Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 141376, 17 pages
doi:10.1155/2010/141376
Research Article
Convergence Analysis for a System of Equilibrium
Problems and a Countable Family of Relatively
Quasi-Nonexpansive Mappings in Banach Spaces
Prasit Cholamjiak and Suthep Suantai
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
Correspondence should be addressed to Suthep Suantai, scmti005@chiangmai.ac.th
Received 11 April 2010; Revised 9 July 2010; Accepted 16 July 2010
Academic Editor: Simeon Reich
Copyright q 2010 P. Cholamjiak and S. Suantai. 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 new hybrid iterative scheme for finding a common element in the solutions set of a
system of equilibrium problems and the common fixed points set of an infinitely countable family
of relatively quasi-nonexpansive mappings in the framework of Banach spaces. We prove the
strong convergence theorem by the shrinking projection method. In addition, the results obtained
in this paper can be applied to a system of variational inequality problems and to a system of
convex minimization problems in a Banach space.
1. Introduction
Let E be a real Banach space, and let E∗ be the dual of E. Let C be a closed and convex subset
of E. Let {fj }j∈Λ be bifunctions from C × C to R, where R is the set of real numbers and Λ is
an arbitrary index set. The system of equilibrium problems is to find x ∈ C such that
fj x,
y ≥ 0,
∀y ∈ C, j ∈ Λ.
1.1
If Λ is a singleton, then problem 1.1 reduces to find x ∈ C such that
f x,
y ≥ 0,
∀y ∈ C.
The set of solutions of the equilibrium problem 1.2 is denoted by EPf.
1.2
2
Abstract and Applied Analysis
Combettes and Hirstoaga 1 introduced an iterative scheme for finding a common
element in the solutions set of problem 1.1 in a Hilbert space and obtained a weak
convergence theorem.
In 2004, Matsushita and Takahashi 2 introduced the following algorithm for a
relatively nonexpansive mapping T in a Banach space E: for any initial point x0 ∈ C, define
the sequence {xn } by
xn1 ΠC J −1 αn Jxn 1 − αn JT xn ,
1.3
n ≥ 0,
where J is the duality mapping on E, ΠC is the generalized projection from E onto C, and
{αn } is a sequence in 0, 1. They proved that the sequence {xn } converges weakly to fixed
point of T under some suitable conditions on {αn }.
In 2008, Takahashi and Zembayashi 3 introduced the following iterative scheme
which is called the shrinking projection method for a relatively nonexpansive mapping T
and an equilibrium problem in a Banach space E:
x0 x ∈ C,
C0 C,
yn J −1 αn Jxn 1 − αn JT xn ,
1
un ∈ C such that f un , y y − un , Jun − Jyn ≥ 0
rn
Cn1 z ∈ Cn : φz, un ≤ φz, xn ,
xn1 ΠCn1 x0 ,
∀y ∈ C,
1.4
n ≥ 0.
They proved that the sequence {xn } converges strongly to ΠFT ∩EPf x0 under some
appropriate conditions.
2. Preliminaries and Lemmas
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
< 1.
x/
y implies 2 2.1
It is also said to be uniformly convex if, for each 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
Abstract and Applied Analysis
3
It is known that a uniformly convex Banach space is reflexive and strictly convex. The
function δ : 0, 2 → 0, 1 which is called the modulus of convexity of E is defined as follows:
x y
: x, y ∈ E, x y 1, x − y ≥ ε .
δε inf 1 − 2 2.3
The space 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
t→0
x ty − x
t
2.4
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 duality mapping J : E → 2E is defined by
Jx x∗ ∈ E∗ : x, x∗ x2 x∗ 2
2.5
for all x ∈ E. If E is a Hilbert space, then J I, where I is the identity operator. It is also known
that, if E is uniformly smooth, then J is uniformly norm-to-norm continuous on bounded
subset of E see 4 for more details.
Let E be a smooth Banach space. The function φ : E × E → R is defined by
2
φ x, y x2 − 2 x, Jy y
2.6
for all x, y ∈ E. In a Hilbert space H, we have φx, y x − y2 for all x, y ∈ H.
