Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2009, Article ID 794178, 21 pages
doi:10.1155/2009/794178
Research Article
An Iterative Algorithm Combining
Viscosity Method with Parallel Method for a
Generalized Equilibrium Problem and Strict
Pseudocontractions
Jian-Wen Peng,1 Yeong-Cheng Liou,2 and Jen-Chih Yao3
1
College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China
Department of Information Management, Cheng Shiu University, Kaohsiung, Taiwan 833, Taiwan
3
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 804, Taiwan
2
Correspondence should be addressed to Yeong-Cheng Liou, simplex liou@hotmail.com
Received 5 August 2008; Accepted 4 January 2009
Recommended by Hichem Ben-El-Mechaiekh
We introduce a new approximation scheme combining the viscosity method with parallel method
for finding a common element of the set of solutions of a generalized equilibrium problem and the
set of fixed points of a family of finitely strict pseudocontractions. We obtain a strong convergence
theorem for the sequences generated by these processes in Hilbert spaces. Based on this result,
we also get some new and interesting results. The results in this paper extend and improve some
well-known results in the literature.
Copyright q 2009 Jian-Wen Peng 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.
1. Introduction
Let H be a real Hilbert space with inner product · , · and induced norm · , and let C be
a nonempty-closed convex subset of H. Let ϕ : H → R ∪ {∞} be a function and let F be
a bifunction from C × C to R such that C ∩ dom ϕ /
∅, where R is the set of real numbers
and dom ϕ {x ∈ H : ϕx < ∞}. Flores-Bazán 1 introduced the following generalized
equilibrium problem:
Find x ∈ C such that Fx, y ϕy ≥ ϕx,
∀y ∈ C.
1.1
The set of solutions of 1.1 is denoted by GEP F, ϕ. Flores-Bazán 1 provided some
characterizations of the nonemptiness of the solution set for problem 1.1 in reflexive Banach
spaces in the quasiconvex case. Bigi et al. 2 studied a dual problem associated with the
problem 1.1 with C H Rn .
2
Fixed Point Theory and Applications
Let ϕx δC x, ∀x ∈ H. Here δC denotes the indicator function of the set C; that is,
δC x 0 if x ∈ C and δC x ∞ otherwise. Then the problem 1.1 becomes the following
equilibrium problem:
Finding x ∈ C such that Fx, y ≥ 0,
∀y ∈ C.
1.2
The set of solutions of 1.2 is denoted by EPF. The problem 1.2 includes, as special
cases, the optimization problem, the variational inequality problem, the fixed point problem,
the nonlinear complementarity problem, the Nash equilibrium problem in noncooperative
games, and the vector optimization problem. For more detail, please see 3–5 and the
references therein.
If Fx, y gy−gx for all x, y ∈ C, where g : C → R is a function, then the problem
1.1 becomes a problem of finding x ∈ C which is a solution of the following minimization
problem:
min ϕy gy .
1.3
y∈C
The set of solutions of 1.3 is denoted by Argming, ϕ.
If ϕ : H → R ∪ {∞} is replaced by a real-valued function φ : C → R, the problem
1.1 reduces to the following mixed equilibrium problem introduced by Ceng and Yao 6:
Find x ∈ C such that Fx, y φy − φx ≥ 0,
∀y ∈ C.
1.4
Recall that a mapping T : C → C is said to be a κ-strict pseudocontraction 7 if there
exists 0 ≤ κ < 1, such that
2
T x − T y2 ≤ x − y2 κI − T x − I − T y ,
∀x, y ∈ C,
1.5
where I denotes the identity operator on C. When κ 0, T is said to be nonexpansive. Note
that the class of strict pseudocontraction mappings strictly includes the class of nonexpansive
mappings. We denote the set of fixed points of S by FixS.
Ceng and Yao 6, Yao et al. 8, and Peng and Yao 9, 10 introduced some iterative
schemes for finding a common element of the set of solutions of the mixed equilibrium
problem 1.4 and the set of common fixed points of a family of finitely infinitely
nonexpansive mappings strict pseudocontractions in a Hilbert space and obtained some
strong convergence theoremsweak convergence theorems. Some methods have been
proposed to solve the problem 1.2; see, for instance, 3–5, 11–18 and the references
therein. Recently, S. Takahashi and W. Takahashi 12 introduced an iterative scheme by
the viscosity approximation method for finding a common element of the set of solutions
of problem 1.2 and the set of fixed points of a nonexpansive mapping in a Hilbert space
and proved a strong convergence theorem. Su et al. 13 introduced an iterative scheme by
the viscosity approximation method for finding a common element of the set of solutions
of problem 1.2 and the set of fixed points of a nonexpansive mapping and the set of
solutions of the variational inequality problem for an α-inverse strongly monotone mapping
in a Hilbert space. Tada and Takahashi 14 introduced two iterative schemes for finding
Fixed Point Theory and Applications
3
a common element of the set of solutions of problem 1.2 and the set of fixed points of
a nonexpansive mapping in a Hilbert space and obtained both strong convergence theorem
and weak convergence theorem. Ceng et al. 15 introduced an iterative algorithm for finding
a common element of the set of solutions of problem 1.2 and the set of fixed points of a strict
pseudocontraction mapping. Chang et al. 16 introduced some iterative processes based on
the extragradient method for finding the common element of the set of fixed points of a family
of infinitely nonexpansive mappings, the set of problem 1.2, and the set of solutions of
a variational inequality problem for an α-inverse strongly monotone mapping. Colao et al.
