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J. Nonlinear Sci. Appl. 5 (2012), 466–474
Research Article
Hybrid projection algorithms for approximating fixed
points of asymptotically quasi-pseudocontractive
mappings
Shin Min Kanga , Sun Young Chob,∗, Xiaolong Qinc
a
Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Korea.
Department of Mathematics, Gyeongsang National University, Jinju 660-701, Korea.
c
Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China.
b
Dedicated to George A Anastassiou on the occasion of his sixtieth birthday
Communicated by Professor R. Saadati
Abstract
The purpose of this paper is to modify Ishikawa iterative process to have strong convergence without any
compact assumptions for asymptotically quasi-pseudocontractive mappings in the framework of real Hilbert
spaces.
Keywords: Asymptotically pseudocontractive mapping; asymptotically nonexpansive mapping; fixed
point; hybrid projection algorithm.
2010 MSC: 47H09, 47J25.
1. Introduction and Preliminaries
Throughout this paper, we always assume that H is a real Hilbert space with inner product h·, ·i, and
norm k · k. Assume that C is a nonempty closed convex subset of H and T : C → C is a nonlinear mapping.
We use F (T ) to denote the set of fixed points of T .
T is said to be nonexpansive if
kT x − T yk ≤ kx − yk,
∗
∀x, y ∈ C.
Corresponding author
Email addresses: smkang@gnu.ac.kr (Shin Min Kang ), ooly61@yahoo.co.kr (Sun Young Cho ), qxlxajh@163.com
(Xiaolong Qin )
Received 2011-9-11
S.M. Kang, S.Y. Cho and X. Qin, J. Nonlinear Sci. Appl. 5 (2012), 466–474
467
T is said to be asymptotically nonexpansive [3] if there exists a sequence {kn } ⊂ [1, ∞) with limn→∞ kn =
1 such that
kT n x − T n yk ≤ kn kx − yk, ∀x, y ∈ C, n ≥ 1.
(1.1)
T is said to be asymptotically quasi-nonexpansive if F (T ) 6= ∅ and (1.1) holds for every x ∈ C but
y ∈ F (T ). We remark here that the class of asymptotically nonexpansive mappings was introduced by
Goebel and Kirk; see [3] for more details. They proved that, if C is a nonempty bounded closed convex
subset of a uniformly convex Banach space E, then every asymptotically nonexpansive self-mapping T on
C has a fixed point. Further, the set F (T ) of fixed points of T is closed and convex.
T is said to be pseudocontractive if
hT x − T y, x − yi ≤ kx − yk2 ,
∀x, y ∈ C.
T is said to be asymptotically pseudocontractive if there exists a sequence {kn } ⊂ [1, ∞) with limn→∞ kn =
1 such that
hT n x − T n y, x − yi ≤ kn kx − yk2 , ∀x, y ∈ C.
(1.2)
We remark here that the class of asymptotically pseudocontractive mappings was introduced by Schu; see
[16] for more details.
It is clear that (1.2) is equivalent to
kT n x − T n yk2 ≤ (2kn − 1)kx − yk2 + k(I − T n )x − (I − T n )yk2 ,
∀x, y ∈ C.
(1.3)
The class of asymptotically pseudocontractive mappings contains properly the class of asymptotically nonexpansive mappings as a subclass, which can be seen from the following example.
Example. ([15]) For x ∈ [0, 1], define a mapping T : [0, 1] → [0, 1] by
2
3
T x = (1 − x 3 ) 2 .
Then T is asymptotically pseudocontractive but it is not asymptotically nonexpansive.
T : C → C is said to be asymptotically quasi-pseudocontractive if F (T ) 6= ∅ and there exists a sequence
{kn } ⊂ [1, ∞) with limn→∞ kn = 1 such that
hT n x − p, x − pi ≤ kn kx − pk2 ,
∀x ∈ C, p ∈ F (T ).
(1.4)
It is clear that (1.4) is equivalent to
kT n x − pk2 ≤ (2kn − 1)kx − pk2 + kx − T n xk2 ,
∀x ∈ C, p ∈ F (T ).
(1.5)
In 1991, Schu [16] proved the following results for asymptotically pseudocontractive mappings in the
framework of Hilbert spaces.
Theorem Schu. Let C be a nonempty closed bounded convex subset of a Hilbert space H. Let L > 0
and T : C → C be completely continuous, uniformly L-Lipschitzian
and asymptotically pseudo-contractive
P∞
2 − 1) < ∞, {α } and {β } ⊂ [0, 1],
with sequence {kn } ⊂ [1, ∞), qn = 2kn − 1 for all n ≥ 1,
(q
n
n
n=1 n
√
≤ αn ≤ βn ≤ b for all n ≥ 1 and for some > 0 and some b ∈ (0, L−2 [ 1 + L2 − 1]). For given x1 ∈ K,
define a sequence {xn } in C by the following algorithm:
(
yn = (1 − βn )xn + βn T n xn ,
xn+1 = (1 − αn )xn + αn T n yn , ∀n ≥ 1.
Then {xn } converges strongly to some fixed point of T.
S.M. Kang, S.Y. Cho and X. Qin, J. Nonlinear Sci. Appl. 5 (2012), 466–474
468
Two classical iteration processes are often used to approximate a fixed point of a nonexpansive mapping
and its extensions. The first one was introduced by Mann [7], which is defined as follows:
(
x0 ∈ C arbitrary choosen,
(1.6)
xn+1 = (1 − αn )xn + αn T xn , ∀n ≥ 0,
where {αn } is a sequence in the interval (0, 1).
The second one was referred to as Ishikawa iteration process [4], which is defined recursively as follows:


