Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 876819, 17 pages
doi:10.1155/2010/876819
Research Article
A Strong Convergence Theorem for
a Common Fixed Point of Two Sequences
of Strictly Pseudocontractive Mappings in
Hilbert Spaces and Applications
Kasamsuk Ungchittrakool1, 2
1
2
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
Correspondence should be addressed to Kasamsuk Ungchittrakool, kasamsuku@nu.ac.th
Received 17 July 2010; Accepted 27 October 2010
Academic Editor: Chaitan Gupta
Copyright q 2010 Kasamsuk Ungchittrakool. 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 prove a strong convergence theorem for a common fixed point of two sequences of strictly
pseudocontractive mappings in Hilbert spaces. We also provide some applications of the main
theorem to find a common element of the set of fixed points of a strict pseudocontraction and the
set of solutions of an equilibrium problem in Hilbert spaces. The results extend and improve the
recent ones announced by Marino and Xu 2007 and others.
1. Introduction
Let H be a real Hilbert space and C a nonempty closed convex subset of H. Let T : C → C
be a self-mapping of C. Recall that T is said to be a strict pseudocontraction if there exists a
constant 0 k < 1 such that
T x − T y2 x − y kI − T x − I − T y2
1.1
for all x, y ∈ C. We also say that T is a k-strict pseudocontraction if T satisfies 1.1. We use
FT to denote the set of fixed points of T i.e., FT {x ∈ C : T x x}. Note that the class
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Abstract and Applied Analysis
of strict pseudocontractions strictly includes the class of nonexpansive mappings which are
mappings T on C such that
T x − T y x − y
1.2
for all x, y ∈ C. That is, T is nonexpansive if and only if T is a 0-strict pseudocontraction.
In 1953, Mann 1 introduced the following iterative scheme:
x0 ∈ C
chosen arbitrarily,
1.3
xn1 αn xn 1 − αn T xn ,
n 0, 1, 2, . . . ,
where the sequence {αn } is chosen in 0, 1. Mann’s iteration process 1.3 has been
extensively investigated for nonexpansive mappings. One of the fundamental convergence
results was proved by Reich 2. In an infinite-dimensional Hilbert space, the Mann’s iteration
1.3 can conclude only weak convergence 3, 4. In 1967, Browder and Petryshyn 5
established the first convergence result for a k-strict pseudocontraction in a real Hilbert
space. They proved weak and strong convergence theorems by using 1.3 with a constant
control sequence {αn } ≡ α for all n. However, this scheme has only weak convergence
even in a Hilbert space. Therefore, many authors try to modify the normal Mann’s
iteration process to have strong convergence; see, for example, 6–10 and the references
therein.
Attempts to modify 1.3 so that strong convergence is guaranteed have been made.
In 2003, Nakajo and Takahashi 9 proposed the following modification of 1.3 for a single
nonexpansive mapping T by using the hybrid projection method in a Hilbert space H
x0 ∈ C
chosen arbitrarily,
yn αn xn 1 − αn T xn ,
Cn z ∈ C : z − yn z − xn ,
1.4
Qn {z ∈ C : xn − z, x − xn 0},
xn1 PCn ∩Qn x,
n 0, 1, 2, . . . ,
where PC denotes the metric projection from H onto a closed convex subset C of H. They
proved that if the sequence {αn } is bounded above from one, then {xn } defined by 1.4
converges strongly to PFT x.
Abstract and Applied Analysis
3
In 2007, Marino and Xu 11 proved the following strong convergence theorem by
using the hybrid projection method for a strict pseudocontraction. They defined a sequence
as follows:
x0 ∈ C
chosen arbitrarily,
yn αn xn 1 − αn T xn ,
2
Cn z ∈ C : yn − z xn − z2 1 − αn k − αn xn − T xn 2 ,
1.5
Qn {z ∈ C : xn − z, x0 − xn 0},
xn1 PCn ∩Qn x0 ,
n 0, 1, 2, . . . ,
They proved that if 0 αn < 1, then {xn } defined by 1.5 converges strongly to PFT x0 .