17 introduced an iterative method for finding a common element of the set of solutions
of problem 1.2 and the set of fixed points of a finite family of nonexpansive mappings
in a Hilbert space and proved the strong convergence of the proposed iterative algorithm
to the unique solution of a variational inequality, which is the optimality condition for a
minimization problem. To the best of our knowledge, there is not any algorithms for solving
problem 1.1.
On the other hand, Marino and Xu 19 and Zhou 20 introduced and researched
some iterative scheme for finding a fixed point of a strict pseudocontraction mapping. Acedo
and Xu 21 introduced some parallel and cyclic algorithms for finding a common fixed point
of a family of finite strict pseudocontraction mappings and obtained both weak and strong
convergence theorems for the sequences generated by the iterative schemes.
In the present paper, we introduce a new approximation scheme combining the
viscosity method with parallel method for finding a common element of the set of solutions
of the generalized equilibrium problem and the set of fixed points of a family of finitely strict
pseudocontractions. We obtain a strong convergence theorem for the sequences generated by
these processes. Based on this result, we also get some new and interesting results. The results
in this paper extend and improve some well-known results in the literature.
2. Preliminaries
Let H be a real Hilbert space with inner product · , · and norm · . Let C be a nonemptyclosed convex subset of H. Let symbols → and denote strong and weak convergences,
respectively. 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 C, denoted by PC x, such that
x − PC x ≤ x − y for all y ∈ C. The mapping PC is called the metric projection of H onto
C. We know that PC is a nonexpansive mapping from H onto C. It is also known that PC x ∈ C
and
x − PC x, PC x − y ≥ 0
2.2
for all x ∈ H and y ∈ C.
For each B ⊆ H, we denote by convB the convex hull of B. A multivalued mapping
G : B → 2H is said to be a KKM map if, for every finite subset {x1 , x2 , . . . , xn } ⊆ B,
conv{x1 , x2 , . . . , xn } ⊆ ∞
n1 Gxi .
4
Fixed Point Theory and Applications
We will use the following results in the sequel.
Lemma 2.1 see 22. Let B be a nonempty subset of a Hausdorff topological vector space X and let
G : B → 2X be a KKM map. If Gx is closed for all x ∈ B and is compact for at least one x ∈ B, then
∅.
x∈B Gx /
For solving the generalized equilibrium problem, let us give the following assumptions for the bifunction F, ϕ, and the set C:
A1 Fx, x 0 for all x ∈ C;
A2 F is monotone, that is, Fx, y Fy, x ≤ 0 for any x, y ∈ C;
A3 for each y ∈ C, x → Fx, y is weakly upper semicontinuous;
A4 for each x ∈ C, y → Fx, y is convex;
A5 for each x ∈ C, y → Fx, y is lower semicontinuous;
B1 For each x ∈ H and r > 0, there exist a bounded subset Dx ⊆ C and yx ∈ C ∩ dom ϕ
such that for any z ∈ C \ Dx ,
1
F z, yx ϕ yx yx − z, z − x < ϕz;
r
2.3
B2 C is a bounded set.
Lemma 2.2. Let C be a nonempty-closed convex subset of H. Let F be a bifunction from C × C to
R satisfying (A1)–(A4) and let ϕ : H → R ∪ {∞} be a proper lower semicontinuous and convex
function such that C ∩ dom ϕ /
∅. For r > 0 and x ∈ H, define a mapping Sr : H → C as follows:
1
Sr x z ∈ C : Fz, y ϕy y − z, z − x ≥ ϕz, ∀y ∈ C
r
2.4
for all x ∈ H. Assume that either (B1) or (B2) holds. Then, the following conclusions hold:
1 for each x ∈ H, Sr x / ∅;
2 Sr is single-valued;
3 Sr is firmly nonexpansive, that is, for any x, y ∈ H,
Sr x − Sr y2 ≤ Sr x − Sr y, x − y ;
2.5
4 FixSr GEPF, ϕ;
5 GEPF, ϕ is closed and convex.
Proof. Let x0 be any given point in E. For each y ∈ C, we define
1
Gy z ∈ C : Fz, y ϕy y − z, z − x0 ≥ ϕz .
r
2.6
Fixed Point Theory and Applications
5
Note that for each y ∈ C∩dom ϕ, Gy is nonempty since y ∈ Gy and for each y ∈ C\dom ϕ,
Gy C. We will prove that G is a KKM map on C ∩ dom ϕ. Suppose that there exists a finite
subset {y1 , y2 , . . . , yn } of C ∩ dom ϕ and μi ≥ 0 for all i 1, 2, . . . , n with ni1 μi 1 such that
∈Gyi for each i 1, 2, . . . , n. Then we have
z ni1 μi yi /
1
F z, yi ϕ yi − ϕ
z yi − z, z − x0 < 0
r
2.7
for each i 1, 2, . . . , n. By A4 and the convexity of ϕ, we have
1
z − z, z − x0
r
n
n
1 ≤
μi F z, yi ϕ yi − ϕ
μi yi − z, z − x0 < 0,
z r i1
i1
0 F
z, z ϕ
z − ϕ
z 2.8
w
which is a contradiction. Hence, G is a KKM map on C ∩ dom ϕ. Note that Gy the weak
closure of Gy is a weakly closed subset of C for each y ∈ C. Moreover, if B2 holds, then
w
Gy is also weakly compact for each y ∈ C. If B1 holds, then for x0 ∈ E, there exists a
bounded subset Dx0 ⊆ C and yx0 ∈ C ∩ dom ϕ such that for any z ∈ C \ Dx0 ,
1
F z, yx0 ϕ yx0 yx0 − z, z − x0 < ϕz.
