Zhang et al. Journal of Inequalities and Applications 2015, 2015:1
http://www.journalofinequalitiesandapplications.com/content/2015/1/1
RESEARCH
Open Access
The strong convergence theorems for split
common ﬁxed point problem of
asymptotically nonexpansive mappings in
Hilbert spaces
Xin-Fang Zhang1 , Lin Wang1* , Zhao Li Ma2 and Li Juan Qin3
Dedicated to Professor SS Chang on the occasion of his 80th birthday.
*
Correspondence:
[email protected]
1
College of Statistics and
Mathematics, Yunnan University of
Finance and Economics, Long Quan
Road, Kunming, China
Full list of author information is
available at the end of the article
Abstract
In this paper, an iterative algorithm is introduced to solve the split common ﬁxed
point problem for asymptotically nonexpansive mappings in Hilbert spaces. The
iterative algorithm presented in this paper is shown to possess strong convergence
for the split common ﬁxed point problem of asymptotically nonexpansive mappings
although the mappings do not have semi-compactness. Our results improve and
develop previous methods for solving the split common ﬁxed point problem.
MSC: 47H09; 47J25
Keywords: split common ﬁxed point problem; asymptotically nonexpansive
mapping; strong convergence; Hilbert space; algorithm
1 Introduction and preliminaries
Throughout this paper, let H and H be real Hilbert spaces whose inner product and
norm are denoted by ·, · and · , respectively; let C and Q be nonempty closed convex subsets of H and H , respectively. A mapping T : C → C is said to be nonexpansive if
Tx – Ty ≤ x – y for any x, y ∈ C. A mapping T : C → C is said to be quasi-nonexpansive
if Tx – p ≤ x – p for any x ∈ C and p ∈ F(T), where F(T) is the set of ﬁxed points
of T. A mapping T : C → C is called asymptotically nonexpansive if there exists a sequence {kn } ⊂ [, ∞) satisfying limn→∞ kn =  such that T n x – T n y ≤ kn x – y for any
x, y ∈ C. A mapping T : C → C is semi-compact if, for any bounded sequence {xn } ⊂ C
with limn→∞ xn – Txn = , there exists a subsequence {xnj } ⊂ {xn } such that {xnj } converges strongly to some point x∗ ∈ C.
The split feasibility problem (SFP) is to ﬁnd a point q ∈ H with the property
q∈C
and
Aq ∈ Q,
(.)
where A : H → H is a bounded linear operator.
Assuming that SFP (.) is consistent (i.e., (.) has a solution), it is not hard to see that
x ∈ C solves (.) if and only if it solves the following ﬁxed point equation:
x = PC I – γ A∗ (I – PQ )A x,
x ∈ C,
(.)
©2015 Zhang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Zhang et al. Journal of Inequalities and Applications 2015, 2015:1
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where PC and PQ are the (orthogonal) projections onto C and Q, respectively, γ >  is any
positive constant, and A∗ denotes the adjoint of A.
The SFP in ﬁnite-dimensional Hilbert spaces was ﬁrst introduced by Censor and Elfving
[] for modeling inverse problems which arise from phase retrievals and in medical image
reconstruction []. Recently, it has been found that the SFP can also be used in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment
planning [–].
Let S : H → H and T : H → H be two mappings satisfying F(S) = {x ∈ H : Sx = x} =
φ and F(T) = {x ∈ H : Tx = x} =
φ, respectively; let A : H → H be a bounded linear
operator. The split common ﬁxed point problem (SCFP) for mappings S and T is to ﬁnd
a point q ∈ H with the property
q ∈ F(S) and Aq ∈ F(T).
(.)
We use to denote the set of solutions of SCFP (.), that is, = {q ∈ F(S) : Aq ∈ F(T)}.
Since each closed and convex subset may be considered as a ﬁxed point set of a projection on the subset, hence the split common ﬁxed point problem (SCFP) is a generalization
of the split feasibility problem (SFP) and the convex feasibility problem (CFP) [].
Split feasibility problems and split common ﬁxed point problems have been studied by
some authors [–]. In , Moudaﬁ [] proposed the following iteration method to
approximate a split common ﬁxed point of demi-contractive mappings: for arbitrarily chosen x ∈ H ,
un = xn + γβA∗ (T – I)Axn ,
xn+ = ( – αn )un + αn Uun , n ∈ N,
and he proved that {xn } converges weakly to a split common ﬁxed point x∗ ∈ , where U :
H → H and T : H → H are two demi-contractive mappings, A : H → H is a bounded
linear operator.
Using the iterative algorithm above, in , Moudaﬁ [] also obtained a weak convergence theorem for the split common ﬁxed point problem of quasi-nonexpansive mappings in Hilbert spaces. After that, some authors also proposed some iterative algorithms to approximate a split common ﬁxed point of other nonlinear mappings, such
as nonspreading type mappings [], asymptotically quasi-nonexpansive mappings [],
κ-asymptotically strictly pseudononspreading mappings [], asymptotically strictly pseudocontraction mappings [] etc., but they just obtained weak convergence theorems when
those mappings do not have semi-compactness. This naturally brings us to the following
question.
Can we construct an iterative scheme which can guarantee the strong convergence for split
common ﬁxed point problems without assumption of semi-compactness?
In this paper, we introduce the following iterative scheme. Let x ∈ H , C = H , the sequence {xn } is deﬁned as follows:
⎧
yn = αn zn + ( – αn )Tn zn ,
⎪
⎪
⎪
⎨ z = x + λA∗ (T n – I)Ax ,
n
n
n

