Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 941090, 20 pages
doi:10.1155/2011/941090
Research Article
Convergence Analysis for a System of
Generalized Equilibrium Problems and a Countable
Family of Strict Pseudocontractions
Prasit Cholamjiak1 and Suthep Suantai1, 2
1
2
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
Correspondence should be addressed to Suthep Suantai, scmti005@chiangmai.ac.th
Received 18 October 2010; Accepted 27 December 2010
Academic Editor: Jen Chih Yao
Copyright q 2011 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 iterative algorithm for a system of generalized equilibrium problems and
a countable family of strict pseudocontractions in Hilbert spaces. We then prove that the sequence
generated by the proposed algorithm converges strongly to a common element in the solutions set
of a system of generalized equilibrium problems and the common fixed points set of an infinitely
countable family of strict pseudocontractions.
1. Introduction
Let H be a real Hilbert space with the inner product ·, · and inducted norm · . Let C
be a nonempty, closed, and convex subset of H. Let {fk }k∈Λ : C × C → Ê be a family of
bifunctions, and let {Ak }k∈Λ : C → H be a family of nonlinear mappings, where Λ is an
arbitrary index set. The system of generalized equilibrium problems is to find x ∈ C such that
y Ak x,
y − x ≥ 0,
fk x,
∀y ∈ C, k ∈ Λ.
1.1
If Λ is a singleton, then 1.1 reduces to find x ∈ C such that
f x,
y Ax,
y − x ≥ 0,
∀y ∈ C.
1.2
The solutions set of 1.2 is denoted by GEPf, A. If f ≡ 0, then the solutions set of 1.2
is denoted by VIC, A, and if A ≡ 0, then the solutions set of 1.2 is denoted by EPf.
2
Fixed Point Theory and Applications
The problem 1.2 is very general in the sense that it includes, as special cases, optimization
problems, variational inequalities, minimax problems, and the Nash equilibrium problem
in noncooperative games; see also 1, 2. Some methods have been constructed to solve the
system of equilibrium problems see, e.g., 3–7. Recall that a mapping A : C → H is
said to be
1 monotone if
Ax − Ay, x − y ≥ 0,
∀x, y ∈ C,
1.3
2 α-inverse-strongly monotone if there exists a constant α > 0 such that
2
Ax − Ay, x − y ≥ αAx − Ay ,
∀x, y ∈ C.
1.4
It is easy to see that if A is α-inverse-strongly monotone, then A is monotone and
1/α-Lipschitz.
For solving the equilibrium problem, let us assume that 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 each x, y, z ∈ C, limt → 0 ftz 1 − tx, y ≤ fx, y,
A4 for each x ∈ C, y → fx, y is convex and lower semicontinuous.
Throughout this paper, we denote the fixed points set of a nonlinear mapping
T : C → C by FT {x ∈ C : Tx x}. Recall that T is said to be a κ-strict pseudocontraction if
there exists a constant 0 ≤ κ < 1 such that
Tx − Ty2 ≤ x − y2 κI − Tx − I − Ty2 .
1.5
It is well known that 1.5 is equivalent to
2 1 − κ I − Tx − I − Ty2 .
Tx − Ty, x − y ≤ x − y −
2
1.6
It is worth mentioning that the class of strict pseudocontractions includes properly the
class of nonexpansive mappings. It is also known that every κ-strict pseudocontraction is
1 κ/1 − κ-Lipschitz; see 8.
In 1953, Mann 9 introduced the iteration as follows: a sequence {xn } defined by
x0 ∈ C and
xn1 αn xn 1 − αn Sxn ,
n ≥ 0,
1.7
where {αn }∞
n0 ⊂ 0, 1. If S is a nonexpansive mapping with a fixed point and the control
∞
sequence {αn }∞
n0 is chosen so that
n0 αn 1 − αn ∞, then the sequence {xn } defined
Fixed Point Theory and Applications
3
by 1.7 converges weakly to a fixed point of S this is also valid in a uniformly convex
Banach space with the Fréchet differentiable norm 10.
In 1967, Browder and Petryshyn 11 introduced the class of strict pseudocontractions
and proved existence and weak convergence theorems in a real Hilbert setting by using Mann
iterative algorithm 1.7 with a constant sequence αn α for all n ≥ 0. Recently, Marino
and Xu 8 and Zhou 12 extended the results of Browder and Petryshyn 11 to Mann’s
iteration process 1.7. Since 1967, the construction of fixed points for pseudocontractions via
the iterative process has been extensively investigated by many authors see, e.g., 13–22.
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let
S : C → C be a nonexpansive mapping, f : C × C → Ê a bifunction, and let A : C → H be
an inverse-strongly monotone mapping.