Motivated and inspired by the above-mentioned results, it is the purpose of this paper
to improve and generalize the algorithm 1.5 to the new general process of two sequences
of strictly pseudocontractive mappings in Hilbert spaces. Let C be a closed convex subset of
a Hilbert space H and Tn , Sn : C → C two sequences of strictly pseudocontractive mappings
∞
∅. Define {xn } in the following ways:
such that ∞
n0 FTn ∩ n0 FSn /
x0 ∈ C
chosen arbitrarily,
yn αn xn 1 − αn zn ,
zn βn Tn xn 1 − βn Sn xn ,
n z ∈ C : yn − z2 xn − z2 1 − αn βn kn − αn xn − Tn xn 2
C
T
1 − αn 1 − βn kSn − αn xn − Sn xn 2 − 1 − αn 2 βn 1 − βn Tn xn − Sn xn 2 ,
Qn {z ∈ C : xn − z, x0 − xn 0},
xn1 PC
n ∩Qn x0 ,
1.6
where {αn }, {βn } are sequences in 0, 1.
We prove that the algorithm 1.6 converges strongly to a common fixed point of two
sequences of strictly pseudocontractive mappings {Tn } and {Sn } provided that {Tn }, {Sn },
{αn } and {βn } satisfy some appropriate conditions, and then we apply the result for finding
a common element of the set of fixed points of a strict pseudocontraction and the set of
solutions of an equilibrium problem in Hilbert spaces. Our results extend and improve the
corresponding ones announced by Marino and Xu 11 and others.
Throughout the paper, we will use the following notation:
i → for strong convergence and for weak convergence,
ii ωw xn {x : ∃xnr x} denotes the weak ω-limit set of {xn }.
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Abstract and Applied Analysis
2. Preliminaries
This section collects some definitions and lemmas which will be used in the proofs for the
main results in the next section. Some of them are known; others are not hard to derive.
Lemma 2.1. Let H be a real Hilbert space. There holds the following identity:
i x − y2 x2 − y2 − 2x − y, y for all x, y ∈ H.
ii tx 1 − ty2 tx2 1 − ty2 − t1 − tx − y2 for all t ∈ 0, 1, for all x, y ∈ H.
Lemma 2.2. Let H be a real Hilbert space. Given a closed convex subset C ⊂ H and x, y, z ∈ H.
Given also a real number a ∈ R. The set
2
v ∈ C : y − v x − v2 z, v a
2.1
is convex (and closed).
Recall that given a closed convex subset C of a real Hilbert space H, the nearest point
projection PC from H onto C assigns to each x ∈ H its nearest point denoted by PC x which is
a unique point in C with the property
x − PC x x − z
∀z ∈ C.
2.2
Lemma 2.3. Let C be a closed convex subset of real Hilbert space H. Given x ∈ H and z ∈ C. Then,
z PC x if and only if there holds the relation
x − z, z − y 0 ∀y ∈ C.
2.3
Lemma 2.4 Martinez-Yanes and Xu 8. Let C be a closed convex subset of real Hilbert space H.
Let {xn } be a sequence in H and u ∈ H. Let q PC u. If {xn } is such that ωw xn ⊂ C and satisfies
the condition
xn − u u − q ∀n.
2.4
Then, xn → q.
Given a closed convex subset C of a real Hilbert space H and a mapping T : C → C.
Recall that T is said to be a quasistrict pseudocontraction if FT is nonempty and there exists
a constant 0 k < 1 such that
T x − p2 x − p2 kx − T x2
for all x ∈ C and p ∈ FT .
2.5
Abstract and Applied Analysis
5
Proposition 2.5 Marino and Xu 11, Proposition 2.1. Assume C is a closed convex subset of a
Hilbert space H, and let T : C → C be a self-mapping of C.
i If T is a k-strict pseudocontraction, then T satisfies Lipschitz condition
T x − T y 1 k x − y
1−k
∀x, y ∈ C.
2.6
ii If T is a k-strict pseudocontraction, then the mapping I − T is demiclosed (at 0). That is, if
{xn } is a sequence in C such that xn x
and I − T xn → 0, then I − T x
0.
iii If T is a k-quasistrict pseudocontraction, then the fixed-point set FT of T is closed and
convex so that the projection PFT is well defined.
Lemma 2.6 Plubtieng and Ungchittrakool 12, Lemma 3.1. Let C be a nonempty subset of a
Banach space E and {Tn } a sequence of mappings from C into E. Suppose that for any bounded subset
B of C there exists continuous increasing function hB from R into R such that hB 0 0 and
lim ρlk 0,
2.7
ρlk : sup{hB Tk z − Tl z : z ∈ B} < ∞,
2.8
k,l → ∞
where
for all k, l ∈ N. Then, for each x ∈ C, {Tn x} converges strongly to some point of E. Moreover, let T be
a mapping from C into E defined by
T x lim Tn x
n→∞
∀x ∈ C.