⎪ Cn+ = {v ∈ Cn : yn – v ≤ kn zn – v, zn – v ≤ kn xn – v},
⎪
⎪
⎩
xn+ = PCn+ (x ), n ≥ ,
(.)
Zhang et al. Journal of Inequalities and Applications 2015, 2015:1
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where T : H → H and T : H → H are two asymptotically nonexpansive mappings,
A : H → H is a bounded linear operator, A∗ denotes the adjoint of A. Under some suitable
conditions on parameters, the iterative scheme {xn } is shown to converge strongly to a
split common ﬁxed point of asymptotically nonexpansive mappings T and T without
the assumption of semi-compactness on T and T .
The following lemma and results are useful for our proofs.
Lemma . [] Let E be a real uniformly convex Banach space, K be a nonempty closed
subset of E, and let T : K → K be an asymptotically nonexpansive mapping. Then I – T is
demiclosed at zero, that is, if {xn } ⊂ K converges weakly to a point p ∈ K and limn→∞ xn –
Txn = , then p = Tp.
Let C be a closed convex subset of a real Hilbert space H. PC denotes the metric projection of H onto C. It is well known that PC is characterized by the properties: for x ∈ H
and z ∈ C,
z = PC (x)
⇔
x – z, z – y ≥ ,
∀y ∈ C
(.)
and
y – PC (x) + x – PC (x) ≤ x – y ,
∀y ∈ C, ∀x ∈ H.
(.)
In a real Hilbert space H, it is also well known that
λx + ( – λ)y = λx + ( – λ)y – λ( – λ)x – y ,
∀x, y ∈ H, ∀λ ∈ R
(.)
and
x, y = x + y – x – y ,
∀x, y ∈ H.
(.)
2 Main results
Theorem . Let H and H be two Hilbert spaces, A : H → H be a bounded linear
operator, T : H → H be an asymptotically nonexpansive mapping with the sequence
{kn() } ⊂ [, ∞) satisfying limn→∞ kn() = , and T : H → H be an asymptotically nonexpansive mapping with the sequence {kn() } ⊂ [, ∞) satisfying limn→∞ kn() = , F(T ) = ∅ and
F(T ) = ∅, respectively. Let x ∈ H , C = H , and let the sequence {xn } be deﬁned as follows:
⎧
zn = xn + λA∗ (Tn – I)Axn ,
⎪
⎪
⎪
⎨ y = α z + ( – α )T n z ,
n
n n
n  n
⎪
Cn+ = {v ∈ Cn : yn – v ≤ kn zn – v, zn – v ≤ kn xn – v},
⎪
⎪
⎩
xn+ = PCn+ (x ), n ≥ ,
(.)
where A∗ denotes the adjoint of A, λ ∈ (, A∗  ) and {αn } ⊂ (, η] ⊂ (, ) satisﬁes
lim infn→∞ αn ( – αn ) > , kn = max{kn() , kn() }, n ≥ . If = {p ∈ F(T ) : Ap ∈ F(T )} = ∅,
then {xn } converges strongly to x∗ ∈ .
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Proof We will divide the proof into ﬁve steps.
Step . We ﬁrst show that Cn is closed and convex for any n ≥ .
Since C = H , so C is closed and convex. Assume that Cn is closed and convex. For any
v ∈ Cn , since
yn – v ≤ kn zn – v
⇔
zn – v ≤ kn xn – v
⇔

