Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 575014, 13 pages
doi:10.1155/2012/575014
Research Article
Strong Convergence Theorems for the Generalized
Split Common Fixed Point Problem
Cuijie Zhang
College of Science, Civil Aviation University of China, Tianjin 300300, China
Correspondence should be addressed to Cuijie Zhang, zhang cui jie@126.com
Received 9 January 2012; Accepted 17 February 2012
Academic Editor: Rudong Chen
Copyright q 2012 Cuijie Zhang. 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 the generalized split common fixed point problem GSCFPP and show that the
GSCFPP for nonexpansive operators is equivalent to the common fixed point problem. Moreover,
we introduce a new iterative algorithm for finding a solution of the GSCFPP and obtain some
strong convergence theorems under suitable assumptions.
1. Introduction
Let H1 and H2 be real Hilbert spaces and let A : H1 → H2 be a bounded linear operator.
Given intergers p, r ≥ 1, let us recall that the multiple-set split feasibility problem MSSFP
was recently introduced 1 and is to find a point:
x∗ ∈
p
Ci ,
i1
Ax∗ ∈
r
j1
Qj ,
1.1
p
where {Ci }i1 and {Qj }rj1 are nonempty closed convex subsets of H1 and H2 , respectively. If
p r 1, the MSSFP 1.1 becomes the so-called split feasibility problem SFP 2 which is
to find a point:
x∗ ∈ C,
Ax∗ ∈ Q,
1.2
where C and Q are nonempty closed convex subsets of H1 and H2 , respectively. Recently, the
SFP 1.2 and MSSFP 1.1 have been investigated by many researchers; see, 3–10.
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Since every closed convex subset in a Hilbert space is looked as the fixed point set of
its associating projection, the MSSFP 1.1 becomes a special case of the split common fixed
point problem SCFPP, which is to find a point:
p
x∗ ∈
FixUi ,
Ax∗ ∈
i1
r
Fix Tj ,
j1
1.3
where Ui : H1 → H1 i 1, 2, . . . , p and Tj : H2 → H2 j 1, 2, . . . , r are nonlinear
operators. If p r 1, the problem 1.3 reduces to the so-called two-set SCFPP, which is to
find a point:
x∗ ∈ FixU,
Ax∗ ∈ FixT .
1.4
Censor and Segal in 11 firstly introduced the concept of SCFPP in finite-dimensional
Hilbert spaces and considered the following iterative algorithm for the two-set SCFPP 1.4
for Class- operators:
xn1 U xn − γA∗ I − T Axn ,
n ≥ 0,
1.5
where x0 ∈ H1 , 0 < γ < 2/A2 and I is the identity operator. They proved the convergence
of the algorithm 1.5 to a solution of problem 1.4. Moreover, they introduced a parallel
iterative algorithm, which converges to a solution of the SCFPP 1.3. However, the parallel
iterative algorithm does not include the algorithm 1.5 as a special case.
Very recently, Wang and Xu in 12 considered the SCFPP 1.3 for Class- operators
and introduced the following iterative algorithm for solving the SCFPP 1.3:
xn1 Un xn − γA∗ I − Tn Axn ,
n ≥ 0.
1.6
Under some mild conditions, they proved some weak and strong convergence theorems.
Their iterative algorithm 1.6 includes Censor and Segal’s algorithm 1.5 as a special case
for the two-set SCFPP 1.4. Moreover, they prove that the SCFPP 1.3 for the Class-
operators is equivalent to a common fixed point problem. This is also a classical method.
Many problems eventually converted to a common fixed point problem; see 13–15.
Motivated and inspired by the aforementioned research works, we introduce a
generalized split common fixed point problem GSCFPP which is to find a point:
x∗ ∈
∞
FixUi ,
Ax∗ ∈
i1
∞
Fix Tj .
j1
1.7
Then, we show that the GSCFPP 1.7 for nonexpansive operators is equivalent to the
following common fixed point problem:
x∗ ∈
∞
i1
FixUi ,
x∗ ∈
∞
j1
Fix Vj ,
1.8
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3
where Vj I − γA∗ I − Tj A 0 < γ ≤ 1/A2 for every j ∈ N. Moreover, we give a new
iterative algorithm for solving the GSCFPP 1.7 for nonexpansive operators and obtain some
strong convergence theorems.