r
2.9
This shows that
G yx0 1
z ∈ C : F z, yx0 ϕ yx0 yx0 − z, z − x0 ≥ ϕz ⊆ Dx0 .
r
2.10
w
Hence, Gyx0 is weakly compact. Thus, in both cases, we can use Lemma 2.1 and have
w
∅.
y∈C∩dom ϕ Gy /
w
Next, we will prove that Gy Gy for each y ∈ C; that is, Gy is weakly closed.
w
Let z ∈ Gy and let zm be a sequence in Gy such that zm z. Then,
1
F zm , y ϕy y − zm , zm − x0 ≥ ϕ zm .
r
2.11
Since · 2 is weakly lower semicontinuous, we can show that
lim sup y − zm , zm − x0 ≤ z − y, x0 − z .
m→∞
2.12
6
Fixed Point Theory and Applications
It follows from A3 and the weak lower semicontinuity of ϕ that
1
ϕz ≤ lim infϕ zm ≤ lim sup F zm , y ϕy y − zm , zm − x0
m→∞
r
m→∞
1
≤ lim sup F zm , y ϕy lim sup y − zm , zm − x0
r
m→∞
m→∞
1
≤ Fz, y ϕy z − y, x0 − z .
r
2.13
This implies that z ∈ Gy. Hence, Gy is weakly closed. Hence, Sr x0 y∈C Gy w
/ ∅. Hence, from the arbitrariness of x0 , we conclude that
y∈C∩dom ϕ Gy y∈C∩dom ϕ Gy ∅,
∀x
∈
H.
Sr x /
We observe that Sr x ⊆ dom ϕ. So by similar argument with that in the proof of
Lemma 2.3 in 9, we can easily show that Sr is single-valued and Sr is a firmly nonexpansivetype map. Next, we claim that FixSr GEPF, ϕ. Indeed, we have the following:
u ∈ Fix Sr ⇐⇒ u Sr u
1
⇐⇒ Fu, y ϕy y − u, u − u ≥ ϕu,
r
⇐⇒ Fu, y ϕy ≥ ϕu, ∀y ∈ C
∀y ∈ C
2.14
⇐⇒ u ∈ GEPF, ϕ.
At last, we claim that GEPF, ϕ is a closed convex. Indeed, Since Sr is firmly nonexpansive,
Sr is also nonexpansive. By 23, Proposition 5.3, we know that GEPF, ϕ FixSr is closed
and convex.
Remark 2.3. It is easy to see that Lemma 2.2 is a generalization of 9, Lemma 2.3.
Lemma 2.4 see 24, 25. Assume that {αn } is a sequence of nonnegative real numbers such that
αn1 ≤ 1 − γn αn δn ,
2.15
where γn is a sequence in 0, 1 and {δn } is a sequence such that
i
∞
γn ∞;
n1
ii
δn
lim sup
≤ 0 or
n → ∞ γn
∞ δn < ∞.
2.16
n1
Then, limn → ∞ αn 0.
Lemma 2.5. In a real Hilbert space H, there holds the following inequality:
x y2 ≤ x2 2y, x y
for all x, y ∈ H.
2.17
Fixed Point Theory and Applications
7
3. Strong Convergence Theorems
In this section, we show a strong convergence of an iterative algorithm based on both
viscosity approximation method and parallel method which solves the problem of finding
a common element of the set of solutions of a generalized equilibrium problem and the set of
fixed points of a family of finitely strict pseudocontractions in a Hilbert space.
n
We need the following assumptions for the parameters {γn }, {rn }, {αn }, {ζ1 },
n
n
{ζ2 }, . . . , {ζN }, and {βn }:
C1 limn → ∞ αn 0 and
∞
n1 αn
∞;
C2 1 > lim supn → ∞ βn ≥ lim infn → ∞ βn > 0;
C3 {γn } ⊂ c, d for some c, d ∈ ε, 1 and limn → ∞ |γn1 − γn | 0;
C4 lim infn → ∞ rn > 0 and limn → ∞ |rn1 − rn | 0;
n1
C5 limn → ∞ |ζj
n
− ζj | 0 for all j 1, 2, . . . , N.
Theorem 3.1. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let F be a
bifunction from C × C to R satisfying (A1)–(A5), and let ϕ : C → R ∪ {∞} be a proper
lower semicontinuous and convex function such that C ∩ dom ϕ /
∅. Let N ≥ 1 be an integer.
For each 1 ≤ j ≤ N, let Tj : C → C be an εj -strict pseudocontraction for some 0 ≤ εj < 1
n N
such that Ω N
/ ∅. Assume for each n, {ζj }j1 is a finite sequence of
j1 FixTj ∩ GEPF, ϕ n
n
1 for all n and infn≥1 ζj > 0 for all 0 ≤ j ≤ N. Let
positive numbers such that N
j1 ζj
ε max{εj : 1 ≤ j ≤ N}. Assume that either (B1) or (B2) holds. Let f be a contraction of C
into itself and let {xn }, {un }, and {yn } be sequences generated by
x1 x ∈ C,
1
F un , y ϕy y − un , un − xn ≥ ϕ un ,
rn
N
n
ζ j T j un ,
yn γn un 1 − γn
∀y ∈ C,
3.1
j1
xn1 αn f xn βn xn 1 − αn − βn yn
n
n
n
for every n 1, 2, . . ., where {γn }, {rn }, {αn }, {ζ1 }, {ζ2 }, . . . , {ζN }, and {βn } are sequences of
numbers satisfying the conditions (C1)–(C5). Then, {xn }, {un }, and {yn } converge strongly to w PΩ fw.