x0 ∈ C arbitrary choosen,
(1.7)
yn = (1 − βn )xn + βn T xn ,


xn+1 = (1 − αn )xn + αn T yn , ∀n ≥ 0,
where {αn } and {βn } are sequences in the interval (0, 1).
But both (1.6) and (1.7) have only weak convergence, in general; see [2] and [19]. Reich [14] shows
that,
P∞ if E is a uniformly convex and has a Fréchet differentiable norm, and the sequence {αn } is such that
n=0 αn (1 − αn ) = ∞, then the sequence {xn } generated by the process (1.6) converges weakly to a point
in F (T ) (an extension of the results to the process (1.7) can be found in [19]). Therefore, many authors
have attempted to modify (1.6) and (1.7) to have strong convergence.
In 2006, Martinez-Yanes and Xu [9] modified (1.7) to have strong convergence by hybrid projection
algorithms in Hilbert spaces. To be more precise, They proved the following result.
Theorem MYX. Let C be a closed convex subset of a Hilbert space H and T : C → C be a nonexpansive
mapping such that F (T ) 6= ∅. Assume that {αn } and {βn } are sequences in [0, 1] such that αn ≤ 1 − δ for
some δ ∈ (0, 1] and βn → 1. Define a sequence {xn } in C by the following algorithm:


x0 ∈ C chosen arbitrarily,





zn = βn xn + (1 − βn )T xn ,



y = α x + (1 − α )T z ,
n
n n
n
n
2
Cn = {v ∈ C : kyn − vk ≤ kxn − vk2 + (1 − αn )(kzn k2 − kxn k2 + 2hxn − zn , vi)},





Qn = {v ∈ C : hx0 − xn , xn − vi ≥ 0},



x
n+1 = PCn ∩Qn x0 .
Then {xn } converges in norm to PF (T ) x0 .
Recently, Qin, Su and Shang [13] improved the results of Martinez-Yanes and Xu [9] from nonexpansive
mappings to asymptotically nonexpansive mappings. More precisely, They proved the following theorem.
Theorem QSS. Let C be a bounded closed convex subset of a Hilbert space H and T : C → C be an
asymptotically nonexpansive mapping with a sequence {kn } such that kn → 1 as n → ∞. Assume that {αn }
is a sequence in (0, 1) such that αn ≤ 1 − δ for all n and for some δ ∈ (0, 1] and βn → 1. Define a sequence
{xn } in C by the following algorithm:


x0 ∈ C chosen arbitrarily,





zn = βn xn + (1 − βn )T n xn ,



n


yn = αn xn + (1 − αn )T zn ,
Cn = {v ∈ C : kyn − vk2 ≤ kxn − vk2 + (1 − αn )[kn2 kzn k2 − kxn k2