In 2008, Moudafi 23 introduced an iterative method for approximating a common
element of the fixed points set of a nonexpansive mapping S and the solutions set of a
generalized equilibrium problem GEPf, A as follows: a sequence {xn } defined by x0 ∈ C
and
1
f yn , y Axn , y − yn y − yn , yn − xn ≥ 0,
rn
xn1 αn xn 1 − αn Syn ,
∀y ∈ C,
1.8
n ≥ 1,
∞
where {αn }∞
n0 ⊂ 0, 1 and {rn }n0 ⊂ 0, ∞. He proved that the sequence {xn } generated by
1.8 converges weakly to an element in GEPf, A ∩ FS under suitable conditions.
Due to the weak convergence, recently, S. Takahashi and W. Takahashi 24 introduced
another modification iterative method of 1.8 for finding a common element of the fixed
points set of a nonexpansive mapping and the solutions set of a generalized equilibrium
problem in the framework of a real Hilbert space. To be more precise, they proved the
following theorem.
Theorem 1.1 see 24. Let C be a closed convex subset of a real Hilbert space H, and let
f : C × C → Ê be a bifunction satisfying (A1)–(A4). Let A be an α-inverse-strongly monotone
mapping of C into H, and let S be a nonexpansive mapping of C into itself such that FS ∩
GEPf, A /
∅. Let u ∈ C and x1 ∈ C, and let {yn } ⊂ C and {xn } ⊂ C be sequences generated by
1
f yn , y Axn , y − yn y − yn , yn − xn ≥ 0, ∀y ∈ C,
rn
xn1 βn xn 1 − βn S αn u 1 − αn yn , n ≥ 1,
∞
∞
where {αn }∞
n1 ⊂ 0, 1, {βn }n1 ⊂ 0, 1 and {rn }n1 ⊂ 0, 2α satisfy
i limn → ∞ αn 0 and ∞
n1 αn ∞,
ii 0 < c ≤ βn ≤ d < 1,
iii 0 < a ≤ rn ≤ b < 2α,
iv limn → ∞ rn − rn1 0.
Then, {xn } converges strongly to z PFS ∩ GEPf,A u.
1.9
4
Fixed Point Theory and Applications
Recently, Yao et al. 25 introduced a new modified Mann iterative algorithm which is
different from those in the literature for a nonexpansive mapping in a real Hilbert space. To
be more precise, they proved the following theorem.
Theorem 1.2 see 25. Let C be a nonempty, closed, and convex subset of a real Hilbert space H.
∞
Let S : C → C be a nonexpansive mapping such that FS /
∅. Let {αn }∞
n0 , and let {βn }n0 be
two real sequences in 0, 1. For given x0 ∈ C arbitrarily, let the sequence {xn }, n ≥ 0, be generated
iteratively by
yn PC 1 − αn xn ,
xn1 1 − βn xn βn Syn .
1.10
Suppose that the following conditions are satisfied:
i limn → ∞ αn 0 and ∞
n0 αn ∞,
ii 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1,
then, the sequence {xn } generated by 1.10 strongly converges to a fixed point of S.
We know the following crucial lemmas concerning the equilibrium problem in Hilbert
spaces.
Lemma 1.3 see 1. Let C be a nonempty, closed, and convex subset of a real Hilbert space H, let f
be a bifunction from C × C to Ê satisfying (A1)–(A4). Let r > 0 and x ∈ H. Then, there exists z ∈ C
such that
1
f z, y y − z, z − x ≥ 0,
r
∀y ∈ C.
1.11
Lemma 1.4 see 26. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let
f be a bifunction from C × C to Ê satisfying (A1)–(A4). For x ∈ H and r > 0, define the mapping
f
Tr : H → 2C as follows:
f
Tr x
1
z ∈ C : f z, y y − z, z − x ≥ 0, ∀y ∈ C .
r
1.12
Then, the following statements hold:
f
1 Tr is single-valued,
f
2 Tr is firmly nonexpansive, that is, for any x, y ∈ H,
f
f 2
f
f
Tr x − Tr y ≤ Tr x − Tr y, x − y ,
f
3 FTr EPf,
4 EPf is closed and convex.
1.13
Fixed Point Theory and Applications
5
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let rk > 0
M
for each k ∈ {1, 2, . . . , M}. Let {fk }M
k1 : C × C → Ê be a family of bifunctions, let {Ak }k1 :
C → H be a family of αk -inverse-strongly monotone mappings, and let {Tn }∞
n1 : C → C
be a countable family of κ-strict pseudocontractions. For each k ∈ {1, 2, . . . , M}, denote the
f ,A
f ,A
f
f
mapping Trkk k : C → C by Trkk k : Trkk I − rk Ak , where Trkk : H → C is the mapping
defined as in Lemma 1.4.
Motivated and inspired by Marino and Xu 8, Moudafi 23, S. Takahashi and W.
