
∗
1 − αn 2rn xn − un,i Ψi xn − Ψi x .
k i1
3.25
Next, we will consider the following two cases.
Case A. Put Γn ωn − x∗ 2 for all n ∈ N. Suppose that Γn1 ≤ Γn for all n ∈ N. In this case
limn → ∞ Γn exists and then limn → ∞ Γn1 − Γn 0. By C1, C3, and 3.12, we have
lim ωn − T1 ωn 0.
n→∞
3.26
Similarly by C1, C2, and 3.13, we also have
lim yn − T2 ωn 0.
n→∞
3.27
12
Abstract and Applied Analysis
So, we have from 3.14, 3.26, and 3.27 that
lim ωn − T2 ωn 0.
3.28
n→∞
Since limn → ∞ ωn − x∗ exists, we have from 3.11 and 3.26
lim ωn − x∗ lim xn − x∗ .
n→∞
3.29
n→∞
We also have from C1, 3.16, 3.26, and 3.27 that
lim xn1 − ωn 0.
3.30
n→∞
Since limn → ∞ xn − x∗ exists we have from C1 and 3.20 that
lim Ψi xn − Ψi x∗ 0,
n→∞
∀i 1, 2, . . . , k.
3.31
This together with 3.25 and the existence of limn → ∞ xn − x∗ implies that
lim un,i − xn 0,
n→∞
∀i 1, 2, . . . , k,
3.32
which gives that
ωn − xn ≤
k
1
un,i − xn −→ 0
k i1
as n −→ ∞.
3.33
So, from 3.30, limn → ∞ xn1 − xn 0. Furthermore, we have from 3.33 that
ωn1 − ωn ≤ ωn1 − xn1 xn1 − xn xn − ωn −→ 0
as n −→ ∞;
3.34
that is
lim ωn1 − ωn 0.
n→∞
3.35
Now, since {ωn } is a bounded sequence, there exists a subsequence {ωnj } of {ωn } such that
lim supu − x∗ , ωn − x∗ lim u − x∗ , ωnj − x∗ .
n→∞
j →∞
3.36
Without loss of generality, we may assume that ωnj v. Since T1 is demiclosed at zero and
by 3.26, we conclude that v ∈ FT1 . Similarly, since T2 is demiclosed at zero and by 3.28,
we have v ∈ FT2 . Therefore, we get that
v ∈ FT1 ∩ FT2 .
3.37
Abstract and Applied Analysis
13
Next, we show that v ∈ ∩ki1 GEPGi , Ψi . For each i ∈ {1, 2, . . . , k}, since un,i Trn,i xn −
rn Ψi xn , we have
1
Gi un,i , y Ψi xn , y − un,i y − un,i , un,i − xn ≥ 0,
rn
∀y ∈ C.
3.38
From A2, we also have
1
Ψi xn , y − un,i y − un,i , un,i − xn ≥ Gi y, un,i .
rn
3.39
Replacing n by nj , we have
Ψi xnj , y − unj ,i y − unj ,i ,
unj ,i − xnj
rnj
≥ Gi y, unj ,i .
3.40
Put yt ty 1 − tv for all t ∈ 0, 1 and y ∈ C. Since v ∈ C, then yt ∈ C and
yt − unj ,i , Ψi yt ≥ yt − unj ,i , Ψi yt − yt − unj ,i , Ψi xnj
−
yt − unj ,i ,
unj ,i − xnj
Gi yt , unj ,i
rnj
yt − unj ,i , Ψi yt − Ψi unj ,i yt − unj ,i , Ψi unj ,i − Ψi xnj
−
yt − unj ,i ,
unj ,i − xnj
rnj
3.41
Gi yt , unj ,i .
Since unj ,i −xnj → 0 as j → ∞, we obtain that Ψi unj ,i −Ψi xnj → 0 as j → ∞. Furthermore,
by the monotonicity of Ψi , we obtain that
yt − unj ,i , Ψi yt − Ψi unj ,i ≥ 0.
3.42
Taking j → ∞ in 3.41, we have from A4 that
yt − v, Ψi yt ≥ Gi yt , v .