2. Preliminaries
Throughout this paper, we write xn x and xn → x to indicate that {xn } converges weakly
to x and converges strongly to x, respectively.
An operator T : H → H is said to be nonexpansive if T x − T y ≤ x − y for all
x, y ∈ H. The set of fixed points of T is denoted by FT . It is known that FT is closed and
convex. An operator f : H → H is called contraction if there exists a constant ρ ∈ 0, 1 such
that fx − fy ≤ ρx − y for all x, y ∈ H. Let C be a nonempty closed convex subset
of H. For each x ∈ H, there exists a unique nearest point in C, denoted by PC x, such that
x − PC x ≤ x − y for every y ∈ C. PC is called a metric projection of H onto C. It is known
that for each x ∈ H,
x − PC x, y − PC x ≤ 0
2.1
for all y ∈ C.
Let {Tn } be a sequence of operators of H into itself. The set of common fixed points
of {Tn } is denoted by F{Tn }, that is, F{Tn } ∞
n1 FTn . A sequence {Tn } is said to be
strongly nonexpansive if each {Tn } is nonexpansive and
xn − yn − Tn xn − Tn yn −→ 0
2.2
whenever {xn } and {yn } are sequences in C such that {xn − yn } is bounded and xn − yn −
Tn xn − Tn yn → 0; see 16, 17. A sequence {zn } in H is said to be an approximate fixed
point sequence of {Tn } if zn − Tn zn → 0. The set of all bounded approximate fixed point
n }; see 16, 17. We know that if {Tn } has a common
sequences of {Tn } is denoted by F{T
n } is nonempty; that is, every bounded sequence in the common fixed
fixed point, then F{T
point set is an approximate fixed point sequence. A sequence {Tn } with a common fixed point
is said to satisfy the condition Z if every weak cluster point of {xn } is a common fixed point
n }. A sequence {Tn } of nonexpansive mappings of H into itself is said
whenever {xn } ∈ F{T
to satisfy the condition R if
lim supTn1 y − Tn y 0
n → ∞ y∈D
2.3
for every nonempty bounded subset D of H; see 18.
In order to prove our main results, we collect the following lemmas in this section.
Lemma 2.1 see 16. Let C be a nonempty subset of a Hilbert space H. Let {Tn } be a sequence of
nonexpansive mappings of C into H. Let {λn } be a sequence in 0, 1 such that lim infn → ∞ λn > 0.
Let {Un } be a sequence of mappings of C into H defined by Un λn I 1 − λn Tn for n ∈ N, where
I is the identity mapping on C. Then {Un } is a strongly nonexpansive sequence.
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Journal of Applied Mathematics
Lemma 2.2 see 16. Let H be a Hilbert space, C a nonempty subset of H, and {Sn } and {Tn }
sequences of nonexpansive self-mappings of C. Suppose that {Sn } or {Tn } is a strongly nonexpansive
sequence and F{S
n } ∩ F{Tn } is nonempty. Then F{Sn } ∩ F{Tn } F{Sn Tn }.
Lemma 2.3 see 17. Let H be a Hilbert space, and C a nonempty subset of H. Both {Sn } and
{Tn } satisfy the condition R and {Tn y : n ∈ N, y ∈ D} is bounded for any bounded subset D of C.
Then {Sn Tn } satisfies the condition R.
Lemma 2.4 see 19. Let {xn } and {yn } be bounded sequences in a Banach space X and let {βn }
be a sequence in 0, 1 with 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1. Suppose xn1 1 − βn yn βn xn for all integers n ≥ 0 and
lim sup yn1 − yn − xn1 − xn ≤ 0.
n→∞
2.4
Then limn → ∞ yn − xn 0.
Lemma 2.5 see 20. Assume that {an } is a sequence of nonnegative real numbers such that
an1 ≤ 1 − γn an δn ,
n ≥ 0,
2.5
where {γn } is a sequence in 0, 1 and {δn } is a sequence such that
i
∞
n1 γn
∞,
ii lim supn → ∞ δn /γn ≤ 0 or
∞
n1
|δn | < ∞.
Then limn → ∞ an 0.
3. Main Results
Now we state and prove our main results of this paper.