Takahashi 24, and Yao et al. 25, we consider the following iteration: x1 ∈ C and
yn PC 1 − αn xn ,
f ,A
xn1
f
,A
f ,A
f ,A
M−1
M−1
un TrMM M TrM−1
· · · Tr22 2 Tr11 1 yn ,
βn xn 1 − βn γ un 1 − γ Tn un , n ≥ 1,
1.14
∞
where {αn }∞
n1 ⊂ 0, 1, {βn }n1 ⊂ 0, 1 and γ ∈ 0, 1.
In this paper, we first prove a path convergence result for a nonexpansive mapping
and a system of generalized equilibrium problems. Then, we prove a strong convergence
theorem of the iteration process 1.14 for a system of generalized equilibrium problems and
a countable family of strict pseudocontractions in a real Hilbert space. Our results extend the
main results obtained by Yao et al. 25 in several aspects.
2. Preliminaries
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. For each x ∈ H,
there exists a unique nearest point in C, denoted by PC x, such that x − PC x miny∈C x − y.
PC is called the metric projection of H onto C. It is also known that for x ∈ H and z ∈ C,
z PC x is equivalent to x − z, y − z ≤ 0 for all y ∈ C. Furthermore,
y − PC x2 x − PC x2 ≤ x − y2 ,
2.1
for all x ∈ H, y ∈ C. In a real Hilbert space, we also know that
λx 1 − λy2 λx2 1 − λy2 − λ1 − λx − y2 ,
2.2
for all x, y ∈ H and λ ∈ 0, 1.
In the sequel, we need the following lemmas.
Lemma 2.1 see 27, 28. Let E be a real uniformly convex Banach space, and let C be a nonempty,
closed, and convex subset of E, and let S : C → C be a nonexpansive mapping such that FS / ∅,
then I − S is demiclosed at zero.
Lemma 2.2 see 29. Let {xn } and {zn } be two sequences in a Banach space E such that
xn1 βn xn 1 − βn zn ,
n ≥ 1,
2.3
6
Fixed Point Theory and Applications
where {βn }∞
n1 satisfies conditions: 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1. If lim supn → ∞ zn1 −
zn − xn1 − xn ≤ 0, then xn − zn → 0 as n → ∞.
Lemma 2.3 see 30. Assume that {an }∞
n1 is a sequence of nonnegative real numbers such that
an1 ≤ 1 − γn an γn δn ,
n ≥ 1,
2.4
∞
where {γn }∞
n1 is a sequence in 0, 1 and {δn }n1 is a sequence in Ê such that
∞
a n1 γn ∞; b lim supn → ∞ δn ≤ 0 or ∞
n1 |γn δn | < ∞.
Then, limn → ∞ an 0.
Lemma 2.4 see 31. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let
the mapping A : C → H be α-inverse-strongly monotone, and let r > 0 be a constant. Then, we have
I − rAx − I − rAy2 ≤ x − y2 rr − 2αAx − Ay2 ,
2.5
for all x, y ∈ C. In particular, if 0 ≤ r ≤ 2α, then I − rA is nonexpansive.
To deal with a family of mappings, the following conditions are introduced: let C
be a subset of a real Hilbert space H, and let {Tn }∞
n1 be a family of mappings of C such
∅. Then, {Tn } is said to satisfy the AKTT-condition 32 if for each bounded
that ∞
n1 FTn /
subset B of C,
∞
sup{Tn1 z − Tn z : z ∈ B} < ∞.
2.6
n1
Lemma 2.5 see 32. Let C be a nonempty and closed subset of a Hilbert space H, and let {Tn } be
a family of mappings of C into itself which satisfies the AKTT-condition. Then, for each x ∈ C, {Tn x}
converges strongly to a point in C. Moreover, let the mapping T be defined by
Tx lim Tn x,
n→∞
∀x ∈ C.
2.7
Then, for each bounded subset B of C,
lim sup{Tz − Tn z : z ∈ B} 0.
n→∞
2.8
The following results can be found in 33, 34.
Lemma 2.6 see 33, 34. Let C be a closed, and convex subset of a Hilbert space H. Suppose that
∞
{Tn }∞
∅ and
n1 is a family of κ-strictly pseudocontractive mappings from C into H with
n1 FTn /
∞
∞
{μn }n1 is a real sequence in 0, 1 such that n1 μn 1. Then, the following conclusions hold:
1 G : ∞
n1 μn Tn : C → H is a κ-strictly pseudocontractive mapping,
2 FG ∞
n1 FTn .
Fixed Point Theory and Applications
7
Lemma 2.7 see 34. Let C be a closed and convex subset of a Hilbert space H. Suppose that {Si }∞
i1
is a countable family of κ-strictly pseudocontractive mappings of C into itself with ∞
∅.
i1 FSi /
For each n ∈ Æ , define Tn : C → C by
n
Tn x μin Si x,
x ∈ C,
2.9
i1
where {μin } is a family of nonnegative numbers satisfying
i ni1 μin 1 for all n ∈ Æ ,
ii μi : limn → ∞ μin > 0 for all i ∈ Æ ,
n
i
i
iii ∞
n1
i1 |μn1 − μn | < ∞.