3.43
Now, from A1, A4, and 3.43, we also have
0 Gi yt , yt ≤ tGi yt , y 1 − tGi yt , v
≤ tGi yt , y 1 − t yt − v, Ψi yt
tGi yt , y 1 − tt y − v, Ψi yt ,
3.44
14
Abstract and Applied Analysis
which yields that
Gi yt , y 1 − t y − v, Ψi yt ≥ 0.
3.45
Taking t → 0, we have, for each y ∈ C
Gi v, y y − v, Ψi v ≥ 0,
∀i ∈ {1, 2, . . . , k}.
3.46
This shows v ∈ GEPGi , Ψi , for all i 1, 2, . . . , k. Then, v ∈ ∩ki1 GEPGi , Ψi . Hence we have
v ∈ FT1 ∩ FT2 ∩ ∩ki1 GEPGi , Ψi : Ω. So, we have from 3.36 that
lim supu − x∗ , ωn − x∗ u − x∗ , v − x∗ ≤ 0.
n→∞
3.47
By C1, C4, 3.15, 3.30, 3.47, and Lemma 2.5, we obtain that limn → ∞ ωn − x∗ 0.
Hence we have from 3.29 that {xn } converges to x∗ , where x∗ PΩ u.
Case B. Assume that there exists a subsequence {Γni }i≥0 of {Γn }n≥0 such that Γni < Γni 1 for all
i ∈ N. In this case, it follows from Lemma 2.4 that there exists a subsequence {Γτn } of {Γn }
such that Γτn1 > Γτn , where τ : N → N is defined by
τn max{k ≤ n : Γk < Γk1 },
∀n ∈ N.
3.48
So, from 3.12, that
ωτn1 − x∗ 2 − ωτn − x∗ 2 γτn 1 − γτn ωτn − T1 ωτn 2 ≤ ατn uτn − x∗ 2 .
3.49
Since ωτn − x∗ 2 : Γτn < Γτn1 : ωτn1 − x∗ 2 , we have
2
2
γτn 1 − γτn ωτn − T1 ωτn ≤ ατn uτn − x∗ .
3.50
By C1 and C3, we have
lim ωτn − T1 ωτn 0.
n→∞
3.51
By 3.15, we have
ωτn1 − x∗ 2 ≤ 1 − ατn ωτn − x∗ 2 2ατn uτn − x∗ , xτn1 − x∗ .
3.52
Abstract and Applied Analysis
15
Now, in view of Γτn < Γτn1 , we see that
ωτn − x∗ 2 ≤ 2 uτn − x∗ , xτn1 − x∗
2 uτn − u, xτn1 − x∗ 2 u − x∗ , xτn1 − ωτn
2 u − x∗ , ωτn − x∗ .
3.53
Furthermore, we also have from 3.13 that
2
2 2
βτn 1 − βτn yτn − T2 ωτn ≤ ατn uτn − x∗ ωτn − x∗ 2
− ωτn1 − x∗ 2
≤ ατn uτn − x∗ .
3.54
Applying C1 and C2 to the last inequality, we get that
lim yτn − T2 ωτn 0.
3.55
By C1, 3.16, 3.51, and 3.55, we have
lim xτn1 − ωτn 0.
n→∞
3.56
lim ωτn1 − xτn1 0.
3.57
n→∞
By 3.33, we have
n→∞
It follows from 3.56 and 3.57 that
lim ωτn1 − ωτn 0.
n→∞
3.58
Since {ωτn } is a bounded sequence, there exists a subsequence {ωτnj } such that
lim sup u − x∗ , ωτn − x∗ lim u − x∗ , ωτnj − x∗ .
j →∞
n→∞
3.59
Following the same argument as the proof of Case A for {ωτnj }, we have that
lim sup u − x∗ , ωτn − x∗ ≤ 0.
n→∞
3.60
Using C4, 3.53, 3.56, and 3.60, we have that
lim ωτn − x∗ 0.
n→∞
3.61
16
Abstract and Applied Analysis
By 3.58 and 3.61, we have that
lim ωτn1 − x∗ 0.