2.9
Then, limn → ∞ sup{hB T z − Tn z : z ∈ B} 0.
Lemma 2.7. Let H be a real Hilbert space, let C be a nonempty closed convex subset of H, and
let {Tn } be a sequence such that for each n, Tn is kn -strict pseudo contraction from C into H with
lim supn → ∞ kn < 1 and
T x lim Tn x
n→∞
∀x ∈ C.
Then, FT is closed and convex so that the projection PFT is well defined.
2.10
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Abstract and Applied Analysis
Proof. To see that FT is closed, assume that {pn } is a sequence in FT such that pn → p
.
Since Tn is a kn -quasistrict pseudocontraction, we get, for each n,
2
2 Tm p
− Tm pn 2
T p
− pn T p
− T pn lim Tm p
− lim Tm pn mlim
m→∞
m→∞
→∞
2
2
2 lim sup p
− pn km p
− Tm p
− pn − Tm pn m→∞
2
2
lim sup p
− Tm p
− pn − Tm pn p
− pn lim sup km
m→∞
m→∞
2.11
2
2
p
−
lim
T
p
−
p
−
lim
T
p
p
− pn lim sup km m
n
m n m→∞
m→∞
m→∞
2
2
p
− pn lim sup km p
− T p
.
m→∞
Taking the limit as n → ∞ yields T p
− p
2 κ
p − T p
2 , where κ : lim supm → ∞ km . Since
0 κ < 1, we have T p
p
.
3. Main Result
In this section, we prove a strong convergence theorem by using the hybrid projection method
some authors call this the CQ method for finding a common element of the set of fixed
points of two sequences of strictly pseudocontractive mappings in Hilbert spaces.
Theorem 3.1. Let C be a closed convex subset of a Hilbert space H. For each n, let Tn , Sn : C → C be
kTn , kSn -strict pseudocontractions for some 0 kTn , kSn < 1 with lim supn → ∞ kTn , lim supn → ∞ kSn < 1,
∞
∅. Let {xn }∞
respectively, and assume that ∞
n0 be the sequence generated by
n0 FTn ∩ n0 FSn /
x0 ∈ C
chosen arbitrarily,
yn αn xn 1 − αn zn ,
zn βn Tn xn 1 − βn Sn xn ,
n z ∈ C : yn − z2 xn − z2 1 − αn βn kn − αn xn − Tn xn 2
C
T
1 − αn 1 − βn kSn − αn xn − Sn xn 2 − 1 − αn 2 βn 1 − βn Tn xn − Sn xn 2 ,
Qn {z ∈ C : xn − z, x0 − xn 0},
xn1 PC
n ∩Qn x0 .
3.1
Abstract and Applied Analysis
7
Assume that {αn } and {βn } are chosen so that 0 αn < 1 and 0 < a βn b < 1 for all n. Suppose
that for any bounded subset B of C there exists an increasing, continuous, and convex function hB
from R into R such that hB 0 0, and
lim sup{hB Tk z − Tl z : z ∈ B} 0 lim sup{hB Sk z − Sl z : z ∈ B}.
k,l → ∞
k,l → ∞
3.2
Let T, S : C → C such that T x limn → ∞ Tn x and Sx limn → ∞ Sn x for all x ∈ C, respectively,
∞
and suppose that FT ∞
n0 FTn and FS n0 FSn . Then, {xn } converges strongly to a
common fixed point q PFT ∩FS x0 .
n and Qn are closed and convex for all n via Lemma 2.2
Proof. It is not hard to check that C
∩ Qn is nonempty for all n, the sequence
and the properties of the inner product. Then, if C
∞n
{xn } is well defined. Now, we will show that n0 FTn ∩ ∞
n0 FSn ⊂ Cn for all n. Let
∞
∞
p ∈ n0 FTn ∩ n0 FSn , we observe that
zn − p2 βn Tn xn − p 1 − βn Sn xn − p 2
2 2
βn Tn xn − p 1 − βn Sn xn − p − βn 1 − βn Tn xn − Sn xn 2
2
2
βn xn − p kTn xn − Tn xn 2 1 − βn xn − p kSn xn − Sn xn 2
− βn 1 − βn Tn xn − Sn xn 2
2
xn − p βn kTn xn − Tn xn 2 1 − βn kSn xn − Sn xn 2
− βn 1 − βn Tn xn − Sn xn 2 ,
2
xn − zn 2 βn xn − Tn xn 1 − βn xn − Sn xn βn xn − Tn xn 2 1 − βn xn − Sn xn 2 − βn 1 − βn Tn xn − Sn xn 2 .
