Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 136134, 12 pages
doi:10.1155/2012/136134
Research Article
Strong Convergence Theorems for a Countable
Family of Total Quasi-φ-Asymptotically
Nonexpansive Nonself Mappings
Liang-cai Zhao1 and Shih-sen Chang2
1
2
College of Mathematics, Yibin University, Yibin 644000, China
College of Statistics and Mathematics, Yunnan University of Finance and Economics, Yunnan,
Kunming 650221, China
Correspondence should be addressed to Shih-sen Chang, changss@yahoo.cn
Received 3 June 2012; Revised 22 July 2012; Accepted 23 July 2012
Academic Editor: Naseer Shahzad
Copyright q 2012 L.-c. Zhao and S.-s. Chang. 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.
The purpose of this paper is to introduce a class of total quasi-φ-asymptotically nonexpansivenonself mappings and to study the strong convergence under a limit condition only in the
framework of Banach spaces. As an application, we utilize our results to study the approximation
problem of solution to a system of equilibrium problems. The results presented in the paper extend
and improve the corresponding results announced by some authors recently.
1. Introduction
Throughout this paper, we assume that E is a real Banach space, C is a nonempty closed and
∗
convex subset of E, E∗ is the dual space of E, and J : E → 2E is the normalized duality
mapping defined by
2 1.1
Jx f ∗ ∈ E∗ , x, f ∗ x2 f , x ∈ E.
Recall that a Banach space E is said to be strictly convex if x y/2 < 1 for all x, y ∈
U {z ∈ E : ||z|| 1} with x /
y. E is said to be uniformly convex, if for each ∈ 0, 2, there
exists δ > 0 such that x y/2 < 1 − δ for all x, y ∈ U with x − y ≥ . E is said to be smooth,
if the limit
x ty − x
∗∗
lim
t→0
t
2
Journal of Applied Mathematics
exists for all x, y ∈ U. And E is said to be uniformly smooth, if the above limit is exists uniformly
for x, y ∈ U.
In the sequel, we shall denote the fixed point set of a mapping T by FT . When {xn }
is a sequence in E, then xn → x xn x will denote strong weak convergence of the
sequence {xn } to x.
A mapping T : C → C is said to be nonexpansive, if
T x − T y ≤ x − y,
∀x, y ∈ C.
1.2
A mapping T : C → C is said to be asymptotically nonexpansive if there exists a sequence
{kn } ⊂ 1, ∞ such that
n
T x − T n y ≤ kn x − y,
∀x, y ∈ C.
1.3
Recall that a subset C of E is said to be retract of E, if there exists a continuous mapping
P : E → C such that P x x, for all x ∈ C.
It is well known that every nonempty closed and convex subset of a uniformly convex
Banach space is a retract of E. A mapping P : E → C is said to be a retraction, if P 2 P . It
follows that if a mapping P is a retraction, then P y y for all y in the range of P . A mapping
P : E → C is said to be a nonexpansive retraction, if it is nonexpansive and it is a retraction
from E to C.
In the sequel, we assume that E is a smooth, strictly convex, and reflexive Banach
space and C is a nonempty closed convex subset of E. Throughout this paper we assume that
φ : E × E → R is the Lyapunov function which is defined by
2
φ x, y x − 2x, Jy y ,
∀x, y ∈ E.
1.4
It is obvious from the definition of φ that
2
2
x − y ≤ φ x, y ≤ x y , ∀x, y ∈ E,
φ x, J −1 λJy 1 − λJz ≤ λφ x, y 1 − λφx, z, ∀x, y ∈ E.
1.5
1.6
Following Alber 1, the generalized projection ΠC : E → C is defined by
ΠC x arg inf φ y, x ,
y∈C
∀x ∈ E.
1.7
Lemma 1.1 see 1. Let E be a smooth, strictly convex, and reflexive Banach space and C be a
nonempty closed convex subset of E. Then the following conclusions hold:
1 φx, ΠC y φΠc y, y ≤ φx, y for all x ∈ C and y ∈ E;
2 If x ∈ E and z ∈ C, then z ΠC x ⇔ z − y, Jx − Jz ≥ 0, for all y ∈ C;
3 For x, y ∈ E, φx, y 0 if and only if x y.
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Remark 1.2. If E is a real Hilbert space H, then φx, y ||x − y||2 and ΠC PC the metric
projection of H onto C.