Let C be a closed and convex subset of E, and let T be a mapping from C into itself.
A point p in C is said to be an asymptotic fixed point of T 5 if C contains a sequence {xn }
which converges weakly to p such that limn → ∞ xn − T xn 0. The set of asymptotic fixed
. A mapping T is said to be relatively nonexpansive 6–8
points of T will be denoted by FT
FT and φp, T x ≤ φp, x for all p ∈ FT and x ∈ C. The asymptotic behavior
if FT
of a relatively nonexpansive mapping was studied in 6, 7. T is said to be relatively quasinonexpansive if FT / ∅ and φp, T x ≤ φp, x for all p ∈ FT and x ∈ C. It is obvious that the
class of relatively quasi-nonexpansive mappings is more general than the class of relatively
nonexpansive mappings. The class of relatively quasi-nonexpansive mappings was studied
by many authors see, for example, 9–12. Recall that T is closed if
xn −→ x,
T xn −→ y imply T x y.
2.7
The aim of this paper is to introduce a new hybrid projection algorithm for finding a
common element in the solutions set of a system of equilibrium problems and the common
fixed points set of an infinitely countable family of closed and relatively quasi-nonexpansive
mappings in the frameworks of Banach spaces.
We will need the following lemmas.
4
Abstract and Applied Analysis
Lemma 2.1 Kamimura and Takahashi 8. Let E be a uniformly convex and smooth Banach
space, and let {xn }, {yn }be two sequences of E. If φxn , yn → 0 and either {xn } or {yn } is bounded,
then xn − yn → 0 as n → ∞.
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 13, is a mapping ΠC : E → C that assigns to an arbitrary point x ∈ E the minimum
point of the function φy, x; that is, ΠC x x, where x is the solution of the minimization
problem
φx, x min φ y, x : y ∈ C .
2.8
The existence and uniqueness of the operator ΠC follows from the properties of the functional
φ and strict monotonicity of the duality mapping J see, for instance, 4, 8, 13–15. In a
Hilbert space, ΠC is coincident with the metric projection.
Lemma 2.2 Alber 13, Kamimura and Takahashi 8. Let C be a nonempty, closed, and convex
subset of a smooth, strictly convex, and reflexive Banach space E, let x ∈ E, and let z ∈ C. Then
z ΠC x if and only if
y − z, Jx − Jz ≤ 0,
∀y ∈ C.
2.9
Lemma 2.3 Alber 13, Kamimura and Takahashi 8. Let C be a nonempty, closed, and convex
subset of a smooth, strictly convex, and reflexive Banach space E. Then
φ x, ΠC y φ ΠC y, y ≤ φ x, y
∀x ∈ C, y ∈ E.
2.10
Lemma 2.4 Qin et al. 16. Let E be a uniformly convex, smooth Banach space, and let C be a
closed and convex subset of E. Let T be a closed and relatively quasi-nonexpansive mapping from C
into itself. Then FT is closed and convex.
For solving the equilibrium problem, let us assume that a bifunction f satisfies the
following conditions:
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.
Lemma 2.5 Blum and Oettli 17. 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 y − z, Jz − Jx ≥ 0,
r
∀y ∈ C.
2.11
Abstract and Applied Analysis
5
Lemma 2.6 Takahashi and Zembayashi 18. Let C be a closed and convex subset of a uniformly
smooth, strictly convex, and reflexive Banach space E, and let f be a bifunction from C × C to R which
f
satisfies conditions (A1)–(A4). For all r > 0 and x ∈ E, define the mapping Tr : E → C as follows:
f
Tr x
1
z ∈ C : f z, y y − z, Jz − Jx ≥ 0, ∀y ∈ C .
r
2.12
Then, the following statements hold:
f
1 Tr is single valued;
f
2 Tr is of firmly nonexpansive type [19]; that is, for all x, y ∈ E,
f
f
f
f
f
f
Tr x − Tr y, JTr x − JTr y ≤ Tr x − Tr y, Jx − Jy ;
2.13
f
3 FTr EPf;
4 EPf is closed and convex.
Lemma 2.7 Takahashi and Zembayashi 18. Let C be a closed and convex subset of a smooth,
strictly, and reflexive Banach space E, let f be a bifunction from C × C to R which satisfies conditions
f
(A1)–(A4), and let r > 0. Then, for all x ∈ E and q ∈ FTr ,
f
f
φ q, Tr x φ Tr x, x ≤ φ q, x .