+(kn2 − 1)M + 2hxn − kn2 zn , vi]},





Qn = {v ∈ C : hx0 − xn , xn − vi ≥ 0},




xn+1 = PCn ∩Qn x0 ,
where M is a appropriate constant such that M > kvk2 for each v ∈ Cn , then {xn } converges to PF (T ) x0 .
S.M. Kang, S.Y. Cho and X. Qin, J. Nonlinear Sci. Appl. 5 (2012), 466–474
469
Very recently, Zhou [20] improved the results of Martinez-Yanes and Xu [9] from nonexpansive mappings
to Lipschitz pseudo-contractions. To be more precise, he proved the following theorem.
Theorem Zhou. Let C be a closed convex subset of a real Hilbert space H and T : C → C be a Lipschitz
pseudo-contraction such that F (T ) 6= ∅. Suppose that {αn } and {βn } are two real sequences in (0, 1)
satisfying the conditions:
(a) βn ≤ αn , ∀n ≥ 0;
(b) lim inf n→∞ αn > 0;
1
(c) lim supn→∞ αn ≤ α ≤ √1+L
, ∀n ≥ 0, where L ≥ 1 is the Lipschitzian constant of T .
2 +1
Let a sequence {xn } generated by


x0 ∈ C,





yn = (1 − αn )xn + αn T xn ,



z = (1 − β )x + β T y ,
n
n n
n
n
2
Cn = {z ∈ C : kzn − zk ≤ kxn − zk2 − αn βn (1 − 2αn − L2 αn2 )kxn − T n xn k2 },





Qn = {z ∈ C : hxn − z, x0 − xn i ≥ 0},



x
n+1 = PCn ∩Qn x0 .
Then {xn } converges strongly to a fixed point v of T , where v = PF (T ) x0 .
In this paper, motivated by Acedo and Xu [1], Kim and Xu [5, 6], Marino and Xu [8], Martinez-Yanes
and Xu [9], Nakajo and Takahashi [10], Qin et al. [11], Qin, Cho and Zhou [12], Qin, Su and Shang [13], Su
and Qin [17, 18] and Zhou [20, 21], we modify Ishikawa iterative process (1.7) to have strong convergence for
asymptotically quasi-pseudocontractive mappings in the framework of Hilbert spaces without any compact
assumption.
In order to prove our main results, we need the following lemmas.
Lemma 1.1. ([8]) Let H be a real Hilbert space. Then the following equations hold:
(a) kx − yk2 = kxk2 − kyk2 − 2hx − y, yi for all x, y ∈ H.
(b) ktx + (1 − t)yk2 = tkxk2 + (1 − t)kyk2 − t(1 − t)kx − yk2 for all t ∈ [0, 1] and x, y ∈ H.
Lemma 1.2. Let C be a closed convex subset of real Hilbert space H and PC be the metric projection from
H onto C (i.e., for x ∈ H, PC x is the only point in C such that kx − PC xk = inf{kx − zk : z ∈ C}). Given
x ∈ H and z ∈ C, z = PC x if and only if there holds the relations: hx − z, y − zi ≤ 0 for any y ∈ C.
The following lemma can be found in Zhou and Su [22], we still give the proof for the completeness of
the paper.
Lemma 1.3. Let C be a nonempty bounded closed convex subset of H and T : C → C be a uniformly
L-Lipschitzian and asymptotically quasi-pseudocontractive mapping. Then F (T ) is a closed convex subset
of C.
Proof. From the continuity of T , we can conclude that F (T ) is closed.
Next, we show that F (T ) is convex. If F (T ) = ∅, then the conclusion is always true. Let p1 , p2 ∈ F (T ).
1
We prove p ∈ F (T ), where p = tp1 +(1−t)p2 , for t ∈ (0, 1). Put y(α,n) = (1−α)p+αT n p, where α ∈ (0, 1+L
).
S.M. Kang, S.Y. Cho and X. Qin, J. Nonlinear Sci. Appl. 5 (2012), 466–474
470
For all w ∈ F (T ), we see that
kp − T n pk2
= hp − T n p, p − T n pi
1
= hp − y(α,n) , p − T n pi
α
1
1
= hp − y(α,n) , p − T n p − (y(α,n) − T n y(α,n) )i + hp − y(α,n) , y(α,n) − T n y(α,n) i
α
α
1
1
n
n
= hp − y(α,n) , p − T p − (y(α,n) − T y(α,n) )i + hp − w + w − y(α,n) , y(α,n) − T n y(α,n) i
α
α
1
1
1+L
kp − y(α,n) k2 + hp − w, y(α,n) − T n y(α,n) i + hw − y(α,n) , y(α,n) − T n y(α,n) i
≤
α
α
α
1
1
n 2
n
≤ (1 + L)αkp − T pk + hp − w, y(α,n) − T y(α,n) i + (kn − 1)kw − y(α,n) k2 .
α
α
This implies that
α[1 − (1 + L)α]kp − T n pk2 ≤ hp − w, y(α,n) − T n y(α,n) i + (kn − 1)kw − y(α,n) k2 ,
∀w ∈ F (T ).
(1.8)
Taking w = pi , i = 1, 2 in (1.8), multiplying t and (1 − t) on the both sides of (1.8), respectively and adding
up, we see that
α[1 − (1 + L)α]kp − T n pk2 ≤ (kn − 1)kw − y(α,n) k2 .
This shows that T n p − p → 0 as n → ∞. Note that T is uniformly L-Lipschitzian. It follows that
T n+1 p − T p → 0 as n → ∞. This is, p ∈ F (T ). This completes the proof.
2. Main Results
Theorem 2.1. Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → C be a
uniformly L-Lipschitz and asymptotically quasi-pseudocontractive mapping such that F (T ) is nonempty and
bounded. Let {xn } be a sequence generated in the following algorithm:


x0 ∈ H chosen arbitrarily,





C1 = C,





x1 = PC1 x0 ,
yn = (1 − αn )xn + αn T n xn ,



zn = (1 − βn )xn + βn T n yn ,





Cn+1 = {z ∈ Cn : kzn − zk2 ≤ kxn − zk2 + βn θn − αn βn (1 − 2αn − L2 αn2 )kxn − T n xn k2 },




xn+1 = PCn+1 x0 ,
where
2
θn = 2(kn − 1)[2kn + 1 + (1 + L)2 ] sup kxn − zk → 0.
z∈F (T )
Assume that the control sequences {αn } and {βn } in (0, 1) satisfy the restrictions:
(a) βn ≤ αn , ∀n ≥ 1;
(b) lim inf n→∞ αn > 1;
1
(c) lim supn→∞ αn ≤ α < √1+L
, ∀n ≥ 0.
2 +1
Then the sequence {xn } converges strongly to PF (T ) x0 .
S.M. Kang, S.Y. Cho and X. Qin, J. Nonlinear Sci. Appl. 5 (2012), 466–474
471
Proof. We divide the proof into five parts.
Step 1. Show that Cn is closed and convex for all n ≥ 1.
It is obvious that C1 is closed and convex. Assume that Cm is closed and convex. Next, we show that
Cm+1 is closed and convex for the same m. For all z ∈ Cm , we see that
2
kzm − zk2 ≤ kxm − zk2 + βm θm − αm βm (1 − 2αm − L2 αm
)kxm − T m xm k2
is equivalent to the following inequality
2
2hxm − zm , zi ≤ kxm k2 − kzm k2 + βm θm − αm βm (1 − 2αm − L2 αm
)kxm − T m xm k2 .
This shows that Cm+1 is closed and convex. We, therefore, obtain that Cn is convex for every n ≥ 1.
Step 2. Show that F (T ) ⊂ Cn , ∀n ≥ 1.
It is obvious that F (T ) ⊂ C1 . Assume that F (T ) ⊂ Cm for some m. Next, we show that F (T ) ⊂ Cm+1
for the same m. In view of Lemma 1.1, for all u ∈ F (T ) ⊂ Cm , we see from (1.3) that
kzm − uk2 = k(1 − βm )(xm − u) + βm (T m ym − u)k2
= (1 − βm )kxm − uk2 + βm kT m ym − uk2 − βn (1 − βm )kxm − T m ym k2
≤ (1 − βm )kxm − uk2 + βm (2km − 1)kym − uk2 + kym − T m ym k2
(2.1)
− βm (1 − βm )kxm − T m ym k2
and
kym − T m ym k2
= k(1 − αm )(xm − T m ym ) + αm (T m xm − T m ym )k2
= (1 − αm )kxm − T m ym k2 + αm kT m xn − T m ym k2 − αm (1 − αm )kxm − T m xm k2
(2.2)
≤ (1 − αm )kxm − T m ym k2 + L2 αm kxm − ym k2 − αm (1 − αm )kxm − T m xm k2
2
≤ (1 − αm )kxm − T m ym k2 + αm (L2 αm
+ αm − 1)kxm − T m xm k2 .
Note that
kym − uk2 = (1 − αm )kxm − uk2 + αm kT m xm − uk2 − αm (1 − αm )kxm − T m xm k2
≤ (1 − αm )kxm − uk2 + αm (2km − 1)kxm − uk2 + αm kxm − T m xm k2
− αm (1 − αm )kxm − T m xm k2
2
≤ [1 + 2αm (km − 1)]kxm − uk2 + αm
kxm − T m xm k2 .
Substituting (2.2) and (2.3) into (2.1), we arrive at
kzm − uk2 ≤ (1 − βm )kxm − uk2 + βm (2km − 1)[1 + 2αm (km − 1)]kxm − uk2
2
2
+ (2km − 1)αm
βm kxm − T m xm k2 + αm βm (L2 αm
+ αm − 1)kxm − T m xm k2
+ βm (βm − αm )kxm − T m ym k2
≤ (1 − βm )kxm − uk2 + βm (2km − 1)[1 + 2αm (km − 1)]kxm − uk2
2
2
+ 2(km − 1)αm
βm kxm − T m xm k2 + αm βm (L2 αm
+ 2αm − 1)kxm − T m xm k2
+ βm (βm − αm )kxm − T m ym k2
2