kn zn – yn – kn v + v, v ≤ kn zn  – yn  ,

kn xn – zn – kn v + v, v ≤ kn xn  – zn  ,
we know that Cn+ is closed and convex. Therefore Cn is closed and convex for any n ≥ .
Step . We prove ⊂ Cn for any n ≥ .
Let p ∈ , then from (.) we have

zn – p = xn – p + λA∗ Tn – I Axn 
= xn – p + λA∗ Tn – I Axn + λ xn – p, A∗ Tn – I Axn ,
(.)
where
λ xn – p, A∗ Tn – I Axn
= λ Axn – Ap, Tn – I Axn
= λ A(xn – p) + Tn – I Axn – Tn – I Axn , Tn – I Axn
 = λ Tn Axn – Ap, Tn – I Axn – Tn – I Axn   

= λ Tn Axn – Ap + Tn – I Axn 

n 

 – Axn – Ap – T – I Axn

 
 
≤ λ kn Axn – Ap – Tn – I Axn – Axn – Ap



n 
 = –λ T – I Axn + λ kn –  Axn – Ap .
(.)
Substituting (.) into (.), we can obtain that
 

zn – p ≤ xn – p + λ A∗ Tn – I Axn – λ Tn – I Axn + λ kn –  Axn – Ap
 
= xn – p – λ  – λA∗ Tn – I Axn + λA kn –  xn – p
 
≤ kn xn – p – λ  – λA∗ Tn – I Axn .
(.)
In addition, it follows from (.) that
yn – p = αn (zn – p) + ( – αn ) Tn zn – p ≤ kn zn – p.
Therefore, from (.) and (.), we know that p ∈ Cn and ⊂ Cn for any n ≥ .
(.)
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Step . We will show that {xn } is a Cauchy sequence.
Since ⊂ Cn+ ⊂ Cn and xn+ = PCn+ (x ) ⊂ Cn , then
xn+ – x ≤ p – x for n ≥  and p ∈ .
(.)
It means that {xn } is bounded. For any n ≥ , by using (.), we have
 
xn+ – xn  + x – xn  = xn+ – PCn (x ) + x – PCn (x )
≤ xn+ – x  ,
which implies that  ≤ xn – xn+  ≤ xn+ – x  – xn – x  . Thus {xn – x } is nondecreasing. Therefore, by the boundedness of {xn }, limn→∞ xn –x exists. For some positive
integers m, n with m ≤ n, from xn = PCn (x ) ⊂ Cm and (.), we have
 
xm – xn  + x – xn  = xm – PCn (x ) + x – PCn (x ) ≤ xm – x  .
(.)
Since limn→∞ xn – x exists, it follows from (.) that limn→∞ xn – xm = . Therefore
{xn } is a Cauchy sequence.
Step . We will show that limn→∞ zn – T zn = limn→∞ Axn – T Axn = .
Since xn+ = PCn+ (x ) ∈ Cn+ ⊂ Cn , we have
zn – xn ≤ zn – xn+ + xn+ – xn ≤ ( + kn )xn+ – xn → ,
yn – xn ≤ yn – xn+ + xn+ – xn ≤  + kn xn+ – xn → ,
yn – zn ≤ yn – xn + xn – zn → .
(.)
(.)
(.)
Notice that λ( – λA∗  ) > , it follows from (.) that



n T – I Axn  ≤ kn xn – p – zn – p

∗

λ( – λA )
≤
(kn – )xn – p + (xn – p + zn – p)(xn – p – zn – p)
λ( – λA∗  )
≤
(kn – )xn – p + xn – zn (xn – p + zn – p)
,
λ( – λA∗  )
thus, since {xn } is bounded and limn→∞ kn = , from (.) we have
lim Tn – I Axn = .
n→∞
On the other hand, since

yn – p = αn (zn – p) + ( – αn ) Tn zn – p 

= αn zn – p + ( – αn )Tn zn – p – αn ( – αn )Tn zn – zn 
≤  + kn –  zn – p – αn ( – αn )Tn zn – zn ,
(.)
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we have