2.14
3. Strong Convergence Theorems
Theorem 3.1. Let E be a uniformly convex and uniformly smooth Banach space, and let C be a
nonempty, closed, and convex subset of E. Let {fj }M
j1 be bifunctions from C × C to R which satisfies
be
an
infinitely
countable family of closed and relatively quasiconditions (A1)–(A4), and let {Ti }∞
i1
M
∅. For
nonexpansive mappings from C into itself. Assume that F : ∞
i1 FTi ∩ j1 EPfj /
any initial point x0 ∈ E with x1 ΠC1 x0 and C1 C, define the sequence {xn } as follows:
yn,i J −1 αn Jxn 1 − αn JTi xn ,
f
f
f
M
M−1
TrM−1,n
· · · Tr1,n1 yn,i ,
un,i TrM,n
Cn1 z ∈ Cn : sup φz, un,i ≤ φz, xn ,
3.1
i≥1
xn1 ΠCn1 x0 ,
n ≥ 1.
Assume that {αn } and {rj,n } for j 1, 2, . . . , M are sequences which satisfy the following conditions:
B1 lim supn → ∞ αn < 1;
B2 lim infn → ∞ rj,n > 0.
Then the sequence {xn } converges strongly to ΠF x0 .
6
Abstract and Applied Analysis
Proof. We divide our proof into six steps as follows.
Step 1. F ⊂ Cn for all n ≥ 1.
From Lemma 2.4 we know that FTi is closed, and convex for all i ≥ 1. From
Lemma 2.64, we also know that EPfj is closed and convex for each j 1, 2, . . . , M. Hence
M
F : ∞
i1 FTi ∩ j1 EPfj is a nonempty, closed and convex subset of C. Clearly C1 C
is closed and convex. Suppose that Ck is closed and convex for some k ∈ N. For each z ∈ Ck
and i ≥ 1, we see that φz, uk,i ≤ φz, xk is equivalent to
2
z, Jxk − 2
z, Juk,i ≤ xk 2 − uk,i 2 .
3.2
By the construction of the set Ck1 , we see that
Ck1 z ∈ Ck : sup φz, uk,i ≤ φz, xk i≥1
∞
z ∈ Ck : φz, uk,i ≤ φz, xk .
3.3
i1
Hence Ck1 is also closed and convex.
It is obvious that F ⊂ C1 C. Now, suppose that F ⊂ Ck for some k ∈ N, and let
M
p ∈ F : ∞
i1 FTi ∩ j1 EPfj . Then
fM fM−1
f
φ p, uk,i φ p, TrM,n
TrM−1,n · · · Tr1,n1 yk,i
fM−1 fM−2
f
≤ φ p, TrM−1,n
TrM−2,n · · · Tr1,n1 yk,i
..
.
f
≤ φ p, Tr1,n1 yk,i
≤ φ p, yk,i
φ p, J −1 αk Jxk 1 − αk JTi xk 2
p − 2 p, αk Jxk 1 − αk JTi xk
αk Jxk 1 − αk JTi xk 2
2
≤ p − 2αk p, Jxk − 21 − αk p, JTi xk αk xk 2 1 − αk Ti xk 2
αk φ p, xk 1 − αk φ p, Ti xk
≤ φ p, xk .
Hence F ⊂ Ck1 . By induction, we can conclude that F ⊂ Cn for all n ≥ 1.
3.4
Abstract and Applied Analysis
7
Step 2. limn → ∞ φxn , x0 exists.