A mapping T : C → C is said to be closed, if for any sequence {xn } ⊂ C with xn → x
and T xn → y, then T x y.
Definition 1.3. Let P : E → C be the nonexpansive retraction.
1 T : C → E is said to be quasi-φ-nonexpansive nonself mapping, if FT /
∅ and
φu, T x ≤ φu, x,
∀x ∈ C, u ∈ FT .
1.8
2 T : C → E is said to be quasi-φ-asymptotically nonexpansive nonself mapping, if
FT /
∅ and there exists a real sequence {kn } ⊂ 1, ∞ with kn → 1 such that
φ u, T P T n−1 x ≤ kn φu, x,
∀x ∈ C, u ∈ FT , n ≥ 1.
1.9
3 T : C → E is said to be total quasi-φ-asymptotically nonexpansive nonself mapping, if
FT /
∅ and there exists nonnegative real sequence {νn }, {μn } with νn → 0, μn →
0 as n → ∞ and a strictly increasing continuous function ρ : R → R with
ρ0 0 such that for all x ∈ C, u ∈ FT φ u, T P T n−1 x ≤ φu, x νn ρ φu, x μn ,
∀n ≥ 1.
1.10
4 A countable family of nonself mappings {Ti } : C → E is said to be uniformly total
∅ and there exists nonnegative real
quasi-φ-asymptotically nonexpansive, if ∞
i1 FTi /
sequence {νn }, {μn } with νn → 0, μn → 0 as n → ∞ and a strictly increasing
continuous function ρ : R → R with ρ0 0 such that for each i ≥ 1 and all
x ∈ C, u ∈ ∞
i1 FTi φ u, Ti P Ti n−1 x ≤ φu, x νn ρ φu, x μn ,
∀n ≥ 1.
1.11
Remark 1.4. From the definitions, it is easy to know that
1 If T is a quasi-φ-nonexpansive nonself mapping, then it must be a quasi-φasymptotically nonexpansive nonself mapping with {kn 1}.
2 Taking ρt t, t > 0, νn kn − 1 and μn 0, then 1.9 can be rewritten as
φ u, T P T n−1 x ≤ φu, x νn ρ φu, x μn ,
∀n ≥ 1, x ∈ C, u ∈ FT .
1.12
This implies that each quasi-φ-asymptotically nonexpansive nonself mapping must be a total
quasi-φ-asymptotically nonexpansive nonself mapping, but the converse is not true.
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A nonself mapping T : C → E is said to be uniformly L-Lipschitz continuous, if there
exists a constant L > 0 such that
1.13
T P T n−1 x − T P T n−1 y ≤ Lx − y, ∀x, y ∈ C, n ≥ 1.
Lemma 1.5 see 2. Let E be a smooth and uniformly convex Banach space and let {xn }, {yn }
be two sequences of E. If φxn , yn → 0 as n → ∞ and either {xn } or {yn } is bounded, then
||xn − yn || → 0 as n → ∞).
Lemma 1.6. Let E be a smooth, strictly convex, and reflexive Banach space and C be a nonempty
closed and convex subset E. Let T : C → E be a closed and total quasi-φ-asymptotically nonexpansive
nonself mapping with nonnegative real sequence {νn }, {μn } and a strictly increasing continuous
function ρ : R → R such that νn → 0, μn → 0 and ρ0 0. Then the fixed point set FT is a
closed and convex subset of C.
Proof. Let {xn } be a sequence in FT such that xn → u as n → ∞. Since T xn xn → u, by
the closeness of T , we have u T u, that is, u ∈ FT . This shows that FT is a closed set in C.
Next, we prove that FT is convex. For any x, y ∈ FT , t ∈ 0, 1, putting q tx 1 −
ty, we prove that q ∈ FT . Indeed, let {un } be a sequence generated by
u1 T q,
u2 T P T q T P u1 ,
u3 T P T 2 q T P u2 , . . . ,
un T P T n−1 q T P un−1 , . . . ,
1.14
we have
2
φ q, un q − 2 q, Jun un 2
2
q − 2tx, Jun − 21 − t y, Jun un 2
2
2
q tφx, un 1 − tφ y, un − tx2 − 1 − ty .