Then,
1 Each Tn is a κ-strictly pseudocontractive mapping.
2 {Tn } satisfies AKTT-condition.
3 If T : C → C is defined by
Tx ∞
μi Si x,
x ∈ C,
2.10
i1
then Tx limn → ∞ Tn x and FT ∞
n1
FTn ∞
i1
FSi .
In the sequel, we will write {Tn }, T satisfies the AKTT-condition if {Tn } satisfies the
AKTT-condition and T is defined by Lemma 2.5 with FT ∞
n1 FTn .
3. Path Convergence Results
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let S : C → C be
M
a nonexpansive mapping. Let {fk }M
k1 : C × C → Ê be a family of bifunctions, let {Ak }k1 :
C → H be a family of αk -inverse-strongly monotone mappings, and let rk ∈ 0, 2αk . For each
f ,A
k ∈ {1, 2, . . . , M}, we denote the mapping Trkk k : C → C by
f ,Ak
Trkk
f
3.1
: Trkk I − rk Ak ,
f
where Trkk is the mapping defined as in Lemma 1.4. For each t ∈ 0, 1, we define the mapping
St : C → C as follows:
f ,AM
St x STrMM
f
M−1
TrM−1
,AM−1
f ,A1
· · · Tr11
f
PC 1 − tx,
∀x ∈ C.
3.2
By Lemmas 1.42 and 2.4, we know that Trkk and I − rk Ak are nonexpansive for each
f ,A
k ∈ {1, 2, . . . , M}. So, the mapping Trkk k is also nonexpansive for each k ∈ {1, 2, . . . , M}.
8
Fixed Point Theory and Applications
Moreover, we can check easily that St is a contraction. Then, the Banach contraction principle
ensures that there exists a unique fixed point xt of St in C, that is,
f ,AM
xt STrMM
f
M−1
TrM−1
,AM−1
f ,A1
· · · Tr11
PC 1 − txt ,
t ∈ 0, 1.
3.3
Theorem 3.1. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let
S : C → C be a nonexpansive mapping. Let {fk }M
k1 : C × C → Ê be a family of bifunctions,
M
let {Ak }k1 : C → H be a family of αk -inverse-strongly monotone mappings, and let rk ∈
f ,A
0, 2αk . For each k ∈ {1, 2, . . . , M}, let the mapping Trkk k be defined by 3.1. Assume that
∞
F : M
/ ∅. For each t ∈ 0, 1, let the net {xt } be generated by
k1 GEPfk , Ak ∩ n1 FTn 3.3. Then, as t → 0, the net {xt } converges strongly to an element in F.
Proof. First, we show that {xt } is bounded. For each t ∈ 0, 1, let yt PC 1 − txt and
f ,AM
ut TrMM
f
M−1
TrM−1
,AM−1
f ,A1
· · · Tr11
yt . From 3.3, we have for each p ∈ F that
xt − p Sut − Sp ≤ ut − p ≤ yt − p ≤ 1 − txt − p tp.
3.4
It follows that
xt − p ≤ p.
3.5
Hence, {xt } is bounded and so are {yt } and {ut }. Observe that
yt − xt ≤ txt −→ 0,
3.6
as t → 0 since {xt } is bounded.
f ,A fk−1 ,Ak−1
f ,A
Next, we show that ut − xt → 0 as t → 0. Denote Θk Trkk k Trk−1
· · · Tr11 1 for
any k ∈ {1, 2, . . . , M} and Θ0 I. We note that ut ΘM yt for each t ∈ 0, 1. From Lemma 2.4,
we have for each k ∈ {1, 2, . . . , M} and p ∈ F that
2 2
f ,A
k
f ,A
Θ yt − p Trkk k Θk−1 yt − Trkk k Θk−1 p
2
f
f
Trkk Θk−1 yt − rk Ak Θk−1 yt − Trkk Θk−1 p − rk Ak Θk−1 p 2
≤ Θk−1 yt − rk Ak Θk−1 yt − Θk−1 p − rk Ak Θk−1 p 2
2
≤ Θk−1 yt − p rk rk − 2αk Ak Θk−1 yt − Ak p .