Proof. We show that PΩ f is a contraction of C into itself. In fact, there exists a ∈ 0, 1 such
that fx − fy ≤ ax − y for all x, y ∈ C. So, we have
PΩ fx − PΩ fy ≤ fx − fy ≤ ax − y
3.2
for all x, y ∈ C. Since H is complete, there exists a unique element u0 ∈ C such that u0 PΩ fu0 .
8
Fixed Point Theory and Applications
Let u ∈ Ω and let {Srn } be a sequence of mappings defined as in Lemma 2.2. From
un Srn xn ∈ C, we have
un − u Sr xn − Sr u ≤ xn − u.
n
n
3.3
We define a mapping Wn by
Wn x N
n
ζj Tj x,
∀x ∈ C.
3.4
j1
By 21, Proposition 2.6, we know that Wn is an ε-strict pseudocontraction and FWn N
j1 FixTj . It follows from 3.3, yn γn un 1 − γn Wn un and u Wn u such that
yn − u2 γn un − u2 1 − γn Wn un − u2 − γn 1 − γn un − Wn un 2
2 2
2 2
≤ γn un − u 1 − γn un − u εun − Wn un − γn 1 − γn un − Wn un 2 2
un − u 1 − γn ε − γn un − Wn un 2
≤ un − u .
3.5
Put M0 max{x1 −u, 1/1−afu−u}. It is obvious that x1 −u ≤ M0 . Suppose
xn − u ≤ M0 . From 3.3, 3.5, and xn1 αn fxn βn xn 1 − αn − βn yn , we have
xn1 − u αn f xn βn xn 1 − αn − βn yn − u
≤ αn f xn − fu αn fu − u βn xn − u 1 − αn − βn yn − u
≤ αn axn − u αn fu − u βn xn − u 1 − αn − βn un − u
3.6
≤ αn axn − u αn fu − u 1 − αn xn − u
fu − u 1 − aαn
1 − 1 − aαn xn − u,
1−a
≤ 1 − aαn M0 1 − 1 − aαn M0 M0
for every n 1, 2, . . . . Therefore, {xn } is bounded. From 3.3 and 3.5, we also obtain that
{yn } and {un } are bounded.
Following 26, define Bn : C → C by
Bn γn I 1 − γn Wn .
3.7
Fixed Point Theory and Applications
9
As shown in 26, each Bn is a nonexpansive mapping on C. Set M1 supn≥1 {un − Wn un },
we have
yn1 − yn Bn1 un1 − Bn un ≤ Bn1 un1 − Bn1 un Bn1 un − Bn un ≤ un1 − un M1 γn1 − γn 1 − γn1 Wn1 un − Wn un 3.8
N ζn1 − ζn Tj un .
≤ un1 − un M1 γn1 − γn 1 − γn1
j
j
j1
On the other hand, from un Trn xn and un1 Trn1 xn1 , we have
1
F un , y ϕy y − un , un − xn ≥ ϕ un ,
rn
∀y ∈ C,
1 F un1 , y ϕy y − un1 , un1 − xn1 ≥ ϕ un1 ,
rn1
∀y ∈ C.
3.9
3.10
Putting y un1 in 3.9 and y un in 3.10, we have
1
F un , un1 ϕ un1 un1 − un , un − xn ≥ ϕ un ,
rn
1 un − un1 , un1 − xn1 ≥ ϕ un1 .
F un1 , un ϕ un rn1
3.11
So, from the monotonicity of F, we get
un − xn un1 − xn1
−
≥ 0,
un1 − un ,
rn
rn1
3.12
rn un1 − un , un − un1 un1 − xn −
un1 − xn1
≥ 0.
rn1
3.13
hence
Without loss of generality, let us assume that there exists a real number b such that rn > b > 0
for all n ∈ N. Then,
rn un1 − xn1
un1 − un , xn1 − xn 1 −
rn1
rn un1 − xn1 ,
xn1 − xn 1 −
≤ un1 − un
rn1 un1 − un 2 ≤
3.14
10
Fixed Point Theory and Applications
hence
un1 − un ≤ xn1 − xn 1 rn1 − rn un1 − xn1 rn1
1
≤ xn1 − xn rn1 − rn M2 ,
b
3.15
where M2 sup{un − xn : n ≥ 1}.
It follows from 3.8 and 3.15 that
yn1 − yn ≤ xn1 − xn 1 rn1 − rn M2 M1 γn1 − γn b
N ζn1 − ζn Tj un .
1 − γn1
j
j
3.16
j1
Define a sequence {vn } such that
xn1 βn xn 1 − βn vn ,
∀n ≥ 1.
3.17
Then, we have
vn1 − vn xn2 − βn1 xn1 xn1 − βn xn
−
1 − βn1
1 − βn
αn1 fxn1 1 − αn1 − βn1 yn1 αn fxn 1 − αn − βn yn
−
1 − βn1
1 − βn
αn1
αn
αn1
αn
f xn1 −
f xn yn1 − yn yn −
yn1 .