From xn ΠCn x0 and xn1 ΠCn1 x0 ∈ Cn1 ⊂ Cn , we have
φxn , x0 ≤ φxn1 , x0 ,
n ≥ 1.
3.5
From Lemma 2.3 we get that
φxn , x0 φΠCn x0 , x0 ≤ φ p, x0 − φ p, xn ≤ φ p, x0 .
3.6
Combining 3.5 and 3.6, we get that limn → ∞ φxn , x0 exists.
Step 3. {xn } is a Cauchy sequence in C.
Since xm ΠCm x0 ∈ Cm ⊂ Cn for m > n, we obtain from Lemma 2.3 that
φxm , xn φxm , ΠCn x0 ≤ φxm , x0 − φΠCn x0 , x0 φxm , x0 − φxn , x0 .
3.7
We see that φxm , xn → 0 as m, n → ∞, which implies with Lemma 2.1 that xm − xn → 0
as m, n → ∞. Therefore {xn } is a Cauchy sequence. By the completeness of the space E and
the closedness of the set C, we can assume that xn → q ∈ C as n → ∞. Moreover, we get that
lim φxn1 , xn 0.
3.8
n→∞
Since xn1 ΠCn1 x0 ∈ Cn1 , we have for all i ≥ 1 that
φxn1 , un,i ≤ φxn1 , xn −→ 0.
3.9
Applying Lemma 2.1 to 3.8 and 3.9, we derive
lim un,i − xn 0,
n→∞
∀i ≥ 1.
3.10
This shows that un,i → q as n → ∞ for all i ≥ 1. Since J is uniformly norm-to-norm
continuous on bounded subsets of E, we obtain that
lim Jun,i − Jxn 0,
n→∞
Step 4. q ∈
∞
i1
FTi .
j
fj
fj−1
∀i ≥ 1.
3.11
f
Denote Θn Trj,n Trj−1,n · · · Tr1,n1 for any j ∈ {1, 2, . . . , M} and Θ0n I for all n ≥ 1. We note
that un,i ΘM
n yn,i for all i ≥ 1. From 3.4 we observe that
φ p, ΘM−1
yn,i ≤ φ p, ΘM−2
yn,i ≤ · · · ≤ φ p, yn,i ≤ φ p, xn ,
n
n
∀i ≥ 1.
3.12
8
Abstract and Applied Analysis
f
M
for all n ≥ 1, it follows from 3.12 and Lemma 2.7 that
Since p ∈ EPfM FTrM,n
φ un,i , ΘM−1
yn,i ≤ φ p, ΘM−1
yn,i − φ p, un,i
n
n
≤ φ p, xn − φ p, un,i .
3.13
From 3.10 and 3.11, we get that limn → ∞ φun,i , ΘM−1
yn,i 0 for all i ≥ 1. From Lemma 2.1,
n
we have
lim un,i − ΘM−1
yn,i 0,
n
n→∞
∀i ≥ 1.
3.14
∀i ≥ 1,
3.15
From 3.10 and 3.14, we have
lim xn − ΘM−1
y
n,i 0,
n
n→∞
and hence,
lim Jxn − JΘM−1
yn,i 0,
n
n→∞
∀i ≥ 1.
3.16
f
M−1
Again, since p ∈ EPfM−1 FTrM−1,n
for all n ≥ 1, it follows from 3.12 and Lemma 2.7 that
φ ΘM−1
yn,i , ΘM−2
yn,i ≤ φ p, ΘM−2
yn,i − φ p, ΘM−1
yn,i
n
n
n
n
≤ φ p, xn − φ p, ΘM−1
yn,i .
n
3.17
From 3.15 and 3.16, we also have
M−2
lim ΘM−1
y
−
Θ
y
0,
n,i
n,i
n
n
n→∞
∀i ≥ 1.
3.18
Hence, from 3.15 and 3.18, we get
lim xn − ΘM−2
yn,i 0,
n
n→∞
lim Jxn − JΘM−2
yn,i 0,
n
n→∞
∀i ≥ 1,
∀i ≥ 1.