Lemma 3.1. Let A : H1 → H2 be a given bounded linear operator and let Tn : H2 → H2 be a
sequence of nonexpansive operators. Assume
A−1 Fix{Tn } {x ∈ H1 : Ax ∈ Fix{Tn }} /
∅.
3.1
For each constant γ > 0, Vn is defined by the following:
Vn I − γA∗ I − Tn A.
3.2
Then Fix{Vn } A−1 Fix{Tn }. Moreover, for 0 < γ ≤ 1/A2 , Vn is nonexpansive on H1 for
n ∈ N.
Journal of Applied Mathematics
5
Proof. Since the inclusion A−1 Fix{Tn } ⊆ Fix{Vn } is evident, now we only need to
show the converse inclusion. If z ∈ Fix{Vn }, then we have A∗ I − Tn Az 0. Since
∅, we take an arbitrary p ∈ A−1 Fix{Tn }. Hence
A−1 Fix{Tn } /
Az − Tn Az2 Az − Tn Az, Az − Tn Az
Az − Tn Az, Az − Ap Ap − Tn Az
A∗ I − Tn Az, z − p Az − Tn Az, Ap − Tn Az
2 1
1
− Az − Ap Az − Tn Az2 2
2
≤
1
Ap − Tn Az2
2
3.3
1
Az − Tn Az2 .
2
It follows that 1/2Az − Tn Az2 ≤ 0, then Az Tn Az for every n ∈ N, hence z ∈
A−1 Fix{Tn }. Next we turn to show that Vn is a nonexpansive operator for n ∈ N. Since
Tn is nonexpansive, we have
I − Tn Ax − I − Tn Ay2 Ax − Ay2 Tn Ax − Tn Ay2 − 2 Ax − Ay, Tn Ax − Tn Ay
2
≤ 2Ax − Ay − 2 Ax − Ay, Tn Ax − Tn Ay
≤ 2 Ax − Ay, Ax − Ay − Tn Ax − Tn Ay .
3.4
Hence
Vn x − Vn y2 I − γA∗ I − Tn A x − I − γA∗ I − Tn Ay2
2
2
x − y γ 2 A2 I − Tn Ax − I − Tn Ay
− 2γ Ax − Ay, I − Tn Ax − I − Tn Ay
2
2
≤ x − y γ γA2 − 1 I − Tn Ax − I − Tn Ay .
3.5
For 0 < γ ≤ 1/A2 , we can immediately obtain that Vn is a nonexpansive operator for every
n ∈ N.
From Lemma 3.1, we can obtain that the solution set of GSCFPP 1.7 is identical to the
solution set of problem 1.8.
Theorem 3.2. Let {Un } and {Vn } be sequences of nonexpansive operators on Hilbert space H1 . Both
{Un } and {Vn } satisfy the conditions R and Z. Let f : H1 → H1 be a contraction with coefficient
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Journal of Applied Mathematics
ρ ∈ 0, 1. Suppose Ω FixUn {xn } by the following algorithm:
FixVn /
∅. Take an initial guess x1 ∈ H1 and define a sequence
yn λn xn 1 − λn Vn xn ,
xn1 αn fxn βn xn γn Un yn ,
3.6
where {αn }, {βn }, {γn }, and {λn } are sequences in 0, 1. If the following conditions are satisfied:
i αn βn γn 1, for all n ≥ 1;
ii limn → ∞ αn 0 and Σ∞
n1 αn ∞;
iii 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1;
iv 0 < lim infn → ∞ λn ≤ lim supn → ∞ λn < 1;
v limn → ∞ |λn1 − λn | 0,
then {xn } converges strongly to w ∈ Ω where w PΩ fw.
Proof. We proceed with the following steps.
Step 1. First show that there exists w ∈ Ω such that w PΩ fw.
In fact, since f is a contraction with coefficient ρ, we have
PΩ fx − PΩ f y ≤ fx − f y ≤ ρx − y
3.7
for every x, y. Hence PΩ f is also a contraction. Therefore, there exists a unique w ∈ Ω such
that w PΩ fw.
Step 2. Now we show that {xn } is bounded.