3.62
n→∞
By Lemma 2.4 ii, we get limn → ∞ Γn 0; that is limn → ∞ ωn − x∗ 0. We observe that
xn1 − x∗ 2 ≤ αn un − x∗ 2 1 − αn ωn − x∗ 2 .
3.63
Applying C1, C4, and limn → ∞ ωn − x∗ 2 0, we have immediately
lim xn − x∗ 0;
3.64
n→∞
that is, {xn } converges strongly to x∗ , where x∗ PΩ u. This completes the proof.
Setting Ψi ≡ 0 for all i 1, 2, . . . , k in Theorem 3.1, we obtain the following result.
Corollary 3.2. Let C be a nonempty closed convex subset of a Hilbert space H. For each i 1, 2, . . . , k,
let Gi : C × C → R be a bifunction satisfying A1–A4. For each j 1, 2, let Tj : C → H be
two quasi-nonexpansive mappings such that I − Tj are demiclosed at zero with Ω : FT1 ∩ FT2 ∩
∩ki1 EP Gi / ∅. Let the sequences {xn }, {yn }, and {zn } be defined by
x1 ∈ H,
1
G1 un,1 , y y − un,1 , un,1 − xn ≥ 0,
rn
1
G2 un,2 , y y − un,2 , un,2 − xn ≥ 0,
rn
..
.
∀y ∈ C,
∀y ∈ C,
1
y − un,k , un,k − xn ≥ 0,
Gk un,k , y rn
k
1
un,i ,
k i1
yn γn ωn 1 − γn T1 ωn ,
zn βn yn 1 − βn T2 ωn ,
ωn xn1 αn un 1 − αn zn ,
where {αn }, {βn }, {γn } satisfy the following conditions.
C1 limn → ∞ αn 0 and ∞
n1 αn ∞;
C2 lim infn → ∞ βn 1 − βn > 0;
C3 lim infn → ∞ γn 1 − γn > 0;
C4 limn → ∞ un u for some u ∈ H.
∀n ∈ N,
∀y ∈ C,
3.65
Abstract and Applied Analysis
17
Then {xn } converges strongly to x∗ , where x∗ PΩ u.
In the next results, using Theorem 3.1, we have new strong convergence theorems for
two nonexpansive mappings in a Hilbert space.
Corollary 3.3. Let C be a nonempty closed convex subset of a Hilbert space H. For each i 1, 2, . . . , k,
let Gi : C × C → R be a bifunction satisfying A1–A4 and Ψi a μi -inverse strongly monotone
mapping. For each j 1, 2, let Tj : C → H be two nonexpansive mappings such that Ω : FT1 ∩
∅. Let the sequences {xn }, {yn }, and {zn } be defined by 3.1, where
FT2 ∩ ∩ki1 GEP Gi , Ψi /
{αn }, {βn }, {γn } satisfy the following conditions.
C1 limn → ∞ αn 0 and
∞
n1
αn ∞;
C2 lim infn → ∞ βn 1 − βn > 0;
C3 lim infn → ∞ γn 1 − γn > 0;
C4 limn → ∞ un u for some u ∈ H.
Then {xn } converges strongly to x∗ , where x∗ PΩ u.
4. Applications
In this section, we present some convergence theorems deduced from the results in the previous section. Recall that a mapping T : C → H is said to be nonspreading 2 if
2 2 2
2T x − T y ≤ T x − y T y − x
4.1
for all x, y ∈ C. Further, a mapping T : C → H is said to be hybrid 23 if
2 2 2 2
3T x − T y ≤ x − y T x − y T y − x
4.2
for all x, y ∈ C. These mappings are deduced from a firmly nonexpansive mapping in a Hilbert space.
A mapping F : C → H is said to be firmly nonexpansive if
Fx − Fy2 ≤ x − y, Fx − Fy
4.3
for all x, y ∈ C; see, for instance, Browder 24 and Goebel and Kirk 25. We also know that
a firmly nonexpansive mapping F can be deduced from an equilibrium problem in a Hilbert
space.