3.19
3.20
In a similar way, we can verify that
1
M−3
lim ΘM−2
·
·
·
lim
y
−
Θ
y
y
−
y
Θ
n,i
n,i
n,i 0
n
n
n n,i
n→∞
n→∞
3.21
Abstract and Applied Analysis
9
for all i ≥ 1,
lim xn − ΘM−3
yn,i · · · lim xn − yn,i 0
n
3.22
lim Jxn − JΘM−3
y
· · · lim Jxn − Jyn,i 0
n,i
n
3.23
n→∞
n→∞
for all i ≥ 1,
n→∞
n→∞
for all i ≥ 1. Hence, we can conclude that
j−1
j
lim Θn yn,i − Θn yn,i 0
3.24
n→∞
for each j 1, 2, . . . , M and i ≥ 1. Observe that
Jyn,i − Jxn αn Jxn 1 − αn JTi xn − Jxn 3.25
1 − αn JTi xn − Jxn ,
then we obtain from B1 and 3.23 that
lim JTi xn − Jxn 0,
n→∞
∀i ≥ 1.
3.26
Since J −1 is also uniformly norm-to-norm continuous on bounded subsets, we get that
lim Ti xn − xn 0,
n→∞
∀i ≥ 1.
Since Ti is closed for all i ≥ 1 and xn → q, we conclude that q ∈
Step 5. q ∈
M
j1
EPfj .
j
3.27
∞
i1
FTi .
j−1
From 3.24 and B2, we have that JΘn yn,i − JΘn yn,i /rj,n → 0 as n → ∞. Then,
for each j 1, 2, . . . , M, we obtain that
j
1 j
j
j−1
y − Θn yn,i , JΘn yn,i − JΘn yn,i ≥ 0,
fj Θn yn,i , y rj,n
∀y ∈ C.
3.28
From A2 we have that
j
j−1
y
−
JΘ
y
JΘ
n,i n n,i
n
1 j
j
j
j−1
y − Θn yn,i , JΘn yn,i − JΘn yn,i
≥
y − Θn yn,i rj,n
rj,n
j
j
≥ −fj Θn yn,i , y ≥ fj y, Θn yn,i , ∀y ∈ C.
3.29
10
Abstract and Applied Analysis
j
From A4 and the fact that Θn yn,i → q for i ≥ 1, we get fj y, q ≤ 0 for all y ∈ C. For each
0 < t < 1 and y ∈ C, denote yt ty 1 − tq. Then yt ∈ C, which implies that fj yt , q ≤ 0.
From A1 and A4, we obtain that 0 fj yt , yt ≤ tfj yt , y 1 − tfj yt , q ≤ tfj yt , y.
Thus, fj yt , y ≥ 0. From A3, we have fj q, y ≥ 0 for all y ∈ C and j 1, 2, . . . , M. Hence
q∈ M
j1 EPfj .
Step 6. q ΠF x0 .
From xn ΠCn x0 , we have
Jx0 − Jxn , xn − z ≥ 0 ∀z ∈ Cn .
3.30
Jx0 − Jxn , xn − p ≥ 0 ∀p ∈ F.
3.31
Since F ⊂ Cn , we also have
Letting n → ∞ in 3.31, we obtain that
Jx0 − Jq, q − p ≥ 0 ∀p ∈ F.
3.32
From Lemma 2.2 we conclude that q ΠF x0 . This completes the proof.
Remark 3.2. Theorem 3.1 improves and extends Theorem 3.1 of Takahashi and Zembayashi in
3 in the following senses:
i from the case of an equilibrium problem to a finite family of equilibrium problems;
ii from a single relatively nonexpansive mapping to an infinitely countable family of
relatively quasi-nonexpansive mappings;
iii if M 1 and Ti T for all i ≥ 1, then our restriction on {αn } is weaker than that of
Theorem 3.1 of 3.
Remark 3.3. The iteration 3.1 is a modification of 1.4 in the following ways.
fj
i We use the composition of mappings {Trj,n }M
j1 in the second step.
ii We construct the set Cn1 by using the concept of supremum concerning an
infinitely countable family of closed and relatively quasi-nonexpansive mappings
{Ti }∞
i1 . If M 1 and Ti T for all i ≥ 1, then the iteration 3.1 reduces to that of
1.4.