Let p ∈ Ω, then p ∈ Fix{Un } and p ∈ Fix{Vn }. Hence
Un yn − p ≤ yn − p ≤ λn xn − p 1 − λn Vn xn − p ≤ xn − p.
3.8
Then
xn1 − p ≤ αn fxn − p βn xn − p γn Un yn − p
≤ αn ρxn − p αn f p − p βn xn − p γn xn − p
1 f p − p
≤ 1 − αn 1 − ρ xn − p αn 1 − ρ
1−ρ
1 f p − p .
≤ max xn − p,
1−ρ
3.9
By induction on n,
Vn xn − p ≤ xn − p ≤ max x1 − p,
1 f p − p
1−ρ
3.10
Journal of Applied Mathematics
7
for every n ∈ N. This shows that {xn } and {Vn xn } are bounded, and hence, {Un yn }, {yn }, and
{fxn } are also bounded.
Step 3. We claim that F{A
n } F{Vn } and F{Un An } F{Un } ∩ F{Vn }, where An λn I 1 − λn Vn .
We first show the former equality. Let {zn } be a bounded sequence in H1 . If {zn } ∈
F{Vn }, then
An zn − zn λn zn 1 − λn Vn zn − zn 1 − λn Vn zn − zn −→ 0.
3.11
Hence {zn } ∈ F{A
n }. On the other hand, if {zn } ∈ F{An }, combining 3.11 and
lim supn → ∞ λn < 1, we obtain that Vn zn − zn → 0. Hence {zn } ∈ F{V
n }. Therefore,
F{An } F{Vn }.
Next, we show the latter equality. Using Lemma 2.1, we know that {An } is a strongly
∅, from
nonexpansive sequence. Thus, since F{U
n } ∩ F{An } F{Un } ∩ F{Vn } /
Lemma 2.2 we have
F{U
n An } F{Un } ∩ F{An } F{Un } ∩ F{Vn }.
3.12
Step 4. {Sn } satisfies the condition R, where Sn Un An .
Let D be a nonempty bounded subset of H1 . From the definition of {An }, we have, for
all y ∈ D,
An1 y − An y λn1 y 1 − λn1 Vn1 y − λn y − 1 − λn Vn y
≤ |λn1 − λn |y Vn1 y − Vn y λn1 Vn1 y − λn Vn y
≤ |λn1 − λn |y Vn1 y − Vn y λn1 Vn1 y − λn Vn1 y
λn Vn1 y − λn Vn y
|λn1 − λn |y Vn1 y − Vn y |λn1 − λn |Vn1 y
λn Vn1 y − Vn y
|λn1 − λn | y Vn1 y 1 λn Vn1 y − Vn y.
3.13
It follows that
supAn1 y − An y ≤ |λn1 − λn |sup y Vn1 y 1 λn supVn1 y − Vn y.
y∈D
y∈D
y∈D
3.14
Since {Vn } satisfies the condition R and limn → ∞ |λn1 − λn | 0, we have
lim supAn1 y − An y 0,
n → ∞ y∈D
3.15
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that is, {An } satisfies the condition R. Since {An y : n ∈ N, y ∈ D} is bounded for any
bounded subset D of H1 , by using Lemma 2.3, we have that {Vn An } satisfies the condition
R, that is, {Sn } satisfies the condition R.
Step 5. We show xn1 − xn → 0.
We can write 3.6 as xn1 βn xn 1 − βn zn where zn αn fxn γn Sn xn /1 − βn . It
follows that
αn1 fxn1 γn1 Sn1 xn1 αn fxn γn Sn xn
−
1 − βn1
1 − βn
αn1 αn
αn1
−
fxn fxn1 − fxn 1 − βn1
1 − βn1 1 − βn
γn
γn1
γn1
−
Sn xn .
Sn1 xn1 − Sn xn 1 − βn1
1 − βn1 1 − βn
zn1 − zn 3.16
From Step 2, we may assume that {xn } ⊂ D , where D is a bounded set of H1 . Then from
3.16, we obtain
αn1
αn fxn Sn xn αn1 ρxn1 − xn zn1 − zn ≤ −
1 − βn1 1 − βn
1 − βn1
γn1
γn1
Sn1 xn1 − Sn xn1 Sn xn1 − Sn xn 1 − βn1
1 − βn1
αn1
αn fxn Sn xn 1 − αn1 1 − ρ xn1 − xn ≤
−
1 − βn1 1 − βn 1 − βn1
γn1
supSn1 y − Sn y.