3.7
Fixed Point Theory and Applications
9
It follows that
2
ut − p2 ΘM yt − p
M
2
2 ≤ yt − p ri ri − 2αi Ai Θi−1 yt − Ai p
i1
M
2
2 PC 1 − txt − p ri ri − 2αi Ai Θi−1 yt − Ai p
i1
3.8
M
2
2 ≤ xt − p txt ri ri − 2αi Ai Θi−1 yt − Ai p
i1
M
2
2
≤ xt − p tM1 ri ri − 2αi Ai Θi−1 yt − Ai p ,
i1
where M1 sup0<t<1 {2xt − pxt txt 2 }. So, we have
xt − p2 ≤ ut − p2
M
2
2
≤ xt − p tM1 ri ri − 2αi Ai Θi−1 yt − Ai p ,
3.9
i1
which implies that
limAk Θk−1 yt − Ak p 0,
t→0
f
3.10
for each k ∈ {1, 2, . . . , M}. Since Trkk is firmly nonexpansive for each k ∈ {1, 2, . . . , M}, we
have for each p ∈ F and k ∈ {1, 2, . . . , M} that
2 2
f ,A
k
f ,A
Θ yt − p Trkk k Θk−1 yt − Trkk k Θk−1 p
2
f
f
Trkk Θk−1 yt − rk Ak Θk−1 yt − Trkk Θk−1 p − rk Ak Θk−1 p ≤ Θk−1 yt − rk Ak Θk−1 yt − p − rk Ak p , Θk yt − p
2
1 2 k−1
Θ yt − rk Ak Θk−1 yt − p − rk Ak p Θk yt − p
2
2 k
k−1
k−1
−Θ yt − rk Ak Θ yt − p − rk Ak p − Θ yt − p 10
Fixed Point Theory and Applications
2 2 2 1 k−1
≤
Θ yt − p Θk yt − p − Θk−1 yt − Θk yt − rk Ak Θk−1 yt − Ak p 2
2
2 2 1 k−1
≤
Θ yt − p Θk yt − p − Θk−1 yt − Θk yt 2
2rk Θk−1 yt − Θk yt Ak Θk−1 yt − Ak p .
3.11
This implies that
2
2 2 k
Θ yt − p ≤ Θk−1 yt − p − Θk−1 yt − Θk yt 2rk Θk−1 yt − Θk yt Ak Θk−1 yt − Ak p
3.12
2
2 ≤ Θk−1 yt − p − Θk−1 yt − Θk yt M2 Ak Θk−1 yt − Ak p,
where M2 max1≤k≤M sup0<t<1 {2rk Θk−1 yt − Θk yt }. This shows that
2
ut − p 2 ΘM yt − p
M
M
2
2 ≤ yt − p − Θi−1 yt − Θi yt M2 Ai Θi−1 yt − Ai p
i1
i1
3.13
M
M
2
2
≤ xt − p tM1 − Θi−1 yt − Θi yt M2 Ai Θi−1 yt − Ai p.
i1
i1
Hence,
xt − p2 ≤ ut − p2
M
M
2
2
≤ xt − p tM1 − Θi−1 yt − Θi yt M2 Ai Θi−1 yt − Ai p.
i1
3.14
i1
From 3.10, we obtain
M
i−1
Θ yt − Θi yt −→ 0,
3.15
i1
as t → 0. So, we can conclude that
limΘk−1 yt − Θk yt 0,
t→0
3.16
Fixed Point Theory and Applications
11
for each k ∈ {1, 2, . . . , M}. Observing
M
un − yt Θ yt − yt ≤ ΘM yt − ΘM−1 yt ΘM−1 yt − ΘM−2 yt · · · Θ1 yt − yt ,
3.17
it follows by 3.16 that
limut − yt 0.
t→0
3.18
limut − xt 0.
3.19
xt − Sxt Sut − Sxt ≤ ut − xt −→ 0,
3.20
From 3.6 and 3.18, we have
t→0
Hence,
as t → 0.
Next, we show that {xt } is relatively norm compact as t → 0. Let {tn } ⊂ 0, 1 be
a sequence such that tn → 0 as n → ∞. Put xn : xtn . From 3.20, we obtain
lim xn − Sxn 0.
3.21
n→∞
Since {xn } is bounded, without loss of generality, we may assume that {xn } converges weakly
to x∗ ∈ C. Applying Lemma 2.1 to 3.21, we can conclude that x∗ ∈ FS.
M
fk ,Ak k−1
k
Next, we show that x∗ ∈
Θ yn k1 GEPfk , Ak . Note that Θ yn Trk
f
Trkk Θk−1 yn − rk Ak Θk−1 yn for each k ∈ {1, 2, . . . , M}. Hence, for each y ∈ C and k ∈
{1, 2, . . . , M}, we obtain
1
fk Θk yn , y y − Θk yn , Θk yn − Θk−1 yn − rk Ak Θk−1 yn
≥ 0.
rk
3.22
From A2, we have
1
y − Θk yn , Θk yn − Θk−1 yn − rk Ak Θk−1 yn
≥ fk y, Θk yn ,
rk
∀y ∈ C.
3.23
Therefore,
y − Θ ynj ,
k
Θk ynj − Θk−1 ynj
rk
Ak Θ
k−1
ynj
≥ fk y, Θk ynj ,
∀y ∈ C.