If we take αn 0 for all n ∈ N in Theorem 3.1, then we have the following corollary.
Corollary 3.4. Let E be a uniformly convex and uniformly smooth Banach space, and let C be a
nonempty, closed, and convex subset of E. Let {fj }M
j1 be bifunctions from C × C to R which satisfies
Abstract and Applied Analysis
11
family of closed and relatively quasiconditions (A1)–(A4), and let {Ti }∞
i1 be an infinitely countable M
∅. For
nonexpansive mappings from C into itself. Assume that F : ∞
i1 FTi ∩ j1 EPfj /
any initial point x0 ∈ E with x1 ΠC1 x0 and C1 C, define the sequence {xn } as follows:
yn,i Ti xn ,
f
f
f
M
M−1
un,i TrM,n
TrM−1,n
· · · Tr1,n1 yn,i ,
Cn1 z ∈ Cn : sup φz, un,i ≤ φz, xn ,
3.33
i≥1
xn1 ΠCn1 x0 ,
n ≥ 1.
If lim infn → ∞ rj,n > 0 for each j 1, 2, . . . , M, then {xn } converges strongly to ΠF x0 .
4. Applications
In this section, we give several applications of Theorem 3.1 in the framework of Banach spaces
and Hilbert spaces.
Let A : C → E∗ be a nonlinear mapping. The classical variational inequality problem
is to find that x ∈ C such that
Ax,
y − x ≥ 0 ∀y ∈ C.
4.1
The solutions set of 4.1 is denoted by V IC, A. For each r > 0 and x ∈ E, define the mapping
TrA : E → C as follows:
TrA x 1
z ∈ C : Az, y − z y − z, Jz − Jx ≥ 0, ∀y ∈ C .
r
4.2
Theorem 4.1. Let E be a uniformly convex and uniformly smooth Banach space, and let C be a
nonempty, closed, and convex subset of E. Let {Aj }M
j1 be continuous and monotone operators from
be
an
infinitely
countable
family
of closed and relatively quasi-nonexpansive
C to E∗ , and let {Ti }∞
i1
FT
∩
M
mappings from C into itself such that F : ∞
/ ∅. For any initial point
i
i1
j1 V IC, Aj x0 ∈ E with x1 ΠC1 x0 and C1 C, define the sequence {xn } as follows:
yn,i J −1 αn Jxn 1 − αn JTi xn ,
M−1
M
un,i TrAM,n
TrAM−1,n
· · · TrA1,n1 yn,i ,
Cn1 z ∈ Cn : sup φz, un,i ≤ φz, xn ,
i≥1
xn1 ΠCn1 x0 ,
n ≥ 1.
4.3
12
Abstract and Applied Analysis
Assume that {αn } and {rj,n } for j 1, 2, . . . , M are sequences which satisfy conditions (B1) and (B2)
of Theorem 3.1.
Then the sequence {xn } converges strongly to ΠF x0 .
Proof. Define fj x, y Aj x, y − x for all x, y ∈ C and j 1, 2, . . . , M. First, we see that
fj
Aj
FTrj EPfj V IC, Aj FTrj for each j 1, 2, . . . , M.
Next, we show that {fj }M
j1 satisfy conditions A1–A4.
A1 Consider fj x, x Aj x, x − x 0 for all x ∈ C and j 1, 2, . . . , M.
A2 For each x, y ∈ C and j 1, 2, . . . , M, we observe that
fj x, y fj y, x Aj x, y − x Aj y, x − y
Aj x − Aj y, y − x .
4.4
By the monotonicity of Aj , we obtain that fj is monotone. Thus {fj }M
j1 satisfy condition A2.
A3 For each x, y, z ∈ C and j 1, 2, . . . , M, we have by the continuity of Aj that
lim supfj tz 1 − tx, y lim sup Aj tz 1 − tx, y − tz 1 − tx
t↓0
t↓0
Aj x, y − x
fj x, y .