Recently, Kocourek et al. 26 introduced a more broad class of nonlinear mappings
call generalized hybrid if there are α, β ∈ R such that
2
2 2
2
αT x − T y 1 − αx − T y ≤ βT x − y 1 − β x − y
4.4
18
Abstract and Applied Analysis
for all x, y ∈ C. Very recently, they defined a more broad class of mappings than the class of
generalized hybrid mappings in a Hilbert space. A mapping S : C → H is called super hybrid
if there are α, β, γ ∈ R such that
2 2 2
2 αSx−Sy 1−αγ x−Sy ≤ β β−α γ Sx−y 1−β− β−α−1 γ x−y
2
α − β γx − Sx2 γ y − Sy ,
4.5
for all x, y ∈ C. We call such a mapping an α, β, γ-super hybrid mapping. We notice that an
α, β, 0-super hybrid mapping is α, β-generalized hybrid. So, the class of super hybrid mappings contains the class of generalized hybrid mappings. A super hybrid mapping is not
quasi-nonexpansive generally. For more details, see 27. Before proving, we need the
following lemmas.
Lemma 4.1 see 27. Let C be a nonempty subset of a Hilbert space H and let α, β and γ be real numbers with γ /
−1. Let S and T be mappings of C into H such that S 1/1γT γ/1γI. Then,
T is α, β, γ-super hybrid if and only if S is α, β-generalized hybrid. In this case, FS FT .
Lemma 4.2 see 27. Let H be a Hilbert space and let C be a nonempty closed convex subset of H.
Let S : C → H be a generalized hybrid mapping. Then S is demiclosed on C.
Setting Sj : 1/1γj Tj γj /1γj I in Theorem 3.1, where Tj is a super hybrid mapping
and γj is a real number, we obtain the following result.
Theorem 4.3. Let C be a nonempty closed convex subset of a Hilbert space H. For each i 1, 2, . . . , k,
let Gi : C × C → R be a bifunction satisfying A1–A4 and Ψi a μi -inverse strongly monotone
mapping. For each j 1, 2, let Tj : C → H be αj , βj , γj -super hybrid mappings such that Ω :
∅. Let the sequences {xn },{yn }, and {zn } be defined by
FT1 ∩ FT2 ∩ ∩ki1 GEP Gi , Ψi /
x1 ∈ H,
1
y − un,1 , un,1 − xn ≥ 0,
G1 un,1 , y Ψ1 xn , y − un,1 rn
1
G2 un,2 , y Ψ2 xn , y − un,2 y − un,2 , un,2 − xn ≥ 0,
rn
..
.
1
Gk un,k , y Ψk xn , y − un,k y − un,k , un,k − xn ≥ 0,
rn
ωn k
1
un,i ,
k i1
γ1
1
T1 ωn ωn ,
1 γ1
1 γ1
γ2
1
zn βn yn 1 − βn
T2 ωn ωn ,
1 γ2
1 γ2
yn γn ωn 1 − γn
xn1 αn un 1 − αn zn ,
∀n ∈ N,
∀y ∈ C,
∀y ∈ C,
∀y ∈ C,
4.6
Abstract and Applied Analysis
19
where {αn }, {βn }, {γn } are sequences in 0, 1 and {un } ⊂ H is a sequence and {rn } ⊂ a, 2μi for
some a > 0 and for all i ∈ {1, 2, . . . , k}. Suppose the following conditions are satisfied.
C1 limn → ∞ αn 0 and
∞
n1
αn ∞;
C2 lim infn → ∞ βn 1 − βn > 0;
C3 lim infn → ∞ γn 1 − γn > 0;
C4 limn → ∞ un u for some u ∈ H.
Then {xn } converges strongly to x∗ , where x∗ PΩ u.
Proof. For each j 1, 2, setting
Sj γj
1
Tj I,
1 γj
1 γj
4.7
we have from Lemma 4.1 that each Sj is a generalized hybrid mapping and FSj ∅, we have that each Sj is quasi-nonexpansive. Following the proof
FTj . Since FSj /
of Theorem 3.1 and applying Lemma 4.2, we have the desired result. This completes the
proof.
Setting Ψ ≡ 0 in Theorem 4.3, we obtains the following result.