1 − βn1
1 − βn
1 − βn
1 − βn1
3.18
From 3.18 and 3.16, we have
vn1 − vn − xn1 − xn ≤
αn1 f xn1 yn1 αn f xn yn 1 − βn1
1 − βn
yn1 − yn − xn1 − xn αn1 f xn1 yn1 αn f xn yn ≤
1 − βn1
1 − βn
N 1
ζn1 − ζn Tj un .
rn1 − rn M2 M1 γn1 − γn 1 − γn1
j
j
b
j1
3.19
Fixed Point Theory and Applications
11
It follows from C1–C5 that
lim sup vn1 − vn − xn1 − xn ≤ 0.
n→∞
3.20
Hence, by 27, Lemma 2.2, we have limn → ∞ vn − xn 0. Consequently,
lim xn1 − xn lim 1 − βn vn − xn 0.
n→∞
n→∞
3.21
Since xn1 αn fxn βn xn 1 − αn − βn yn , we have
xn − yn ≤ xn1 − xn xn1 − yn ≤ xn1 − xn αn f xn − yn βn xn − yn ,
3.22
thus
xn − yn ≤
1 xn1 − xn αn fxn − yn .
1 − βn
3.23
It follows from C1 and C2 that limn → ∞ xn − yn 0.
Since xn1 αn fxn βn xn 1 − αn − βn yn , for u ∈ Ω, it follows from 3.5 and 3.3
that
xn1 − u2 αn f xn βn xn 1 − αn − βn yn − u2
2 2
2
≤ αn f xn − u βn xn − u 1 − αn − βn yn − u
2
2
≤ αn f xn − u βn xn − u
2 2 1 − αn − βn un − u 1 − γn ε − γn un − Wn un 2
2 ≤ αn f xn − u 1 − αn xn − u
2
1 − αn − βn 1 − γn ε − γn un − Wn un ,
3.24
from which it follows that
αn
f xn − u2 − xn − u2
1 − αn − βn 1 − γn γn − ε
1
xn − u2 − xn1 − u2 .
1 − αn − βn 1 − γn γn − ε
3.25
αn
f xn − u2 − xn − u2
≤
1 − αn − βn 1 − γn γn − ε
1
xn − u xn1 − u xn1 − xn .
1 − αn − βn 1 − γn γn − ε
un − Wn un 2 ≤
12
Fixed Point Theory and Applications
It follows from C1–C3 and xn1 − xn → 0 that
un − Wn un −→ 0.
3.26
For u ∈ Ω, we have from Lemma 2.2,
un − u2 Sr xn − Sr u2 ≤ Sr xn − Sr u, xn − u
n
n
n
n
1 un − u2 xn − u2 − xn − un 2 .
un − u, xn − u 2
3.27
Hence,
un − u2 ≤ xn − u2 − xn − un 2 .
3.28
By 3.24 and 3.28, we have
xn1 − u2 ≤ αn f xn − u2 βn xn − u2 1 − αn − βn un − u2
2 2 2 2
≤ αn f xn − u βn xn − u 1 − αn − βn xn − u − xn − un .
3.29
Hence,
2 2 2
2
2
1 − αn − βn xn − un ≤ αn f xn − u − αn xn − u xn − u − xn1 − u
2
2
≤ αn f xn − u − αn xn − u
xn − u xn1 − u xn − xn1 .
3.30
It follows from C1, C2, and xn − xn1 → 0 that limn → ∞ xn − un 0.
Next, we show that
lim sup fu0 − u0 , xn − u0 ≤ 0,
n→∞
3.31
where u0 PΩ fu0 . To show this inequality, we can choose a subsequence {xni } of {xn } such
that
lim fu0 − u0 , xni − u0 lim sup fu0 − u0 , xn − u0 .
i→∞
n→∞
3.32
Since {xni } is bounded, there exists a subsequence {xnij } of {xni } which converges
weakly to w. Without loss of generality, we can assume that {xni } w. From xn − un → 0,
Fixed Point Theory and Applications
13
we obtain that uni w. From xn − yn → 0, we also obtain that yni w. Since {uni } ⊂ C
and C is closed and convex, we obtain w ∈ C.
We first show that w ∈ N
k1 FixTk . To see this, we observe that we may assume by
n passing to a further subsequence if necessary ζk i → ζk as i → ∞ for k 1, 2, . . . , N. It is
easy to see that ζk > 0 and N
k1 ζk 1. We also have
Wni x −→ Wx
as i −→ ∞ ∀x ∈ C,
3.33
where W N
k1 ζk Tk . Note that by 21, Proposition 2.6, W is an ε-strict pseudocontraction
and FixW N
i1 FixTi . Since
un − Wun ≤ un − Wn un Wn un − Wun i
i
i
i
i
i
i
i
N n ≤ uni − Wni uni ζk i − ζk Tk uni ,
3.34
k1
ni it follows from 3.26 and ζk
→ ζk that
un − Wun −→ 0.
i
i
3.35
So by the demiclosedness principle 21, Proposition 2.6ii, it follows that w ∈ FixW N
i1 FixTi .
We now show w ∈ GEPF, ϕ. By un Trn xn , we know that
1
F un , y ϕy y − un , un − xn ≥ ϕ un ,
rn
∀y ∈ C.
3.36
∀y ∈ C.
3.37
∀y ∈ C.
3.38
It follows from A2 that
ϕy 1
y − un , un − xn ≥ F y, un ϕ un ,
rn
Hence,
un − xni
ϕy y − uni , i
rni
≥ F y, uni ϕ un ,
14
Fixed Point Theory and Applications
It follows from A4, A5 and the weakly lower semicontinuity of ϕ, uni − xni /rni →
0, and uni w that
Fy, w ϕw ≤ ϕy,
∀y ∈ C.