1.15
Since
tφx, un 1 − tφ y, un
≤ t φ x, q νn ρ φ x, q μn 1 − t φ y, q νn ρ φ y, q μn
2
t x2 − 2 x, Jq q νn ρ φ x, q μn
2
2
1 − t y − 2 y, Jq q νn ρ φ y, q μn
1.16
2 2
tx2 1 − ty − q tνn ρ φ x, q 1 − tνn ρ φ y, q μn .
Substituting 1.16 into 1.15, and simplifying we have
φ q, un ≤ tνn ρ φ x, q 1 − tνn ρ φ y, q μn −→ 0 n −→ ∞.
By Lemma 1.5, we have un → q n → ∞. This implies that un1 → q n → ∞.
1.17
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Since un1 T P T n q T P T P T n−1 q T P un and T is closed, we have q T P q. Since
q ∈ C, P q q, thus q T q. this implies that FT is a convex set in C.
Concerning the strong and weak convergence of asymptotically nonexpansive
self or nonself mappings, relatively nonexpansive, quasi-φ-nonexpansive and quasi-φasymptotically nonexpansive self or nonself mappings have been considered extensively by
several authors in the setting of Hilbert or Banach spaces see e.g., 2–19.
The purpose of this paper is to modify the Halpern and Mann-type iteration algorithm
for a family of of total quasi-φ-asymptotically nonexpansive nonself mappings and to have
the strong convergence under removing FT is a convex set of condition and a limit
condition only in the framework of Banach spaces. As an application, we utilize our results
to study the approximation problem of solution to a system of equilibrium problems. The
results presented in the paper extend and improve the corresponding results of Chang et al.
4–7, W. P. Guo and W. Guo 8, Hao et al. 9, Kamimura and Takahashi 10, Kiziltunc
and Temir 11, Nilsrakoo and Saejung 2, Pathak et al. 12, Qin et al. 13, Su et al. 14,
Thianwan 15, Wang et al. 16, Yıldırım and Özdemir 17, Yang and Xie 18, Zegeye et al.
19, Kanjanasamranwong et al. 20, Saewan and Kumam 21–24 and Wattanawitoon and
Kumam 25.
2. Main Results
Theorem 2.1. Let E be a real uniformly convex and uniformly smooth Banach space, and C
be a nonempty closed convex subset E. Let Ti : C → E, i 1, 2, . . . be a family of closed
and uniformly total quasi-φ-asymptotically nonexpansive nonself mappings with nonnegative real
sequence {νn }, {μn } and a strictly increasing continuous function ρ : R → R such that
νn → 0, μn → 0 and ρ0 0, and for each i ≥ 1, Ti be uniformly Li -Lipschitz continuous.
Let {αn } be a sequence in 0, 1 and {βn } be a sequence in 0, 1 satisfying the following conditions:
a limn → ∞ αn 0;
b 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1.
Let {xn } be a sequence generated by
yn,i
x1 ∈ E chosen arbitrarily; C1 C,
J −1 αn Jx1 1 − αn βn Jxn 1 − βn JTi P Ti n−1 xn ,
Cn1 i ≥ 1,
2.1
z ∈ Cn : sup φ z, yn,i ≤ αn φz, x1 1 − αn φz, xn θn ,
i≥1
xn1 ΠCn1 x1 ,
∀n ≥ 1.
where θn νn supu∈F ρφu, xn μn , for all n ≥ 1, F :
subset in C, then {xn } converges strongly to ΠF x1 .
∞
i1
Proof. We divide the proof of Theorem 2.1 into five steps.
I F and Cn , n ≥ 1 are closed and convex subset in C.
FTi . If F is a nonempty-bounded
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In fact, it follows from Lemma 1.6 that FTi , i ≥ 1 is closed and convex subset of C.
Therefore F is a closed and convex subset in C.