3.24
12
Fixed Point Theory and Applications
For each t ∈ 0, 1 and y ∈ C, put zt ty 1 − tx∗ . Then, we have zt ∈ C. From 3.24,
we get that
zt − Θk ynj , Ak zt ≥ zt − Θk ynj , Ak zt
−
zt − Θ ynj ,
k
Θk ynj − Θk−1 ynj
rk
Ak Θ
k−1
ynj
fk zt , Θk ynj
zt − Θk ynj , Ak zt − Ak Θk ynj zt − Θk ynj , Ak Θk ynj − Ak Θk−1 ynj
−
zt − Θ ynj ,
k
Θk ynj − Θk−1 ynj
rk
fk zt , Θk ynj .
3.25
We note that Ak Θk ynj − Ak Θk−1 ynj ≤ 1/αk Θk ynj − Θk−1 ynj → 0, Θk ynj x∗ as j → ∞,
and {Ak }M
k1 is a family of monotone mappings. It follows from 3.25 that
zt − x∗ , Ak zt ≥ fk zt , x∗ .
3.26
So, by A1, A4 and 3.26, we have for each y ∈ C and k ∈ {1, 2, . . . , M} that
0 fk zt , zt ≤ tfk zt , y 1 − tfk zt , x∗ ≤ tfk zt , y 1 − tzt − x∗ , Ak zt 3.27
tfk zt , y t1 − t y − x∗ , Ak zt .
This implies that
fk zt , y 1 − t y − x∗ , Ak zt ≥ 0,
∀y ∈ C.
3.28
Letting t → 0 in 3.28, it follows from A3 that
fk x∗ , y y − x∗ , Ak x∗ ≥ 0,
∀y ∈ C.
3.29
Fixed Point Theory and Applications
Hence x∗ ∈
M
k1
13
GEPfk , Ak ; consequently, x∗ ∈ F. Further, we see that
xt − x∗ 2 Sut − x∗ 2
≤ ut − x∗ 2
2
≤ yt − x∗ 3.30
≤ xt − x∗ − txt 2
xt − x∗ 2 − 2txt , xt − x∗ t2 xt 2
xt − x∗ 2 − 2txt − x∗ , xt − x∗ − 2tx∗ , xt − x∗ t2 xt 2 .
So, we have
t
xt − x∗ 2 ≤ x∗ , x∗ − xt xt 2 .
2
3.31
In particular,
xn − x∗ 2 ≤ x∗ , x∗ − xn tn
xn 2 .
2
3.32
Since xn x∗ , we have xn → x∗ as n → ∞. By using the same argument as in the proof of
Theorem 3.1 of 25, we can show that xt → x∗ ∈ F as t → 0. This completes the proof.
4. Strong Convergence Results
Theorem 4.1. Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let {fk }M
k1 :
C × C → Ê be a family of bifunctions, let {Ak }M
k1 : C → H be a family of αk -inverse-strongly
∞
monotone mappings and let {Tn }n1 : C → C be a countable family of κ-strict pseudocontractions for
∞
∞
some 0 < κ < 1 such that F : M
/ ∅. Assume that {αn }n1 ⊂ 0, 1,
k1 GEPfk , Ak ∩ n1 FTn {βn }∞
n1 ⊂ 0, 1, γ ∈ κ, 1 and rk ∈ 0, 2αk for each k ∈ {1, 2, . . . , M} satisfy the following
conditions:
i limn → ∞ αn 0 and
∞
n1
αn ∞,
ii 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1.
Suppose that {Tn }, T satisfies the AKTT-condition. Then, {xn } generated by 1.14
converges strongly to an element in F.
Proof. For each n ∈ Æ , define Sn : C → C by Sn x γ x 1 − γ Tn x, x ∈ C. Then, FSn FTn FT, since γ ∈ 0, 1. Moreover, we know that {Sn } satisfies the AKTT-condition,
since {Tn } satisfies the AKTT-condition. By Lemma 2.5, we can define the mapping S : C →
C by Sx limn → ∞ Sn x for x ∈ C. Hence, Sx γ x 1 − γ Tx, x ∈ C, since Tn x → Tx for
14
Fixed Point Theory and Applications
x ∈ C. Further, we know that Sn is nonexpansive for each n ∈
and n ∈ Æ , we have
Æ . Indeed, for each x, y
∈C
Sn x − Sn y2 γ x 1 − γ Tn x − γ y − 1 − γ Tn y2
2
γ x − y 1 − γ Tn x − Tn y 2 2
2
γ x − y 1 − γ Tn x − Tn y − γ 1 − γ I − Tn x − I − Tn y
2 2
2 ≤ γ x − y 1 − γ x − y 1 − γ κI − Tn x − I − Tn y
4.1
2
− γ 1 − γ I − Tn x − I − Tn y
2 2
x − y 1 − γ κ − γ I − Tn x − I − Tn y
2
≤ x − y .