3.39
For t with 0 < t ≤ 1 and y ∈ C ∩ dom ϕ, let yt ty 1 − tw. Since y ∈ C ∩ dom ϕ and
w ∈ C ∩ dom ϕ, we obtain yt ∈ C ∩ dom ϕ, and hence Fyt , w ϕw ≤ ϕyt . So by A4 and
the convexity of ϕ, we have
0 F yt , yt ϕ yt − ϕ yt
≤ tF yt , y 1 − tF yt , w tϕy 1 − tϕw − ϕ yt
≤ t F yt , y ϕy − ϕ yt .
3.40
Dividing by t, we get
F yt , y ϕy − ϕ yt ≥ 0.
3.41
Letting t → 0, it follows from A3 and the weakly lower semicontinuity of ϕ that
Fw, y ϕy ≥ ϕw
3.42
for all y ∈ C ∩ dom ϕ. Observe that if y ∈ C \ dom ϕ, then Fw, y ϕy ≥ ϕw holds.
Moreover, hence w ∈ GEPF, ϕ. This implies w ∈ Ω. Therefore, we have
lim sup fu0 − u0 , xn − u0 lim fu0 − u0 , xni − u0 fu0 − u0 , w − u0 ≤ 0.
n→∞
i→∞
3.43
Finally, we show that xn → u0 , where u0 PΩ fu0 .
From Lemma 2.5, we have
xn1 − u0 2 αn f xn − u0 βn xn − u0 1 − αn − βn yn − u0 2
2
≤ βn xn − u0 1 − αn − βn yn − u0 2αn f xn − u0 , xn1 − u0
2
≤ 1 − αn − βn yn − u0 βn xn − u0 2αn f xn − u0 , xn1 − u0
2
2 ≤ 1 − αn xn − u0 2αn f xn − f u0 , xn1 − u0
2αn f u0 − u0 , xn1 − u0
2
2 ≤ 1 − αn xn − u0 2αn axn − u0 xn1 − u0 2αn f u0 − u0 , xn1 − u0
2
2 2 2 ≤ 1 − αn xn − u0 αn a xn − u0 xn1 − u0 2αn f u0 − u0 , xn1 − u0 ,
3.44
Fixed Point Theory and Applications
15
thus
xn1 − u0 2 ≤
21 − aαn xn − u0 2
1 − aαn
3.45
2
21 − aαn
1
αn xn − u0 2f u0 − 2u0 , xn1 − u0 .
1 − aαn
21 − a
1−a
1−
It follows from C1, 3.43, 3.45, and Lemma 2.4 that limn → ∞ xn − u0 0. From
xn − un → 0 and yn − xn → 0, we have un → u0 and yn → u0 . The proof is now
complete.
Theorem 3.2. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let F be a
bifunction from C × C to R satisfying (A1)–(A5), and let ϕ : H → R ∪ {∞} be a proper
lower semicontinuous and convex function such that C ∩ dom ϕ /
∅. Let N ≥ 1 be an integer.
For each 1 ≤ j ≤ N, let Tj : C → C be an εj -strict pseudocontraction for some 0 ≤ εj < 1
n N
∅. Assume for each n, {ζj } is a finite sequence of
such that Ω N
j1 FixTj ∩ GEPF, ϕ /
j1
n
n
positive numbers such that N
ζ
1
for
all
n
and
inf
ζ
>
0
for all 0 ≤ j ≤ N. Let
n≥1
j1 j
j
ε max{εj : 1 ≤ j ≤ N}. Assume that either (B1) or (B2) holds. Let v be an arbitrary point in
C and let {xn }, {un }, and {yn } be sequences generated by
x1 x ∈ C,
1
F un , y ϕy y − un , un − xn ≥ ϕ un ,
rn
N
n
ζ j T j un ,
yn γn un 1 − γn
∀y ∈ C,
3.46
j1
xn1 αn v βn xn 1 − αn − βn yn
n
n
n
for every n 1, 2, . . . , where {γn }, {rn }, {αn }, {ζ1 }, {ζ2 }, . . . , {ζN }, and {βn } are sequences of
numbers satisfying the conditions (C1)–(C5). Then, {xn }, {un }, and {yn } converge strongly to w PΩ v.
Proof. Let fx v for all x ∈ C, by Theorem 3.1, we obtain the desired result.
4. Applications
By Theorems 3.1 and 3.2, we can obtain many new and interesting strong convergence
theorems. Now, give some examples as follows: for j 1, 2, . . . , N, let T1 T2 · · · TN T ,
by Theorems 3.1 and 3.2, respectively, we have the following results.
Theorem 4.1. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let F be a
bifunction from C × C to R satisfying (A1)–(A5), and let ϕ : H → R ∪ {∞} be a proper lower
semicontinuous and convex function such that C ∩ dom ϕ /
∅. Let T : C → C be an ε-strict
pseudocontraction for some 0 ≤ ε < 1 such that FixT ∩ GEPF, ϕ /
∅. Assume that either (B1) or
16
Fixed Point Theory and Applications
(B2) holds. Let f be a contraction of C into itself and let {xn }, {un }, and {yn } be sequences generated
by
x1 x ∈ C,
1
F un , y ϕy y − un , un − xn ≥ ϕ un , ∀y ∈ C,
rn
yn γn un 1 − γn T un ,
xn1 αn f xn βn xn 1 − αn − βn yn
4.1
for every n 1, 2, . . . , where {γn }, {rn }, {αn }, and {βn } are sequences of numbers satisfying the
conditions (C1)–(C4). Then, {xn }, {un }, and {yn } converge strongly to w PFixT ∩GEPF,ϕ fw.