Again by the assumption that C1 C is closed and convex. Suppose that Cn is closed
and convex for some n ≥ 2. In view of the definition of φ we have that
Cn1 z ∈ Cn : sup φ z, yn,i ≤ αn φz, x1 1 − αn φz, xn θn
i≥1
z ∈ Cn : φ z, yn,i ≤ αn φz, x1 1 − αn φz, xn θn ∩ Cn
i≥1
z ∈ Cn : 2αn z, Jx1 21 − αn z, Jxn − 2 z, Jyn,i
i≥1
2.2
2
≤ αn x1 2 1 − αn xn 2 − yn,i θn ∩ Cn .
This implies that Cn1 is closed and convex. The conclusion is proved.
II Now we prove that F ⊂ Cn , n ≥ 1.
In fact, it is obvious that F ⊂ C1 C. Suppose that F ⊂ Cn for some n ≥ 2. Letting
wn,i J −1 βn Jxn 1 − βn JTi P Ti n−1 xn ,
2.3
it follows from 1.6 that for any u ∈ F ⊂ Cn we have
φ u, yn,i φ u, J −1 αn Jx1 1 − αn Jwn,i 2.4
≤ αn φu, x1 1 − αn φu, wn,i ,
φu, wn,i φ u, J −1 βn Jxn 1 − βn JTi P Ti n−1 xn
≤ βn φu, xn 1 − βn φ u, Ti P Ti n−1 xn
≤ βn φu, xn 1 − βn φu, xn νn ρ φu, xn μn
φu, xn 1 − βn νn ρ φu, xn μn
≤ φu, xn νn ρ φu, xn μn ,
2.5
therefore we have
supφ u, yn,i ≤ αn φu, x1 1 − αn φu, xn νn ρ φu, xn μn
i≥1
≤ αn φu, x1 1 − αn φu, xn νn sup ρ φu, xn μn
u∈F
≤ αn φu, x1 1 − αn φu, xn θn ,
2.6
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where θn νn supu∈F ρφu, xn μn . This shows that u ∈ Cn1 , and so F ⊂ Cn1 . The
conclusion is proved.
III Next we prove that {xn } is a Cauchy sequence in C.
In fact, since xn ΠCn x1 , from Lemma 1.12 we have
xn − y, Jx1 − Jxn ≥ 0,
∀y ∈ Cn .
2.7
∀u ∈ F.
2.8
Again since F ⊂ Cn , for all n ≥ 1, we have
xn − u, Jx1 − Jxn ≥ 0,
It follows from Lemma 1.11 that for each u ∈ F and for each n ≥ 1
φxn , x1 φΠCn x1 , x1 ≤ φu, x1 − φu, xn ≤ φu, x1 .
2.9
Therefore {φxn , x1 } is bounded. By virtue of 1.5, {xn } is also bounded.
Since xn ΠCn x1 and xn1 ΠCn1 x1 ∈ Cn1 ⊂ Cn , we have φxn , x1 ≤ φxn1 , x1 ,
for all n ≥ 1. This implies that {φxn , x1 } is nondecreasing. Hence the limit limn → ∞ φxn , x1 exists. By the construction of Cn , for any positive integer m ≥ n, we have Cm ⊂ Cn and
xm ΠCm x1 ∈ Cn . This shows that
φxm , xn φxm , ΠCn x1 ≤ φxm , x1 − φxn , x1 −→ 0,
as n, m −→ ∞.
2.10
It follows from Lemma 1.5 that limm,n → ∞ ||xm − xn || 0. Hence {xn } is a Cauchy sequence
in C. Since C is a nonempty closed subset of Banach space E, it is complete, without loss of
generality, we can assume that xn → x∗ n → ∞.
By the assumption, it is easy to see that
lim θn lim
n→∞
n→∞
νn sup ρ φu, xn μn
u∈F
0.
2.11
IV Now we prove that x∗ ∈ F.