Hence, Sn is nonexpansive for each n ∈ Æ and so is S.
f ,A
f
,A
f ,A
k−1
k−1
Next, we show that {xn } is bounded. Denote Θk Trkk k Trk−1
· · · Tr11 1 for any
k ∈ {1, 2, . . . , M} and Θ0 I. We note that un ΘM yn . From 1.14, we have for each
p ∈ F that
xn1 − p βn xn 1 − βn Sn un ≤ βn xn − p 1 − βn Sn un − p
≤ βn xn − p 1 − βn un − p
βn xn − p 1 − βn ΘM yn − p
≤ βn xn − p 1 − βn yn − p
≤ βn xn − p 1 − βn 1 − αn xn − p αn p
1 − αn 1 − βn xn − p αn 1 − βn p
≤ max xn − p, p .
4.2
Hence, by induction, {xn } is bounded and so are {yn } and {un }.
Next, we show that
lim xn1 − xn 0.
n→∞
4.3
Since un ΘM yn and un1 ΘM yn1 ,
un1 − un ΘM yn1 − ΘM yn ≤ yn1 − yn .
4.4
Fixed Point Theory and Applications
15
Set zn Sn un , n ∈ Æ . So, we have from 1.14 and 4.4 that
zn1 − zn Sn1 un1 − Sn un ≤ Sn1 un1 − Sn1 un Sn1 un − Sn un ≤ un1 − un Sn1 un − Sn un ≤ yn1 − yn Sn1 un − Sn un 4.5
≤ 1 − αn1 xn1 − 1 − αn xn sup Sn1 z − Sn z
z∈{un }
≤ xn1 − xn αn1 xn1 αn xn sup Sn1 z − Sn z.
z∈{un }
Since {Sn } satisfies the AKTT-condition and limn → ∞ αn 0, it follows that
lim supzn1 − zn − xn1 − xn ≤ 0.
n→∞
4.6
So, by Lemma 2.2 and ii, we obtain
lim zn − xn 0.
4.7
lim xn1 − xn lim 1 − βn zn − xn 0.
4.8
yn − xn PC 1 − αn xn − PC xn ≤ αn xn −→ 0,
4.9
n→∞
Hence,
n→∞
n→∞
Observe that
as n → ∞. Similar to the proof of Theorem 3.1, we obtain for each p ∈ F that
M
2
i−1
un − p2 ≤ xn − p2 αn M ri ri − 2αi Θ
y
−
A
p
,
A
i
n
i
1
4.10
i1
M
M
2
i−1
i
i−1
un − p2 ≤ xn − p2 αn M − y
−
Θ
y
M
Θ
y
−
A
p
Θ
A
,
n
n
i
n
i
1
2
i1
i1
4.11
16
Fixed Point Theory and Applications
for some M1 > 0 and M2 > 0. Then, from 4.10, we have
xn1 − p2 ≤ βn xn − p2 1 − βn Sn un − p2
2 2
≤ βn xn − p 1 − βn un − p
2 ≤ βn xn − p 1 − βn
M
2
2
i−1
× xn − p αn M ri ri − 2αi Ai Θ yn − Ai p
4.12
1
i1
M
2
2
≤ xn − p αn M1 1 − βn
ri ri − 2αi Ai Θi−1 yn − Ai p ,
i1
which implies that
M
2 2 2
ri 2αi − ri Ai Θi−1 yn − Ai p ≤ xn1 − p − xn − p αn M1 .
1 − βn
4.13
i1
So, from 4.8, i, ii and 0 < rk < 2αk for each k 1, 2, . . . , M, we have
lim Ak Θk−1 yn − Ak p 0,
4.14
n→∞
for each k ∈ {1, 2, . . . , M}. Similarly, from 4.11, we have
xn1 − p2 ≤ βn xn − p2 1 − βn Sn un − p2
2 2
≤ βn xn − p 1 − βn un − p
2 ≤ βn xn − p 1 − βn
M
M
2
2
i−1
i
i−1
× xn − p αn M1 − Θ yn − Θ yn M2 Ai Θ yn − Ai p
i1
i1
M
2
M
2
i−1
≤ xn − p αn M1 − 1 − βn
Θ yn − Θi yn M2 Ai Θi−1 yn − Ai p.
i1
i1
4.15
This implies that
1 − βn
M
2
i−1
Θ yn − Θi yn i1
M
2 2
≤ xn − p − xn1 − p αn M1 M2 Ai Θi−1 yn − Ai p.
i1
4.16
Fixed Point Theory and Applications
17
From i, ii, 4.8, and 4.14, it follows that
lim Θk−1 yn − Θk yn 0,
4.17
lim xn − Sxn 0.