3.3
3.4
By 3.3 and 3.4 we obtain
yn − p2 αn xn − p 1 − αn zn − p 2
2
2
αn xn − p 1 − αn zn − p − αn 1 − αn xn − zn 2
2
xn − p 1 − αn βn kTn − αn xn − Tn xn 2
1 − αn 1 − βn kSn − αn xn − Sn xn 2
− 1 − αn 2 βn 1 − βn Tn xn − Sn xn 2 .
3.5
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Abstract and Applied Analysis
n for all n. Next, we will show that FT ∩ FS ⊂ Qn for all n.
Thus, we have FT ∩ FS ⊂ C
If n 0, then FT ∩ FS ⊂ C Q0 . Assume that FT ∩ FS ⊂ Qn . Since xn1 is the projection
n ∩ Qn , by Lemma 2.3 we have
of x0 onto C
n ∩ Qn .
xn1 − z, x0 − xn1 0 ∀z ∈ C
3.6
n ∩ Qn by the induction assumption, it implies that FT ∩ FS ⊂
Noting that FT ∩ FS ⊂ C
n ∩ Qn for all n. So,
Qn1 , thus by induction FT ∩ FS ⊂ Qn for all n. Hence, FT ∩ FS ⊂ C
{xn } is well defined.
Notice that the definition of Qn actually implies xn PQn x0 . This together with the
fact FT ∩ FS ⊂ Qn further implies
xn − x0 p − x0 ∀p ∈ FT ∩ FS.
3.7
In particular, {xn } is bounded and
xn − x0 q − x0 ∀n,
3.8
where q PFT ∩FS x0 .
The fact xn1 ∈ Qn asserts that xn1 − xn , xn − x0 0. This together with Lemma 2.1i
and Lemma 2.3 implies
xn1 − xn 2 xn1 − x0 − xn − x0 2
xn1 − x0 2 − xn − x0 2 − 2xn1 − xn , xn − x0 3.9
xn1 − x0 2 − xn − x0 2 .
It turns out that
xn1 − xn −→ 0.
3.10
xn1 − yn 2 xn1 − xn 2 1 − αn βn kn − αn xn − Tn xn 2
T
1 − αn 1 − βn kSn − αn xn − Sn xn 2
− 1 − αn 2 βn 1 − βn Tn xn − Sn xn 2 .
3.11
n , we get
By the fact xn1 ∈ C
Abstract and Applied Analysis
9
Observe that
xn1 − yn 2 αn xn1 − xn 1 − αn xn − zn 2
αn xn1 − xn 2 1 − αn xn1 − zn 2 − αn 1 − αn xn − zn 2 ,
2
xn1 − zn 2 βn xn1 − Tn xn 1 − βn xn1 − Sn xn βn xn1 − Tn xn 2 1 − βn xn1 − Sn xn 2 − βn 1 − βn Tn xn − Sn xn 2 .
3.12
With simple calculation by using 3.12 and 3.4, we have
xn1 − yn 2 αn xn1 − xn 2 1 − αn βn xn1 − Tn xn 2
1 − αn 1 − βn xn1 − Sn xn 2 − αn 1 − αn βn xn − Tn xn 2
− αn 1 − αn 1 − βn xn − Sn xn 2 − 1 − αn 2 βn 1 − βn Tn xn − Sn xn 2 .
3.13
So, when we combine 3.11 and 3.13 and compute, we obtain
1 − αn βn xn1 − Tn xn 2 1 − αn 1 − βn xn1 − Sn xn 2
1 − αn xn1 − xn 2 1 − αn βn kTn xn − Tn xn 2 1 − αn 1 − βn kSn xn − Sn xn 2 .
3.14
Since αn < 1 for all n, we have
βn xn1 − Tn xn 2 1 − βn xn1 − Sn xn 2
xn1 − xn 2 βn kTn xn − Tn xn 2 1 − βn kSn xn − Sn xn 2 .