4.5
This shows that {fj }M
j1 satisfy condition A3.
A4 Let u, v ∈ C and s ∈ 0, 1. Then, for each x ∈ C and j 1, 2, . . . , M, we have
fj x, su 1 − sv Aj x, su 1 − sv − x
s Aj x, u − x 1 − s Aj x, v − x
4.6
sfj x, u 1 − sfj x, v.
Thus fj is convex in the second variable. Let un ∈ C and limn → ∞ un u. Then
fj x, u Aj x, u − x
lim Aj x, un − x
n→∞
4.7
lim fj x, un .
n→∞
This shows that fj is lower semicontinuous in the second variable. Hence {fj }M
j1 satisfy
condition A4. From Theorem 3.1 we obtain the desired result.
If we take αn 0 for all n ∈ N in Theorem 4.1, then we have the following corollary.
Abstract and Applied Analysis
13
Corollary 4.2. Let E be a uniformly convex and uniformly smooth Banach space, and let C be a
nonempty, closed, and convex subset of E. Let {Aj }M
j1 be continuous and monotone operators from
be
an
infinitely
countable
family
of closed and relatively quasi-nonexpansive
C to E∗ , and let {Ti }∞
i1
FT
∩
M
mappings from C into itself such that F : ∞
/ ∅. For any initial point
i
i1
j1 V IC, Aj x0 ∈ E with x1 ΠC1 x0 and C1 C, define the sequence {xn } as follows:
yn,i Ti xn ,
M−1
M
un,i TrAM,n
TrAM−1,n
· · · TrA1,n1 yn,i ,
Cn1 z ∈ Cn : sup φz, un,i ≤ φz, xn ,
4.8
i≥1
xn1 ΠCn1 x0 ,
n ≥ 1.
If lim infn → ∞ rj,n > 0 for each j 1, 2, . . . , M, then {xn } converges strongly to ΠF x0 .
Let ϕ : C → R be a real-valued function. The convex minimization problem is to find that
x ∈ C such that
ϕx
≤ϕ y
∀y ∈ C.
4.9
The solutions set of 4.9 is denoted by CMP ϕ. For each r > 0 and x ∈ E, define the mapping
ϕ
Tr : E → C as follows:
ϕ
Tr x 1
z ∈ C : ϕ y y − z, Jz − Jx ≥ ϕz, ∀y ∈ C .
r
4.10
Theorem 4.3. Let E be a uniformly convex and uniformly smooth Banach space, and let C be
a nonempty, closed, and convex subset of E. Let {ϕj }M
j1 be lower semicontinuous and convex
be
an
infinitely
countable
family of closed and relatively quasifunctions from C to R, and let {Ti }∞
i1
M
∅. For
nonexpansive mappings from C into itself such that F : ∞
i1 FTi ∩ j1 CMP ϕj /
any initial point x0 ∈ E with x1 ΠC1 x0 and C1 C, define the sequence {xn } as follows:
yn,i J −1 αn Jxn 1 − αn JTi xn ,
ϕ
ϕ
ϕ
M
M−1
un,i TrM,n
TrM−1,n
· · · Tr1,n1 yn,i ,
Cn1 z ∈ Cn : sup φz, un,i ≤ φz, xn ,
i≥1
xn1 ΠCn1 x0 ,
n ≥ 1.
4.11
14
Abstract and Applied Analysis
Assume that {αn } and {rj,n } for j 1, 2, . . . , M are sequences which satisfy conditions (B1) and (B2)
of Theorem 3.1.
Then the sequence {xn } converges strongly to ΠF x0 .
fj
Proof. Define fj x, y ϕj y − ϕj x for all x, y ∈ C and j 1, 2, . . . , M. Then FTrj ϕj
EPfj CMP ϕj FTrj for each j 1, 2, . . . , M, and therefore {fj }M
j1 satisfy conditions
A1 and A2.