4.18
n→∞
for each k ∈ {1, 2, . . . , M}.
Next, we show that
n→∞
Observing
M
un − yn Θ yn − yn ≤ ΘM yn − ΘM−1 yn ΘM−1 yn − ΘM−2 yn · · · Θ1 yn − yn ,
4.19
it follows, by 4.17, that
lim un − yn 0.
4.20
lim un − xn 0.
4.21
n→∞
From 4.9 and 4.20, we have
n→∞
We see that
xn − Sxn ≤ xn − Sn un Sn un − Sn xn Sn xn − Sxn ≤ xn − Sn un un − xn sup Sn z − Sz.
4.22
z∈{xn }
So, by 4.7, 4.21, and Lemma 2.5, we have
lim xn − Sxn 0.
n→∞
4.23
Let the net {xt } be defined by 3.3. By Theorem 3.1, we have xt → x∗ ∈ F as t → 0.
Moreover, by proving in the same manner as in Theorem 3.2 of 25, we can show that
lim supx∗ , x∗ − xn ≤ 0.
n→∞
4.24
18
Fixed Point Theory and Applications
Finally, we show that xn → x∗ ∈ F as n → ∞. From 1.14, we have
xn1 − x∗ 2 ≤ βn xn − x∗ 2 1 − βn Sn un − x∗ 2
≤ βn xn − x∗ 2 1 − βn un − x∗ 2
2
≤ βn xn − x∗ 2 1 − βn yn − x∗ ≤ βn xn − x∗ 2 1 − βn 1 − αn xn − x∗ − αn x∗ 2
4.25
≤ βn xn − x∗ 2 1 − βn
× 1 − αn xn − x∗ 2 − 2αn 1 − αn x∗ , xn − x∗ α2n x∗ 2
1 − αn 1 − βn xn − x∗ 2
αn 1 − βn 21 − αn x∗ , x∗ − xn αn x∗ 2 .
By i and 4.24, it follows from Lemma 2.3 that xn → x∗ ∈ F. This completes the proof.
As a direct consequence of Lemmas 2.6 and 2.7 and Theorem 4.1, we obtain the
following result.
Theorem 4.2. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let {fk }M
k1 :
C × C → Ê be a family of bifunctions, let {Ak }M
k1 : C → H be a family of αk -inverse-strongly
∞
monotone mappings, and let {Si }i1 be a sequence of κi -strict pseudocontractions of C into itself such
∞
that F : M
/ ∅ and sup{κi : i ∈ Æ } κ > 0. Assume that
k1 GEPfk , Ak ∩ i1 FSi γ ∈ κ, 1 and rk ∈ 0, 2αk for each k ∈ {1, 2, . . . , M}. Define the sequence {xn } by x1 ∈ C and
yn PC 1 − αn xn ,
f ,AM
un TrMM
xn1 βn xn 1 − βn
f
M−1
TrM−1
,AM−1
f ,A2
· · · Tr22
γ un 1 − γ
f ,A1
Tr11
n
yn ,
4.26
μin Si un ,
n ≥ 1,
i1
∞
i
where {αn }∞
n1 and {βn }n1 are real sequences in 0, 1 which satisfy (i)-(ii) of Theorem 4.1 and {μn }
is a real sequence which satisfies (i)–(iii) of Lemma 2.7. Then, {xn } converges strongly to an element
in F.
Remark 4.3. Theorems 4.1 and 4.2 extend the main results in 25 from a nonexpansive mapping to an infinite family of strict pseudocontractions and a system of generalized equilibrium
problems.
Remark 4.4. If we take Ak ≡ 0 and fk ≡ 0 for each k 1, 2, . . . , M, then Theorems 3.1, 4.1,
and 4.2 can be applied to a system of equilibrium problems and to a system of variational
inequality problems, respectively.
Fixed Point Theory and Applications
19
Remark 4.5. Let S1 , S2 , . . . be an infinite family of nonexpansive mappings of C into itself,
and let ξ1 , ξ2 , . . . be real numbers such that 0 < ξi < 1 for all i ∈ Æ . Moreover, let Wn and
W be the W-mappings 35 generated by S1 , S2 , . . . , Sn and ξ1 , ξ2 , . . . , ξn and S1 , S2 , . . . and
ξ1 , ξ2 , . . .. Then, we know from 7, 35 that {Wn }, W satisfies the AKTT-condition. Therefore,
in Theorem 4.1, the mapping Tn can be also replaced by Wn .
Acknowledgments
The authors would like to thank the referees for valuable suggestions. This research is
supported by the Centre of Excellence in Mathematics, the Commission on Higher Education,
and the Thailand Research Fund. The first author is supported by the Royal Golden Jubilee
Grant no. PHD/0261/2551 and by the Graduate School, Chiang Mai University, Thailand.
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