3.15
Notice that
xn1 − Tn xn 2 xn1 − xn xn − Tn xn 2
xn1 − xn 2 2xn1 − xn , xn − Tn xn xn − Tn xn 2 ,
xn1 − Sn xn 2 xn1 − xn xn − Sn xn 2
xn1 − xn 2 2xn1 − xn , xn − Sn xn xn − Sn xn 2 .
3.16
3.17
By 3.15, 3.16, and 3.17, we have
βn 1 − kTn xn − Tn xn 2 1 − βn 1 − kSn xn − Sn xn 2
−2βn xn1 − xn , xn − Tn xn − 2 1 − βn xn1 − xn , xn − Sn xn .
3.18
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Abstract and Applied Analysis
Since {xn }, {Tn xn }, and {Sn xn } are bounded, 0 < a βn b < 1 for all n and lim supn → ∞ kTn ,
lim supn → ∞ kSn < 1, it follows from 3.10 and 3.18 that
lim xn − Tn xn 0 lim xn − Sn xn .
n→∞
n→∞
3.19
Since {xn } is bounded, there exists a bounded subset B of C such that {xn } ⊂ B. From
Lemma 2.6, we are able to set T x limm → ∞ Tm x for all x ∈ C, and then observe that
1
1
1
xn − T xn xn − Tn xn Tn xn − T xn .
2
2
2
3.20
Since hB is an increasing, continuous, and convex function from R into R such that hB 0 0, we discover that
hB
1
1
1
xn − T xn hB xn − Tn xn hB Tn xn − T xn 2
2
2
1
1
hB xn − Tn xn sup{hB Tn z − T z : z ∈ B}.
2
2
3.21
By Lemma 2.6 and the continuity of hB , we have limn → ∞ hB 1/2xn − T xn 0. And then
the properties of hB yield
lim xn − T xn 0.
3.22
lim xn − Sxn 0.
3.23
n→∞
By the same argument, we have
n→∞
Now Proposition 2.5 guarantees that ωw xn ⊂ FT ∩ FS. This fact, the inequality 3.8,
and Lemma 2.4 ensure the strong convergence of {xn } to q PFT ∩FS x0 .
If Tn T and Sn S for all n, then kTn kT and kSn kS for all n. So, Theorem 3.1
reduces to the following corollary.
Abstract and Applied Analysis
11
Corollary 3.2. Let C be a closed convex subset of a Hilbert space H. Let T, S : C → C be kT , kS ∅. Let
strict pseudocontractions for some 0 kT , kS < 1, respectively, and assume that FT ∩ FS /
be
the
sequence
generated
by
{xn }∞
n0
x0 ∈ C
chosen arbitrarily,
yn αn xn 1 − αn zn ,
zn βn T xn 1 − βn Sxn ,
n z ∈ C : yn − z2 xn − z2 1 − αn βn kT − αn xn − T xn 2
C
3.24
1 − αn 1 − βn kS − αn xn − Sxn 2 − 1 − αn 2 βn 1 − βn T xn − Sxn 2 ,
Qn {z ∈ C : xn − z, x0 − xn 0},
xn1 PC
n ∩Qn x0 ,
where {αn } and {βn } be as in Theorem 3.1. Then, {xn } converges strongly to a common fixed point
q PFT ∩FS x0 .
In particular, if T S, then zn T xn and
n z ∈ C : yn − p2 xn − p2 1 − αn βn kT − αn xn − T xn 2
C
1 − αn 1 − βn kT − αn xn − T xn 2 − 1 − αn 2 βn 1 − βn T xn − T xn 2
3.25
Cn .
So, Corollary 3.2 reduces to the following corollary.
Corollary 3.3 Marino and Xu 11, Theorem 4.1. Let C be a closed convex subset of a Hilbert
space H. Let T : C → C be a k-strict pseudocontraction for some 0 k < 1, and assume that the
∞
fixed-point set FT / ∅. Let {xn }n0 be the sequence generated by
x0 ∈ C
chosen arbitrarily,
yn αn xn 1 − αn T xn ,
2
Cn z ∈ C : yn − z xn − z2 1 − αn k − αn xn − T xn 2
3.26
Qn {z ∈ C : xn − z, x0 − xn 0},
xn1 PCn ∩Qn x0 .
Assume that the control sequence {αn }∞
n0 is chosen so that 0 αn < 1 for all n. Then, {xn } converges
strongly to a fixed-point q PFT x0 .
kSn
If Tn T for all n and {Sn } is a sequences of nonexpansive mappings, then kTn k and
0 for all n. So, Theorem 3.1 reduces to the following corollary.