1 − βn1 y∈D
3.17
It follows that
αn1
αn fxn Sn xn −
zn1 − zn − xn1 − xn ≤ 1 − βn1 1 − βn
γn1
αn1 supSn1 y − Sn y −
1 − ρ xn1 − xn .
1 − βn1 y∈D
1 − βn1
3.18
Since {Sn } satisfies the condition R, combining αn → 0 as n → ∞, we have
lim supzn1 − zn − xn1 − xn ≤ 0.
n→∞
3.19
Hence by Lemma 2.4, we get zn − xn → 0 as n → ∞. Consequently,
lim xn1 − xn lim 1 − βn zn − xn 0.
n→∞
n→∞
3.20
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9
Step 6. We claim that {xn } ∈ F{U
n } ∩ F{Vn }.
From 3.6, we have
Sn xn − xn ≤ Sn xn − xn1 xn1 − xn Sn xn − αn fxn − βn xn − γn Sn xn xn1 − xn ≤ αn Sn xn − fxn βn Sn xn − xn xn1 − xn ,
3.21
1 − βn Sn xn − xn ≤ αn Sn xn − fxn xn1 − xn .
3.22
and hence
Since xn1 − xn → 0, αn → 0 and lim supn → ∞ βn < 1, we derive
Sn xn − xn −→ 0.
3.23
Thus 3.23 and Steps 2 and 3 imply that
{xn } ∈ F{S
n } F{Un } ∩ F{Vn }.
3.24
Step 7. Show lim supn → ∞ fw − w, xn − w ≤ 0, where w PΩ fw.
Since {xn } is bounded, there exist a point v ∈ H1 and a subsequence {xni } of {xn } such
that
lim sup fw − w, xn − w lim fw − w, xni − w
n→∞
i→∞
3.25
and xni v. Since {Un } and {Vn } satisfy the condition Z, from Step 6, we have v ∈
F{Un } ∩ F{Vn }. Using 2.1, we get
lim sup fw − w, xn − w lim fw − w, xni − w
n→∞
Step 8. Show xn → w PΩ fw.
i→∞
fw − w, v − w ≤ 0.
3.26
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Journal of Applied Mathematics
Since w ∈ Ω, using 3.8, we have
xn1 − w2 αn fxn − w βn xn − w γn Un yn − w , xn1 − w
≤ αn fxn − fw, xn1 − w αn fw − w, xn1 − w
βn xn − w · xn1 − w γn yn − w · xn1 − w
≤
≤
1
αn ρ xn − w2 xn1 − w2 αn fw − w, xn1 − w
2
1 1 βn xn − w2 xn1 − w2 γn xn − w2 xn1 − w2
2
2
3.27
1
1
1 − αn 1 − ρ xn − w2 xn1 − w2 αn fw − w, xn1 − w ,
2
2
which implies that
1 fw − w, xn1 − w ,
xn1 − w2 ≤ 1 − αn 1 − ρ xn − w2 2αn 1 − ρ
1−ρ
3.28
for every n ∈ N. Consequently, according to Step 7, ρ ∈ 0, 1, and Lemma 2.5, we deduce that
{xn } converges strongly to w PΩ w. This completes the proof.
Combining Lemma 3.1 and Theorem 3.2, we can obtain the following strong convergence theorem for solving the GSCFPP 1.7.
Theorem 3.3. Let {Un } and {Tn } be sequences of nonexpansive operators on Hilbert space H1 and
H2 , respectively. Both {Un } and {Tn } satisfy the conditions R and Z. Let f : H1 → H1 be a
contraction with coefficient ρ ∈ 0, 1. Suppose that the solution set Ω of GSCFPP 1.7 is nonempty.
Take an initial guess x1 ∈ H1 and define a sequence {xn } by the following algorithm:
yn xn − γ1 − λn A∗ I − Tn Axn ,
xn1 αn fxn βn xn γn Un yn ,
3.29
where γ ∈ 0, 1/A2 , and {αn }, {βn }, {γn }, {λn } are sequences in 0, 1. If the following conditions
are satisfied:
i αn βn γn 1, for all n ≥ 1;
ii limn → ∞ αn 0 and Σ∞
n1 αn ∞;
iii 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1;
iv 0 < lim infn → ∞ λn ≤ lim supn → ∞ λn < 1;
v limn → ∞ |λn1 − λn | 0,
then {xn } converges strongly to w ∈ Ω where w PΩ fw.