≤ kxm − uk2 + 2(km − 1)βm [2αm km + 1 − αm + αm
(1 + L)2 ]kxm − uk2
2
+ αm βm (L2 αm
+ 2αm − 1)kxm − T m xm k2 + βm (βm − αm )kxm − T m ym k2
≤ kxm − uk2 + 2(km − 1)βm [2km + 1 + (1 + L)2 ]kxm − uk2
2
+ αm βm (L2 αm
+ 2αm − 1)kxm − T m xm k2 + βm (βm − αm )kxm − T m ym k2 .
(2.3)
S.M. Kang, S.Y. Cho and X. Qin, J. Nonlinear Sci. Appl. 5 (2012), 466–474
472
From the condition (a), we obtain that
2
kzm − uk2 ≤ kxm − uk2 + βm θm − αm βm (1 − 2αm − L2 αm
)kxm − T m xm k2 .
Therefore, we obtain that u ∈ Cm+1 . This concludes that F (T ) ⊂ Cn , ∀n ≥ 1.
Step 3. Show that {xn } is a Cauchy sequence in C.
In view of xn = PCn x0 and PF (T ) x0 ∈ F (T ) ⊂ Cn for each n ≥ 1, we see that
kx0 − xn k ≤ kx0 − PF (T ) x0 k.
This proves that the sequence {xn } is bounded. From xn = PCn x0 , we see that
hx0 − xn , xn − yi ≥ 0,
∀y ∈ Cn .
(2.4)
In view of xn+1 ∈ Cn+1 ⊂ Cn , we see that
0 ≤ hx0 − xn , xn − xn+1 i
= hx0 − xn , xn − x0 + x0 − xn+1 i
≤ −kx0 − xn k2 + kx0 − xn kkx0 − xn+1 k,
that is, kx0 − xn k ≤ kx0 − xn+1 k. This together with the boundedness of {xn } implies that limn→∞ kx0 − xn k
exists. By the construction of Cn , we see that Cm ⊂ Cn and xm = PCm x0 ∈ Cn for any positive integer
m ≥ n. From xn = PCn x0 , we see that
hx0 − xn , xn − xm i ≥ 0.
It follows that
(2.5)
kxm − xn k2 = kxm − x0 + x0 − xn k2
= kxm − x0 k2 + kx0 − xn k2 − 2hx0 − xn , x0 − xm i
≤ kxm − x0 k2 − kx0 − xn k2 − 2hx0 − xn , xn − xm i
(2.6)
≤ kxm − x0 k2 − kx0 − xn k2 .
Letting m, n → ∞ in (2.6), we have limm,n→∞ kxn − xm k = 0. Hence, {xn } is a Cauchy sequence.
Step 4. Show that T xn − xn → 0 as n → ∞.
Since H is a Hilbert space and C is closed and convex, we may assume that
xn → q ∈ C
as n → ∞.
(2.7)
Next, we show that q = PF (T ) x0 . To end this, we first show that q ∈ F (T ). By taking m = n + 1 in (2.6),
we arrive at
lim kxn − xn+1 k = 0,
(2.8)
n→∞
In view of xn+1 = PCn+1 x0 ∈ Cn+1 , we obtain that
kzn − xn+1 k2 ≤ kxn − xn+1 k2 + βn θn − αn βn (1 − 2αn − L2 αn2 )kxn − T n xn k2 .
(2.9)
On the other hand, we have
kzn − xn+1 k2 = kzn − xn + xn − xn+1 k2
= kzn − xn k2 + 2hxn − zn , xn+1 − xn i + kxn − xn+1 k2 .
Combining (2.9) with (2.10) and noting that zn = (1 − βn )xn + βn T n yn , we see that
βn2 kxn − T n yn k2 + 2βn hxn − T n yn , xn+1 − xn i ≤ βn θn − αn βn (1 − 2αn − L2 αn2 )kxn − T n xn k2 .
(2.10)
S.M. Kang, S.Y. Cho and X. Qin, J. Nonlinear Sci. Appl. 5 (2012), 466–474
473
That is,
βn kxn − T n yn k2 + 2hxn − T n yn , xn+1 − xn i ≤ θn − αn (1 − 2αn − L2 αn2 )kxn − T n xn k2 .
It follows that
αn (1 − 2αn − L2 αn2 )kxn − T n xn k2 ≤ θn − 2hxn − T n yn , xn+1 − xn i.
1
From the assumptions on {αn }, we can choose a ∈ (α, √1+L