Next, we show that {fj }M
j1 satisfy conditions A3 and A4. For each x, y, z ∈ C, we
have by the lower semicontinuity of ϕj that
lim supfj tz 1 − tx, y lim sup ϕj y − ϕj tz 1 − tx
t↓0
t↓0
≤ ϕj y − ϕj x
fj x, y .
4.12
This implies that {fj }M
j1 satisfy condition A3.
Let u, v ∈ C and s ∈ 0, 1. For each x ∈ C, we have by the convexity of ϕj that
fj x, su 1 − sv ϕj su 1 − sv − ϕj x
≤ sϕj u 1 − sϕj v − ϕj x
s ϕj u − ϕj x 1 − s ϕj v − ϕj x
4.13
sfj x, u 1 − sfj x, v.
On the other hand, let un ∈ C and limn → ∞ un u. By the lower semicontinuity of ϕj we have
fj x, u ϕj u − ϕj x
≤ lim inf ϕj un − ϕj x
n→∞
4.14
lim inffj x, un .
n→∞
Thus {fj }M
j1 satisfy condition A4. From Theorem 3.1 we also obtain the desired result.
If we take αn 0 for all n ∈ N in Theorem 4.3, then we have the following corollary.
Corollary 4.4. Let E be a uniformly convex and uniformly smooth Banach space, and let C be
a nonempty, closed, and convex subset of E. Let {ϕj }M
j1 be lower semicontinuous and convex
Abstract and Applied Analysis
15
family of closed and relatively quasifunctions from C to R, and let {Ti }∞
i1 be an infinitely countable
M
∅. For
nonexpansive mappings from C into itself such that F : ∞
i1 FTi ∩ j1 CMP ϕj /
any initial point x0 ∈ E with x1 ΠC1 x0 and C1 C, define the sequence {xn } as follows:
yn,i Ti xn ,
ϕ
ϕ
ϕ
M
M−1
un,i TrM,n
TrM−1,n
· · · Tr1,n1 yn,i ,
Cn1 z ∈ Cn : sup φz, un,i ≤ φz, xn ,
4.15
i≥1
xn1 ΠCn1 x0 ,
n ≥ 1.
If lim infn → ∞ rj,n > 0 for each j 1, 2, . . . , M, then {xn } converges strongly to ΠF x0 .
As a direct consequence of Theorem 3.1, we obtain the following application in a
Hilbert space.
Theorem 4.5. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let
∞
{fj }M
j1 be bifunctions from C × C to R which satisfies conditions (A1)–(A4), and let {Ti }i1 be an
infinitely countable family of closed and quasi-nonexpansive mappings from C into itself such that
M
∅. For any initial point x0 ∈ H with x1 PC1 x0 and C1 C,
F : ∞
i1 FTi ∩ j1 EPfj /
define the sequence {xn } as follows:
yn,i αn xn 1 − αn Ti xn ,
f
f
f
M
M−1
un,i TrM,n
TrM−1,n
· · · Tr1,n1 yn,i ,
Cn1 z ∈ Cn : supz − un,i ≤ z − xn ,
4.16
i≥1
xn1 PCn1 x0 ,
n ≥ 1,
where P is the metric projection. Assume that {αn } and {rj,n } for j 1, 2, . . . , M are sequences which
satisfy conditions (B1) and (B2) of Theorem 3.1.
Then the sequence {xn } converges strongly to PF x0 .
Proof. Taking E H in Theorem 3.1, the result is obtained immediately.
Remark 4.6. Theorem 4.5 improves and extends the main results of 20–22 in the following
senses:
i from the case of an equilibrium problem to a finite family of equilibrium problems;
ii from the class of nonexpansive mappings to the class of an infinitely countable
family of quasi-nonexpansive mappings.
16
Abstract and Applied Analysis
Acknowledgments
The authors would like to thank Professor Simeon Reich and the referee for valuable
suggestions on improving the manuscript and the Thailand Research Fund for financial
support. The first author was supported by the Royal Golden Jubilee Grant PHD/0261/2551
and by the Graduate School, Chiang Mai University, Thailand.
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