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Proof. Set Vn I −γA∗ I −Tn A. By Lemma 3.1, Vn is a nonexpansive operator for every n ∈ N.
We can rewrite 3.29 as
yn λn xn 1 − λn Vn xn ,
3.30
xn1 αn fxn βn xn γn Un yn .
We only need to prove that {Vn } satisfies the conditions R and Z. Assume that D
is a nonempty bounded subset of H1 . For every y ∈ D, we have
I − γA∗ I − Tn1 A y − I − γA∗ I − Tn A y ≤ γ A∗ I − Tn1 Ay − A∗ I − Tn Ay
≤ γATn1 Ay − Tn Ay .
3.31
Since {Tn } satisfies the condition R, and D {Ay : y ∈ D} is bounded, it follows from
3.31 that
sup I − γA∗ I − Tn1 A y − I − γA∗ I − Tn A y ≤ γAsupTn1 Ay − Tn Ay y∈D
y∈D
γAsupTn1 z − Tn z −→ 0.
z∈D
3.32
Therefore, {Vn } satisfies the condition R.
Assume that xn z and xn − Vn xn → 0; we next show that Vn z z. By using xn −
Vn xn → 0, we have A∗ I − Tn Axn → 0. Since A−1 Fix{Tn } / ∅, we choose an arbitrary
−1
point p ∈ A Fix{Tn }; then for every n ∈ N,
Axn − Tn Axn 2 Axn − Tn Axn , Axn − Ap Ap − Tn Axn
A∗ I − Tn Axn , xn − p Axn − Tn Axn , Ap − Tn Axn
2 1
1
A∗ I − Tn Axn , xn − p − Axn − Ap Axn − Tn Axn 2
2
2
1
2
Ap − Tn Axn 2
3.33
1
≤ A∗ I − Tn Axn , xn − p Axn − Tn Axn 2 .
2
Hence
1
Axn − Tn Axn 2 ≤ A∗ I − Tn Axn , xn − p −→ 0.
2
3.34
n }. Since {Tn } satisfies the condition Z and Axn Az, we have
Then we get Axn ∈ F{T
Az ∈ F{Tn }. From Lemma 3.1, we have z ∈ Fix{Vn }.
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Journal of Applied Mathematics
Let T : H → H be a nonexpansive mapping with a fixed point, and define Tn T for
all n ∈ N. Then {Tn } satisfies the conditions R and Z. Thus, one obtains the algorithm for
solving the two-set SCFPP 1.4.
Corollary 3.4. Let U and T be nonexpansive operators on Hilbert space H1 and H2 , respectively.
Let f : H1 → H1 be a contraction with coefficient ρ ∈ 0, 1. Suppose that the solution set Ω of
SCFPP 1.4 is nonempty. Take an initial guess x1 ∈ H1 and define a sequence {xn } by the following
algorithm in 3.29, where γ ∈ 0, 1/A2 , and {αn }, {βn }, {γn }, {λn } are sequences in 0, 1. If the
following conditions are satisfied:
i αn βn γn 1, for all n ≥ 1;
ii limn → ∞ αn 0 and Σ∞
n1 αn ∞;
iii 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1;
iv 0 < lim infn → ∞ λn ≤ lim supn → ∞ λn < 1;
v limn → ∞ |λn1 − λn | 0.
Then {xn } converges strongly to w ∈ Ω where w PΩ fw.
Remark 3.5. By adding more operators to the families {Un } and {Tn } by setting Ui I for
i ≥ p 1 and Tj I for j ≥ r 1, the SCFPP 1.3 can be viewed as a special case of the GSCFPP
1.7.
Acknowledgment
This research is supported by the science research foundation program in Civil Aviation
University of China 07kys09, the Fundamental Research Funds for the Central Universities
Program No. ZXH2011D005, and the NSFC Tianyuan Youth Foundation of Mathematics of
China No. 11126136.
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