In fact, since xn1 ∈ Cn1 and αn → 0, it follows from 2.1 and 2.11 that
supφ xn1 , yn,i ≤ αn φxn1 , x1 1 − αn φxn1 , xn θn −→ 0 as n −→ ∞.
i≥1
2.12
Since xn → x∗ , by virtue of Lemma 1.5 for each i ≥ 1,we have
lim yn,i x∗ .
n→∞
2.13
Since {xn } is bounded, {Ti }∞
i1 is uniformly total quasi-φ-asymptotically nonexpansive nonself
mappings with nonnegative real sequence {νn }, {μn } and a strictly increasing continuous
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Journal of Applied Mathematics
function ρ : R → R such that νn → 0, μn → 0, and ρ0 0, for any given u ∈ F, we
have
φ u, Ti P Ti n−1 xn ≤ φu, xn νn ρ φu, xn μn .
2.14
This implies that {Ti P Ti n−1 xn } is uniformly bounded. Since
wn,i J −1 βn Jxn 1 − βn JTi P Ti n−1 xn ≤ βn xn 1 − βn Ti P Ti n−1 xn 2.15
≤ xn Ti P Ti n−1 xn .
This implies that {wn,i } is also uniformly bounded.
Since αn → 0, from 2.1, for each i ≥ 1 we have
lim Jyn,i − Jwn,i lim αn Jx1 − Jwn,i 0.
n→∞
n→∞
2.16
Since J −1 is uniformly continuous on each bounded subset of E∗ , it follows from 2.13 and
2.16 that
lim wn,i x∗
n→∞
for each i ≥ 1.
2.17
Since J is uniformly continuous on each bounded subset of E, we have
0 lim Jwn,i − Jx∗ n→∞
lim βn Jxn 1 − βn JTi P Ti n−1 xn − Jx∗ n→∞
lim βn Jxn − Jx∗ 1 − βn JTi P Ti n−1 xn − Jx∗ 2.18
n→∞
lim 1 − βn JTi P Ti n−1 xn − Jx∗ .
n→∞
By condition b, we have that
lim JTi P Ti n−1 xn − Jx∗ 0.
n→∞
2.19
Since J is uniformly continuous, this shows that limn → ∞ Ti P Ti n−1 xn x∗ uniformly in i ≥ 1.
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Again by the assumptions that for each i ≥ 1, Ti is uniformly Li -Lipschitz continuous,
thus we have
Ti P Ti n xn − Ti P Ti n−1 xn ≤ Ti P Ti n xn − Ti P Ti n xn1 Ti P Ti n xn1 − xn1 xn1 − xn xn − Ti P Ti n−1 xn 2.20
≤ Li 1xn − xn1 Ti P Ti n xn1 − xn1 xn − Ti P Ti n−1 xn .
Since limn → ∞ Ti P Ti n−1 xn x∗ and xn → x∗ , these together with 2.20 imply that
limn → ∞ ||Ti P Ti n xn − Ti P Ti n−1 xn || 0 and limn → ∞ Ti P Ti n xn x∗ , that is,
lim Ti P P Ti n−1 xn x∗ .
2.21
n→∞
In view continuity of Ti P , it yields that Ti P x∗ x∗ . Since x∗ ∈ C, P x∗ x∗ . This shows that
T x∗ x∗ . By the arbitrariness of i ≥ 1, we have x∗ ∈ F.
V Finally we prove that xn → x∗ ΠF x1 .
Let w ΠF x1 . Since w ∈ F ⊂ Cn and xn ΠCn x1 , we have φxn , x1 ≤ φw, x1 , for all
n ≥ 1. This implies that
φx∗ , x1 lim φxn , x1 ≤ φw, x1 .
2.22
n→∞
In view of the definition of ΠF x1 , from 2.22 we have x∗ w. Therefore xn → x∗ ΠF x1 .
This completes the proof of Theorem 2.1.
Theorem 2.2. Let E, C, {αn }, {βn } be the same as in Theorem 2.1. Let Ti : C → E, i 1, 2, . . . be a
family of closed and uniformly quasi-φ-asymptotically nonexpansive nonself mappings with sequence
{kn } ⊂ 1, ∞, kn → 1, and for each i ≥ 1, Ti be uniformly Li -Lipschitz continuous. Let {xn } be a
sequence generated by
yn,i
x1 ∈ E chosen arbitrarily; C1 C,
J −1 αn Jx1 1 − αn βn Jxn 1 − βn JTi P Ti n−1 xn ,
Cn1 i ≥ 1,
2.23
z ∈ Cn : supφ z, yn,i ≤ αn φz, x1 1 − αn φz, xn θn ,
i≥1
xn1 ΠCn1 x1 ,
where θn kn − 1supu∈F φu, xn , F :
{xn } converges strongly to ΠF x1 .