Theorem 4.2. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let F be a
bifunction from C × C to R satisfying (A1)–(A5), and let ϕ : H → R ∪ {∞} be a proper lower
semicontinuous and convex function such that C ∩ dom ϕ /
∅. Let T : C → C be an ε-strict
pseudocontraction for some 0 ≤ ε < 1 such that FixT ∩ GEPF, ϕ /
∅. Assume that either (B1)
or (B2) holds. Let v be an arbitrary point in C, and let {xn }, {un }, and {yn } be sequences generated
by
x1 x ∈ C,
1
F un , y ϕy y − un , un − xn ≥ ϕ un ,
rn
yn γn un 1 − γn T un ,
xn1 αn v βn xn 1 − αn − βn yn
∀y ∈ C,
4.2
for every n 1, 2, . . . , where {γn }, {rn }, {αn }, and {βn } are sequences of numbers satisfying the
conditions (C1)–(C4). Then, {xn }, {un }, and {yn } converge strongly to w PFixT ∩GEPF,ϕ v.
We need the following two assumptions.
B3 For each x ∈ H and r > 0, there exist a bounded subset Dx ⊆ C and yx ∈ C such
that for any z ∈ C \ Dx ,
1
F z, yx yx − z, z − x < 0.
r
4.3
B4 For each x ∈ H and r > 0, there exist a bounded subset Dx ⊆ C and yx ∈ C ∩ dom ϕ
such that for any z ∈ C \ Dx ,
1
g yx ϕ yx yx − z, z − x < ϕz gz.
r
4.4
Let ϕx δC x, ∀x ∈ H, by Theorems 3.1 and 3.2, respectively, we obtain the
following results.
Fixed Point Theory and Applications
17
Theorem 4.3. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let F be a
bifunction from C × C to R satisfying (A1)–(A5). Let N ≥ 1 be an integer. For each 1 ≤ j ≤ N, let Tj :
C → C be an εj -strict pseudocontraction for some 0 ≤ εj < 1 such that Γ N
∅.
j1 FixTj ∩ EPF /
N n
n N
Assume for each n, {ζj } is a finite sequence of positive numbers such that j1 ζj 1 for all n
n
infn≥1 ζj
j1
and
> 0 for all 0 ≤ j ≤ N. Let ε max{εj : 1 ≤ j ≤ N}. Assume that either (B3) or (B2)
holds. Let f be a contraction of H into itself, and let {xn }, {un }, and {yn } be sequences generated by
x1 x ∈ C,
1
F un , y y − un , un − xn ≥ 0, ∀y ∈ C,
rn
N
n
ζ j T j un ,
yn γn un 1 − γn
4.5
j1
xn1 αn f xn βn xn 1 − αn − βn yn
n
n
n
for every n 1, 2, . . . , where {γn }, {rn }, {αn }, {ζ1 }, {ζ2 }, . . . , {ζN }, and {βn } are sequences of
numbers satisfying the conditions (C1)–(C5). Then, {xn }, {un }, and {yn } converge strongly to w PΓ fw.
Theorem 4.4. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let F be a
bifunction from C × C to R satisfying (A1)–(A5). Let N ≥ 1 be an integer. For each 1 ≤ j ≤ N, let Tj :
∅.
C → C be an εj -strict pseudocontraction for some 0 ≤ εj < 1 such that Γ N
j1 FixTj ∩ EPF /
N n
n N
Assume for each n, {ζj } is a finite sequence of positive numbers such that j1 ζj 1 for all n
n
infn≥1 ζj
j1
and
> 0 for all 0 ≤ j ≤ N. Let ε max{εj : 1 ≤ j ≤ N}. Assume that either (B3) or (B2)
holds. Let v be an arbitrary point in C, and let {xn }, {un }, and {yn } be sequences generated by
x1 x ∈ C,
1
F un , y y − un , un − xn ≥ 0, ∀y ∈ C,
rn
N
n
ζ j T j un ,
yn γn un 1 − γn
4.6
j1
xn1 αn v βn xn 1 − αn − βn yn
n
n
n
for every n 1, 2, . . . , where {γn }, {rn }, {αn }, {ζ1 }, {ζ2 }, . . . , {ζN }, and {βn } are sequences of
numbers satisfying the conditions (C1)–(C5). Then, {xn }, {un }, and {yn } converge strongly to w PΓ v.
Let Fx, y gy − gx for all x, y ∈ C, by Theorems 3.1 and 3.2, respectively, we
obtain the following results.
Theorem 4.5. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let g : C → R
be a lower semicontinuous and convex function, and let ϕ : H → R ∪ {∞} be a proper lower
semicontinuous and convex function such that C ∩ dom ϕ /
∅. Let N ≥ 1 be an integer. For each
18
Fixed Point Theory and Applications
1 ≤ j ≤ N, let Tj : C → C be an εj -strict pseudocontraction for some 0 ≤ εj < 1 such that Θ N
n N
∅. Assume for each n, {ζj } is a finite sequence of positive numbers
j1 FixTj ∩Argming, ϕ /
j1
n
n
such that N
ζ
1
for
all
n
and
inf
ζ
>
0
for
all
0
≤ j ≤ N. Let ε max{εj : 1 ≤ j ≤ N}.
n≥1
j1 j
j
Assume that either (B4) or (B2) holds. Let f be a contraction of H into itself, and let {xn }, {un } and
{yn } be sequences generated by
x1 x ∈ C,
1
gy ϕy y − un , un − xn ≥ g un ϕ un ,
rn
N
n
ζ j T j un ,
yn γn un 1 − γn
∀y ∈ C,
4.7
j1
xn1 αn f xn βn xn 1 − αn − βn yn
n
n
n
for every n 1, 2, . . ., where {γn }, {rn }, {αn }, {ζ1 }, {ζ2 }, . . . , {ζN }, and {βn } are sequences of
numbers satisfying the conditions (C1)–(C5). Then, {xn }, {un }, and {yn } converge strongly to w PΘ fw.