). For such chosen a, there exists a positive
2 +1
integer N ≥ 1 such that αn < a for all n ≥ N . It follows that 1 − 2a − L2 a2 > 0. On the other hand, one
can choose b ∈ (0, c), where c = lim inf n→∞ αn . we obtain that αn > b for n large enough. It follows that
b(1 − 2a − L2 a2 )kxn − T n xn k2 ≤ θn + M kxn+1 − xn k
for n ≥ 0 large enough, where M = 2 supn≥0 {kxn − T n yn k}. From (2.8), we obtain that
lim kxn − T n xn k = 0.
n→∞
(2.11)
On the other hand, we have
kxn − T xn k = kxn − xn+1 k + kxn+1 − T n+1 xn+1 k + kT n+1 xn+1 − T n+1 xn k
+ kT n+1 xn − T xn k
≤ kxn − xn+1 k + kxn+1 − T n+1 xn+1 k + Lkxn+1 − xn k + LkT n xn − xn k.
From (2.8) and (2.11), we arrive at
lim kxn − T xn k = 0.
n→∞
(2.12)
Step 5. Show that xn → q = PF (T ) x0 as n → ∞.
Notice that
kq − T qk ≤ kq − xn k + kxn − T xn k + kT xn − T qk
≤ (1 + L)kq − xn k + kxn − T xn k.
It follows from (2.7) and (2.12) that q ∈ F (T ). From (2.4), we see that
hx0 − xn , xn − yi ≥ 0,
∀y ∈ F (T ) ⊂ Cn .
(2.13)
Taking the limit in (2.13), we obtain that hx0 − q, q − yi ≥ 0, ∀y ∈ F (T ). In view of Lemma 1.2, we see that
q = PF (T ) x0 . This completes the proof.
Remark 2.2. Theorem 2.1 includes Theorem 4.1 of Kim and Xu [6] a as special case. It also improves
the results of Kim and Xu [5] and Qin, Su and Shang [13] from asymptotically nonexpansive mappings to
asymptotically quasi-pseudocontractive mappings.
For the class of Lipschitz quasi-pseudocontractive mappings, we have from Theorem 2.1 the following
result.
Corollary 2.3. Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → C be a
L-Lipschitz and quasi-pseudocontractive mapping such that F (T ) 6= ∅. Let {xn } be a sequence generated in
the following algorithm:


x0 ∈ H chosen arbitrarily,




C1 = C,




x1 = PC1 x0 ,

yn = (1 − αn )xn + αn T xn ,



zn = (1 − βn )xn + βn T yn ,





Cn = {z ∈ Cn : kzn − zk2 ≤ kxn − zk2 − αn βn (1 − 2αn − L2 αn2 )kxn − T xn k2 },




xn+1 = PCn+1 x0 .
S.M. Kang, S.Y. Cho and X. Qin, J. Nonlinear Sci. Appl. 5 (2012), 466–474
474
Assume that the control sequences {αn } and {βn } in (0, 1) satisfy the restrictions:
(a) βn ≤ αn , ∀n ≥ 1;
(b) lim inf n→∞ αn > 1;
1
(c) lim supn→∞ αn ≤ α < √1+L
, ∀n ≥ 0.
2 +1
Then the sequence {xn } converges strongly to PF (T ) x0 .
Remark 2.4. Comparing Corollary 2.3 with Theorem 3.6 of Zhou [20], we do not require that the mapping
I − T is demi-closed at zero. From the computation point of view, we remove the iterative step Qn , see [20]
for more details.
Remark 2.5. Corollary 2.3 also gives an affirmative answer to the problem proposed by Marino and Xu [8].
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