∞
i1
∀n ≥ 1,
FTi . If F is a nonempty bounded subset in C, then
Proof. By Remark 1.4 Ti : C → E, i 1, 2, . . . be a family of closed and uniformly quasiφ-asymptotically nonexpansive nonself mappings that it is a family of closed and uniformly
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Journal of Applied Mathematics
total quasi-φ-asymptotically nonexpansive nonself mappings with taking ρt t, t > 0,
νn kn − 1 and μn 0. Therefore all conditions in Theorem 2.1 are satisfied. By the similar
methods as given in the proof of Theorem 2.1, we can prove that the sequence {xn } defined
by 2.23 converges strongly to ΠF x1 .
Theorem 2.3. Let E, C, {αn }, {βn } be the same as in Theorem 2.2. Let Ti : C → E, i 1, 2, . . . be a
∅ and for each i ≥ 1, Ti
family of quasi-φ-nonexpansive nonself mappings such that F : ∞
i1 FTi /
be uniformly Li -Lipschitz continuous. Let {xn } be a sequence generated by
x1 ∈ E chosen arbitrarily; C1 C,
yn,i J −1 αn Jx1 1 − αn βn Jxn 1 − βn JTi xn , i ≥ 1,
Cn1 z ∈ Cn : supφ z, yn,i ≤ αn φz, x1 1 − αn φz, xn ,
2.24
i≥1
xn1 ΠCn1 x1 ,
∀n ≥ 1.
Then {xn } converges strongly to ΠF x1 .
Proof. By Remark 1.4 Ti : C → E, i 1, 2, . . . be a family of quasi-φ-nonexpansive nonself
mappings that it is a family of uniformly quasi-φ-asymptotically nonexpansive nonself
mappings with sequence {kn } {1}. Hence θn kn − 1supu∈F φu, xn 0 Therefore
all conditions in Theorem 2.2 are satisfied. By the similar methods, we can prove that the
sequence {xn } defined by 2.24 converges strongly to ΠF x1 .
3. Application and Example
In this section we utilize the results presented in Section 2 to prove a strong convergence
theorem concerning maximal monotone operators in Hilbert spaces.
Let E be a real Hilbert space and let A be a maximal monotone operator from E to E.
For each r > 0, we can define a single valued mapping JrA : E → E by JrA I rA−1 and
such a mapping JrA is called the resolvent of A. It is easy to prove that JrA is a nonexpansive
mapping and A−1 0 FJrA for all r > 0. Therefore it is a uniformly 1-Lipschitz continuous
and quasi-φ-nonexpansive mapping. Hence for each p ∈ FJrA and w ∈ E, we have
φ p, JrA w ≤ φ p, w ,
3.1
and FJrA A−1 0. These show that all conditions in Theorem 2.3 are satisfied. Hence from
Theorem 2.3 we have the following.
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11
Theorem 3.1. Let E be a real Hilbert space. Let A1 , A2 be two maximal monotone operators from E
−1
∅. Let JrA1 and JrA2 be the resolvent of A1 and A2 , respectively,
to E such that F A−1
1 0 ∩ A2 0 /
where r > 0. Let {αn }, {βn } be the same as in Theorem 2.3 and {xn } be the sequence defined by
yn,i
x1 ∈ E chosen arbitrarily; C1 E,
J −1 αn Jx1 1 − αn βn Jxn 1 − βn JJrAi xn ,
Cn1 i 1, 2,
z ∈ Cn : maxφ z, yn,i ≤ αn φz, x1 1 − αn φz, xn ,
3.2
i1,2
xn1 PCn1 x1 ,
∀n ≥ 1,
where PC is the metric projection from H onto the subset C ⊂ H. Then the sequence {xn } defined by
3.2 converges strongly to PF x1 .
Acknowledgments
The authors would like to express their thanks to the referees and the Editor for their helpful
comments and suggestions. This Project was supported by the Scientific Research Fund of
SiChuan Provincial Education Department no. 11ZA222.
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