12
Abstract and Applied Analysis
Corollary 3.4. Let C be a closed convex subset of a Hilbert space H. Let T : C → C be a k-strict
pseudocontraction for some 0 k < 1, for each n and Sn : C → C a nonexpansive mapping, and
∅. Let {xn }∞
assume that FT ∩ ∞
n0 be the sequence generated by
n0 FSn /
x0 ∈ C
chosen arbitrarily,
yn αn xn 1 − αn zn ,
zn βn T xn 1 − βn Sn xn ,
n z ∈ C : yn − z2 xn − z2 1 − αn βn k − αn xn − T xn 2
C
−αn 1 − αn 1 − βn xn − Sn xn 2 − 1 − αn 2 βn 1 − βn T xn − Sn xn 2 ,
3.27
Qn {z ∈ C : xn − z, x0 − xn 0},
xn1 PC
n ∩Qn x0 ,
where {αn } and {βn } be as in Theorem 3.1. Suppose that for any bounded subset B of C there exists
an increasing, continuous, and convex function hB from R into R such that hB 0 0, and
lim sup{hB Sk z − Sl z : z ∈ B} 0.
3.28
k,l → ∞
Let S : C → C be such that Sx limn → ∞ Sn x for all x ∈ C, and suppose FS {xn } converges strongly to a common fixed point q PFT ∩FS x0 .
∞
n0
FSn . Then,
4. Equilibrium Problem
In this section, we have an application of the main result for finding a common element of
the set of fixed points of a strict pseudocontraction and the set of solutions of an equilibrium
problem.
Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let
ϕ be a bifunction of C × C into R, where R is the set of real numbers. The equilibrium problem
for ϕ : C × C → R is to find x ∈ C such that
ϕ x, y 0 ∀y ∈ C.
4.1
The set of solution of 4.1 is denoted by EPϕ {x ∈ C : ϕx, y 0 for all y ∈ C}. Many
problems in physics, optimization, and economics reduce to find some elements of EPϕ.
Abstract and Applied Analysis
13
For solving the equilibrium problem for a bifunction ϕ : C × C → R, let us assume that
ϕ satisfies the following conditions:
A1 ϕx, x 0 for all x ∈ C;
A2 ϕ is monotone, that is, ϕx, y ϕy, x 0 for all x, y ∈ C;
A3 for each x, y, z ∈ C,
lim ϕ tz 1 − tx, y ϕ x, y ;
t↓0
4.2
A4 for each x ∈ C, y → ϕx, y is convex and lower semicontinuous.
The following lemma appears implicitly in 13.
Lemma 4.1 Blum and Oettli 13. Let C be a nonempty closed convex subset of H, and let ϕ be a
bifunction of C × C into R satisfying (A1)–(A4). Let r > 0 and x ∈ H. Then, there exists z ∈ C such
that
1
ϕ z, y y − z, z − x 0 ∀y ∈ C.
r
4.3
The following lemma was also given in 14.
Lemma 4.2 Combettes and Hirstoaga 14. Assume that ϕ : C × C → R satisfies (A1)–(A4).
For r > 0 and x ∈ H, define a mapping Sr : H → C as follows:
Sr x 1
z ∈ C : ϕ z, y y − z, z − x 0, ∀y ∈ C
r
4.4
for all z ∈ H. Then, the following hold:
1 Sr is single-valued;
2 Sr is firmly nonexpansive, that is, for any x, y ∈ H, Sr x − Sr y2 Sr x − Sr y, x − y;
3 FSr EP ϕ;
4 EPϕ is closed and convex.
The following corollary is an application of Corollary 3.4 in the case of finding a
common element of the set of fixed points of a strict pseudocontraction and the set of
solutions of an equilibrium problem.
Corollary 4.3. Let C be a closed convex subset of a Hilbert space H. Let T : C → C be a k-strict
pseudocontraction for some 0 k < 1 and ϕ a bifunction from C × C into R satisfying (A1)–(A4).