αn ( – αn )Tn zn – zn ≤ zn – p – yn – p + kn –  zn – p
≤ zn – p + yn – p zn – yn + kn –  zn – p .
Since lim infn→∞ αn ( – αn ) >  and limn→∞ kn = , we know that
lim Tn – I zn = .
(.)
n→∞
In addition, since zn+ – zn ≤ zn+ – xn+ + xn+ – xn + xn – zn , we know that
limn→∞ zn+ – zn = . So from
zn – T zn = zn – zn+ + zn+ – Tn+ zn+ + Tn+ zn+
– Tn+ zn + Tn+ zn – T zn ≤ ( + kn+ )zn – zn+ + zn+ – T n+ zn+ + k T n zn – zn ,


we can obtain that
lim zn – T zn = .
n→∞
(.)
Similarly, we have
lim Axn – T Axn = .
n→∞
(.)
Step . We will show that {xn } converges strongly to an element of .
Since {xn } is a Cauchy sequence, we may assume that xn → x∗ , from (.) we have zn →
∗
x , which implies that zn x∗ . So it follows from (.) and Lemma . that x∗ ∈ F(T ).
In addition, since A is a bounded linear operator, we have that limn→∞ Axn – Ax∗ = .
Hence, it follows from (.) and Lemma . that Ax∗ ∈ F(T ). This means that x∗ ∈ and
{xn } converges strongly to x∗ ∈ . The proof is completed.
In Theorem ., as T = T and H = H , we have the following result.
Corollary . Let H be a Hilbert space, T : H → H be an asymptotically nonexpansive mapping with a sequence {kn } ⊂ [, ∞) satisfying limn→∞ kn = . The sequence {xn } is
deﬁned as follows: x ∈ H , C = H
⎧
zn = xn + λ(T n – I)xn ,
⎪
⎪
⎪
⎨
yn = αn zn + ( – αn )T n zn ,
⎪ Cn+ = {v ∈ Cn : yn – v ≤ kn zn – v, zn – v ≤ kn xn – v},
⎪
⎪
⎩
xn+ = PCn+ (x ), n ≥ ,
(.)
where λ ∈ (, ) and {αn } ⊂ (, η] ⊂ (, ) satisﬁes lim infn→∞ αn ( – αn ) > . If F(T) = {p ∈
H : p = Tp} = ∅, then {xn } converges strongly to a ﬁxed point x∗ of T.
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In Theorem ., when T and T are two nonexpansive mappings, the following result
holds.
Corollary . Let H and H be two Hilbert spaces, A : H → H be a bounded linear
operator, T : H → H and T : H → H be two nonexpansive mappings such that F(T ) =
∅ and F(T ) = ∅, respectively. Let x ∈ H , C = H , and let the sequence {xn } be deﬁned as
follows:
⎧
zn = xn + λA∗ (T – I)Axn ,
⎪
⎪
⎪
⎨
yn = αn zn + ( – αn )T zn ,
⎪
C
n+ = {v ∈ Cn : yn – v ≤ zn – v ≤ xn – v},
⎪
⎪
⎩
xn+ = PCn+ (x ), n ≥ ,
(.)
where A∗ denotes the adjoint of A, λ ∈ (, A∗  ) and {αn } ⊂ (, η] ⊂ (, ) satisﬁes
lim infn→∞ αn ( – αn ) > . If = {p ∈ F(T ) : Ap ∈ F(T )} =
∅, then {xn } converges strongly
∗
to x ∈ .
Remark . When T and T are two quasi-nonexpansive mappings and I – T and I – T
are demiclosed at zero, Corollary . also holds.
Example . Let C be a unit ball in a real Hilbert space l , and let T : C → C be a mapping
deﬁned by
T : (x , x , . . .) → , x , a x , a x , . . . .
It is proved in Goebel and Kirk [] that
(i) T x – T y ≤ x – y, ∀x, y ∈ C;
(ii) Tn x – Tn y ≤  nj= aj x – y, ∀x, y ∈ C, ∀n ≥ .
– 

Taking aj =  j– , j ≥ , it is easy to see that ∞
j= aj =  . So we can take k = , and
n
kn =  j= aj , n ≥ , then
lim kn = lim 
n→∞
n→∞
n


– j–

= .
j=
Therefore T is an asymptotically nonexpansive mapping from C into itself with F(T ) =
{(, , . . . , , . . .)}.
n