Theorem 4.6. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let g : C → R
be a lower semicontinuous and convex function, and let ϕ : H → R ∪ {∞} be a proper lower
semicontinuous and convex function such that C ∩ dom ϕ /
∅. Let N ≥ 1 be an integer. For each
1 ≤ j ≤ N, let Tj : C → C be an εj -strict pseudocontraction for some 0 ≤ εj < 1 such that Θ N
n N
/ ∅. Assume for each n, {ζj }j1 is a finite sequence of positive numbers
j1 FixTj ∩Argming, ϕ n
n
such that N
1 for all n and infn≥1 ζj > 0 for all 0 ≤ j ≤ N. Let ε max{εj : 1 ≤ j ≤ N}.
j1 ζj
Assume that either (B4) or (B2) holds. Let v be an arbitrary point in C, and let {xn }, {un }, and {yn }
be sequences generated by
x1 x ∈ C,
1
gy ϕy y − un , un − xn ≥ ϕ un g un ,
rn
N
n
ζ j T j un ,
yn γn un 1 − γn
∀y ∈ C,
4.8
j1
xn1 αn v βn xn 1 − αn − βn yn
n
n
n
for every n 1, 2, . . . , where {γn }, {rn }, {αn }, {ζ1 }, {ζ2 }, . . . , {ζN }, and {βn } are sequences of
numbers satisfying the conditions (C1)–(C5). Then, {xn }, {un }, and {yn } converge strongly to w PΘ v.
Let ϕx δC x, ∀x ∈ H, and let Fx, y 0 for all x, y ∈ C. Then un PC xn xn . By
Theorems 3.1 and 3.2, we obtain the following results.
Theorem 4.7. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let N ≥ 1 be an
integer. For each 1 ≤ j ≤ N, let Tj : C → C be an εj -strict pseudocontraction for some 0 ≤ εj < 1
Fixed Point Theory and Applications
19
n N
such that N
∅. Assume for each n, {ζj } is a finite sequence of positive numbers such
j1 FixTj /
j1
n
n
ζ
1
for
all
n
and
inf
ζ
>
0
for
all
0
≤ j ≤ N. Let ε max{εj : 1 ≤ j ≤ N}. Let f
that N
n≥1 j
j1 j
be a contraction of H into itself, and let {xn } and {yn } be sequences generated by
x1 x ∈ C,
N
n
yn γn xn 1 − γn
ζj Tj xn ,
4.9
j1
xn1 αn f xn βn xn 1 − αn − βn yn
n
n
n
for every n 1, 2, . . ., where {γn }, {αn }, {ζ1 }, {ζ2 }, . . . , {ζN }, and {βn } are sequences of numbers
satisfying the conditions (C1)–(C3) and (C5). Then, {xn }, and {yn } converge strongly to w P∩Nj1 FixTj fw.
Theorem 4.8. Let C be a nonempty-closed convex subset of a real Hilbert space H. Let N ≥ 1 be an
integer. For each 1 ≤ j ≤ N, let Tj : C → C be an εj -strict pseudocontraction for some 0 ≤ εj < 1
n N
such that N
/ ∅. Assume for each n, {ζj }j1 is a finite sequence of positive numbers such
j1 FixTj n
n
1 for all n and infn≥1 ζj > 0 for all 0 ≤ j ≤ N. Let ε max{εj : 1 ≤ j ≤ N}. Let v
that N
j1 ζj
be an arbitrary point in C, and let {xn } and {yn } be sequences generated by
x1 x ∈ C,
N
n
yn γn xn 1 − γn
ζj Tj xn ,
4.10
j1
xn1 αn v βn xn 1 − αn − βn yn
n
n
n
for every n 1, 2, . . ., where {γn }, {αn }, {ζ1 }, {ζ2 }, . . . , {ζN }, and {βn } are sequences of numbers
satisfying the conditions (C1)–(C3) and (C5). Then, {xn } and {yn } converge strongly to w P∩Nj1 FixTj v.
Remark 4.9. 1 Since the nonexpansive mappings have been replaced by the strict
pseudocontractions, Theorems 3.1, 3.2, 4.1 and 4.2 extend and improve 6, Theorem 3.1, 8,
Theorem 3.5, 9, Theorems 4.1 and 4.2, 18, Theorem 4.1, and the main results in 9–11, 13–
16.
2 Since the weak convergence has been replaced by strong convergence, Theorems
3.1, 3.2, 4.1−4.4 extend and improve 12, Theorem 3.1, 10, Corollary 4.1.
3 Theorems 4.7 and 4.8 are strong convergence theorems for strict pseudocontractions without CQ constraints and hence they improve the corresponding results in 19, 21.
Theorems 3.1 and 3.2 also improve 10, Corollary 3.1.
Acknowledgments
The first author was supported by the National Natural Science Foundation of China
Grants No. 10771228 and No. 10831009, the Science and Technology Research Project of
20
Fixed Point Theory and Applications
Chinese Ministry of Education Grant no. 206123, the Education Committee project Research
Foundation of Chongqing Normal University Grant no. KJ070816; the second and third
authors were partially supported by the Grants NSC97-2221-E-230-017 and NSC96-2628-E110-014-MY3, respectively.
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