14
Abstract and Applied Analysis
Suppose that FT ∩ EP ϕ /
∅. Let {xn }∞
n0 be the sequence generated by
x0 ∈ C
chosen arbitrarily,
yn αn xn 1 − αn zn ,
zn βn T xn 1 − βn un ,
1
such that ϕ un , y y − un , un − xn 0, ∀y ∈ C,
rn
2
z ∈ C : yn − z xn − z2 1 − αn βn k − αn xn − T xn 2
−αn 1 − αn 1 − βn xn − un 2 − 1 − αn 2 βn 1 − βn T xn − un 2 ,
un ∈ C
n
C
4.5
Qn {z ∈ C : xn − z, x0 − xn 0},
xn1 PC
n ∩Qn x0 ,
where {αn } and {βn } be as in Theorem 3.1 and {rn }∞
n0 is chosen so that {rn } ⊂ 0, ∞ with infn rn > 0
|r
−r
|
<
∞.
Then,
{x
}
converges
strongly
to a common fixed point q PFT ∩EP ϕ x0 .
and ∞
n
n
n0 n1
Proof. Obviously, un Srn xn , where Srn are mappings as in Lemma 4.2. Next, we want to
show that for any bounded subset B of C there exists an increasing, continuous, and convex
function hB ·2 from R into R such that hB 0 0, and
lim sup{hB Srk v − Srl v : v ∈ B} 0.
k,l → ∞
4.6
Let B be a bounded subset of C. For each v ∈ B, let vn Srn v. Then, by Lemma 4.2, we have
1
ϕ vl , y y − vl , vl − v 0
rl
1
ϕ vk , y y − vk , vk − v 0
rk
∀y ∈ C,
∀y ∈ C.
4.7
4.8
Put y vk in 4.7 and y vl in 4.8, we have
ϕvl , vk 1
vk − vl , vl − v 0,
rl
ϕvk , vl 1
vl − vk , vk − v 0.
rk
4.9
So, from A2, we have vk − vl , vl − v/rl − vk − v/rk 0 and hence vk − vl , vl − v −
rl /rk vk − v 0. Thus,
rl
vk − v 0.
vk − vl , vl − vk vk − vl , 1 −
rk
4.10
Abstract and Applied Analysis
15
Let b : infn rn . Thus, we have
Srk v − Srl v2 vk − vl , vk − vl rl
vk − vl , 1 −
vk − v
rk
1
vk − vl |rk − rl |vk − v
b
1
Srk v − Srl vSrk v − v|rk − rl |
b
4
v − p|rk − rl | where p ∈ EP ϕ
b
4
M|rk − rl | where M : sup v − p : v ∈ B .
b
4.11
Put k > l. Observe that
Srk v − Srl v2 k−1
∞
4
4 4 M|rk − rl | M |rn1 − rn | M |rn1 − rn | < ∞,
b
b nl
b nl
4.12
for all v ∈ B, and then
∞
4 ρlk sup Srk v − Srl v2 : v ∈ B M |rn1 − rn | < ∞.
b nl
4.13
Let l → ∞, we have limk,l → ∞ ρlk 0. Then, by Lemma 2.6, we can define a mapping S by
Sx lim Srn x
n→∞
∀x ∈ C.
4.14
Next, we will show that
FS ∞
FSrn EP ϕ .
4.15
n0
Since {rn } ⊂ 0, ∞, infn rn > 0 and 4.14, it is easy to see that
∞
EP ϕ FSrn ⊂ FS.
4.16
n0
Let p ∈ FS. By the definition of Sr , we have
1
ϕ Srn p, y y − Srn p, Srn p − p 0
rn
∀y ∈ C.
4.17
16
Abstract and Applied Analysis
By A2, we have
1
y − Srn p, Srn p − p ϕ y, Srn p
rn
∀y ∈ C.
4.18
From 4.14 and the lower semicontinuity of ϕy, ·, we have 0 ϕy, p for all y ∈ C. Let
y ∈ C and set xt ty 1 − tp, for t ∈ 0, 1. Then, we have
0 ϕxt , xt tϕ xt , y 1 − tϕ xt , p tϕ xt , y .
4.19
So ϕxt , y 0. Letting t ↓ 0 and using A3, we get ϕp, y 0 for all y ∈ C, and hence
p ∈ EPϕ ∞
n1 FSrn . Hence, we have 4.15. Then, applying Corollary 3.4, xn → q PFT ∩EPϕ x0 .
Acknowledgments
This research is supported by the Centre of Excellence in Mathematics under the Commission
on Higher Education, Ministry of Education, Thailand. The author would like to thank the
referees for reading this paper carefully and providing valuable suggestions and comments
of this paper.
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