Let D be an orthogonal subspace of Rn with the norm x =
i= xi and the inner
n
product x, y = i= xi yi for x = (x , . . . , xn ) and y = (y , . . . , yn ). For each x = (x , x , . . . , xn ) ∈
D, we deﬁne a mapping T : D → D by
T x =
if ni= xi < ;
(x , x , . . . , xn )
(–x , –x , . . . , –xn ) if ni= xi ≥ .
It is easy to show that Tn x – Tn y = x – (–)n y = x + y = x – y or Tn x –
Tn y = (–)n x – y = x + y = x – y for any x, y ∈ D. Therefore T is an
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asymptotically nonexpansive mapping from D into itself with F(T ) = {(, , . . . , )} ∪
{(x , x , . . . , xn ) : ni= xi < } since Tn x – Tn y ≤ kn x – y for any sequence {kn } ⊂ [, ∞)
with limn→∞ kn = .
Obviously, C and D are closed convex subsets of l and RN , respectively. Let A : C → D
be deﬁned by Ax = (x , x , . . . , xn ) for x = (x , x , . . .) ∈ C. Then A is a bounded linear operator with adjoint operator A∗ z = (x , x , . . . , xn , , , . . .) for z = (x , x , . . . , xn ) ∈ D. Clearly,
= {(, , . . . , , . . .)}, A = A∗ = .

– j–


, n ≥ , γ =  and αn = . – n
, n ≥ . It follows
Taking C = C, k = , kn+ =  n+
j= 
from Theorem . that {xn } converges strongly to (, , . . .) ∈ .
3 Applications and examples
Application to the equilibrium problem
Let H be a real Hilbert space, C be a nonempty closed and convex subset of H, and let the
bifunction F : C × C → R satisfy the following conditions:
(A) F(x, x) = , ∀x ∈ C;
(A) F(x, y) + F(y, x) ≤ , ∀x, y ∈ C;
(A) For all x, y, z ∈ C, limt↓ F(tz + ( – t)x, y) ≤ F(x, y);
(A) For each x ∈ C, the function y −→ F(x, y) is convex and lower semi-continuous.
The so-called equilibrium problem for F is to ﬁnd a point x∗ ∈ C such that F(x∗ , x) ≥ 
for all y ∈ C. The set of its solutions is denoted by EP(F).
Lemma . [] Let C be a nonempty closed convex subset of a Hilbert space H, and let
F : C × C → R be a bifunction satisfying (A)-(A). Let r >  and x ∈ H. Then there exists
z ∈ K such that

F(z, y) + y – z, z – x ≥ ,
r
∀y ∈ C.
Lemma . [] Assume that F : C × C → R satisﬁes (A)-(A). For r >  and x ∈ H, deﬁne
a mapping Tr : H → H as follows:

Tr (x) = z ∈ C : F(z, y) + y – z, z – x ≥ , ∀y ∈ C ,
r
∀x ∈ H.
Then
() Tr is single-valued;
() Tr is ﬁrmly nonexpansive, that is, for all x, y ∈ H,
Tr x – Tr y ≤ Tr x – Tr y, x – y;
() F(Tr ) = EP(F);
() EP(F) is nonempty, closed and convex.
Theorem . Let H and H be two Hilbert spaces, A : H → H be a bounded linear operator, T : H → H be a nonexpansive mapping, F : H × H → R be a bifunction satisfying
(A)-(A). Assume that C := EP(F) = ∅ and Q := F(T) = ∅. Taking C = H , for arbitrarily
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chosen x ∈ H , the sequence {xn } is deﬁned as follows:
⎧
⎪
z = xn + λA∗ (T – I)Axn ,
⎪
⎪ n
⎪

⎪
⎪
⎨ F(un , y) + rn y – un , un – zn ≥ , ∀y ∈ H ,
yn = αn zn + ( – αn )un ,
⎪
⎪
⎪
Cn+ = {v ∈ Cn : yn – v ≤ zn – v ≤ xn – v},
⎪
⎪
⎪
⎩
xn+ = PCn+ (x ), n ≥ ,
(.)
where A∗ denotes the adjoint of A, {rn } ⊂ (, ∞), λ ∈ (, A∗  ) and {αn } ⊂ (, η] ⊂ (, ) satisﬁes lim infn→∞ αn ( – αn ) > . If = {p ∈ C : Ap ∈ Q} = ∅, then the sequence {xn } converges
strongly to a point x∗ ∈ .
Proof It follows from Lemma . that un = Tr (zn ), F(Tr ) = EP(F) is nonempty, closed and
convex and Tr is a ﬁrmly nonexpansive mapping. Hence all conditions in Corollary . are
satisﬁed. The conclusion of Theorem . can be directly obtained from Corollary .. Let E and E be two real Hilbert spaces. Let C be a closed convex subset of E , K be
a closed convex subset of E , A : E → E be a bounded linear operator. Assume that F
is a bi-function from C × C into R and G is a bi-function from K × K into R. The split
equilibrium problem (SEP) is to
ﬁnd an element p ∈ C such that F(p, y) ≥ ,
∀y ∈ C
(.)
and
such that u := Ap ∈ C solves G(u, v) ≥ ,
∀v ∈ K.
(.)
Let = {p ∈ EP(F) : Ap ∈ EP(G)} denote the solution set of the split equilibrium problem
SEP.
Example . [] Let E = E = R, C := [, +∞) and K := (–∞, –]. Let A(x) = –x for all
R, then A is a bounded linear operator. Let F : C × C → R and G : K × K → R be deﬁned
by F(x, y) = y – x and G(u, v) = (u – v), respectively. Clearly, EP(F) = {} and A() = – ∈
EP(G). So = {p ∈ EP(F) : Ap ∈ EP(G)} =
∅.
Example . [] Let E = R with the standard norm | · | and E = R with the norm α =