Corollary 4.4. Let C be a nonempty closed convex subset of a Hilbert space H. For each i 1, 2, . . . , k,
let Gi : C × C → R be a bifunction satisfying A1–A4. For each j 1, 2, let Tj : C → H be
∅. Let the sequences
αj , βj , γj -super hybrid mappings such that Ω : FT1 ∩FT2 ∩∩ki1 EP Gi /
{xn },{yn }, and {zn } be defined by
x1 ∈ H,
1
G1 un,1 , y y − un,1 , un,1 − xn ≥ 0,
rn
1
G2 un,2 , y y − un,2 , un,2 − xn ≥ 0,
rn
∀y ∈ C,
∀y ∈ C,
..
.
1
y − un,k , un,k − xn ≥ 0,
Gk un,k , y rn
ωn ∀y ∈ C,
k
1
un,i ,
k i1
γ1
1
T1 ωn ωn ,
1 γ1
1 γ1
γ2
1
T2 ωn ωn ,
zn βn yn 1 − βn
1 γ2
1 γ2
yn γn ωn 1 − γn
xn1 αn un 1 − αn zn ,
∀n ∈ N,
4.8
20
Abstract and Applied Analysis
where {αn }, {βn }, {γn } are sequences in (0, 1) and {un } ⊂ H is a sequence and {rn } ⊂ a, ∞ for
some a > 0. Suppose the following conditions are satisfied.
C1 limn → ∞ αn 0 and ∞
n1 αn ∞;
C2 lim infn → ∞ βn 1 − βn > 0;
C3 lim infn → ∞ γn 1 − γn > 0;
C4 limn → ∞ un u for some u ∈ H.
Then {xn } converges strongly to x∗ , where x∗ PΩ u.
In Corollary 4.4, put Gi x, y 0 for all x, y ∈ C and rn 1 for all n ∈ N. Then we have
that un,i xn for all i 1, 2, . . . , k, which gives that ωn 1/k ki1 un,i xn . Thus we obtain
the following results from Corollary 4.4.
Corollary 4.5. Let C be a nonempty closed convex subset of a Hilbert space H. For each j 1, 2, let
∅. Let the sequences
Tj : C → H be αj , βj , γj -super hybrid mappings such that FT1 ∩ FT2 /
{xn },{yn }, and {zn } be defined by
x1 ∈ H,
γ1
1
T1 xn xn ,
yn γn xn 1 − γn
1 γ1
1 γ1
γ2
1
T2 xn xn ,
zn βn yn 1 − βn
1 γ2
1 γ2
xn1 αn un 1 − αn zn ,
4.9
∀n ∈ N,
where {αn }, {βn }, {γn } are sequences in 0, 1 and {un } ⊂ H is a sequence. Suppose the following
conditions are satisfied.
C1 limn → ∞ αn 0 and ∞
n1 αn ∞;
C2 lim infn → ∞ βn 1 − βn > 0;
C3 lim infn → ∞ γn 1 − γn > 0;
C4 limn → ∞ un u for some u ∈ H.
Then {xn } converges strongly to x∗ , where x∗ PFT1 ∩FT2 u.
In Corollary 4.5, put T1 I, the identity mapping, and T2 : T , an α, β, γ-super hybrid
mapping. Thus we obtain the following results.
Corollary 4.6. Let C be a nonempty closed convex subset of a Hilbert space H. Let T be an α, β, γsuper hybrid mapping such that FT /
∅. Let the sequences {xn }, {yn }, and {zn } be defined by
x1 ∈ H,
γ
1
T xn xn ,
zn βn xn 1 − βn
1γ
1γ
xn1 αn un 1 − αn zn ,
∀n ∈ N,
4.10
4.11
4.12
Abstract and Applied Analysis
21
where {αn } and {βn } are sequences in 0, 1 and {un } ⊂ H is a sequence. Suppose the following conditions are satisfied.
C1 limn → ∞ αn 0 and ∞
n1 αn ∞;
C2 lim infn → ∞ βn 1 − βn > 0;
C3 limn → ∞ un u for some u ∈ H.
Then {xn } converges strongly to x∗ , where x∗ PFT u.
Acknowledgments
The first author is supported by the National Research Council of Thailand. The authors
would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.
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