(a + a )  for some α = (a , a ) ∈ R . K := [, +∞) and C := {α = (a , a ) ∈ R |a – a ≥ }.
Deﬁne a bi-function F(w, α) = w – w + a – a , where w = (w , w ), α = (a , a ) ∈ C, then
F is a bi-function from C × C into R with EP(F) = {p = (p , p )|p – p = }. For each α =
(a , a ) ∈ E , let Aα = a – a , then A is a bounded linear operator from E into E . In fact, it
√
is also easy to verify that A(aα + bα ) = aA(α ) + bA(α ) and A =  for some α , α ∈ E
and a, b ∈ R. Now deﬁne another bi-function G as follows: G(u, v) = v – u for all u, v ∈ K .
Then G is a bi-function from K × K into R with EP(G) = {}.
Clearly, when p ∈ EP(F), we have Ap =  ∈ EP(G). So = {p ∈ EP(F) : Ap ∈ EP(G)} = ∅.
Corollary . Let H and H be two Hilbert spaces, A : H → H be a bounded linear
operator, F : H × H → R be a bifunction satisfying EP(F) = ∅ and G : H × H → R be a
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bifunction satisfying EP(G) = ∅. Taking C = H , for arbitrarily chosen x ∈ H , the sequence
{xn } is deﬁned as follows:
⎧
F(un , y) + rn y – un , un – xn ≥ , ∀y ∈ H ,
⎪
⎪
⎪
⎪
⎪
G(vn , z) + rn z – vn , vn – Aun ≥ , ∀z ∈ H ,
⎪
⎪
⎪
⎨
zn = un + λA∗ (TrGn – I)Aun ,
⎪
yn = αn zn + ( – αn )TrFn xn ,
⎪
⎪
⎪
⎪
⎪ Cn+ = {v ∈ Cn : yn – v ≤ zn – v ≤ xn – v},
⎪
⎪
⎩
xn+ = PCn+ (x ), n ≥ ,
(.)
where A∗ denotes the adjoint of A, {rn } ⊂ (, ∞), λ ∈ (, A∗  ) and {αn } ⊂ (, η] ⊂ (, )
satisﬁes lim infn→∞ αn ( – αn ) > . If = {p ∈ EP(F) : Ap ∈ EP(G)} = ∅, then the sequence
{xn } converges strongly to a point x∗ ∈ .
Remark . Since Example . and Example . satisfy the conditions of Corollary .,
the split equilibrium problems in Example . and Example . can be solved by algorithm
(.).
Application to the hierarchial variational inequality problem
Let H be a real Hilbert space, T and T be two nonexpansive mappings from H to H such
that F(T ) = ∅ and F(T ) = ∅.
The so-called hierarchical variational inequality problem for nonexpansive mapping T
with respect to a nonexpansive mapping T : H → H is to ﬁnd a point x∗ ∈ F(T ) such that
∗
x – T x∗ , x∗ – x ≤ ,
∀x ∈ F(T ).
(.)
It is easy to see that (.) is equivalent to the following ﬁxed point problem:
ﬁnd x∗ ∈ F(T ) such that x∗ = PF(T ) T x∗ ,
(.)
where PF(T ) is the metric projection from H onto F(T ). Letting C := F(T ) and Q :=
F(PF(T ) T ) (the ﬁxed point set of the mapping PF(T ) T ) and A = I (the identity mapping
on H), then problem (.) is equivalent to the following split feasibility problem:
ﬁnd x∗ ∈ C such that Ax∗ ∈ Q.
(.)
Hence from Theorem . we have the following theorem.
Theorem . Let H, T , T , C and Q be the same as above. Let x ∈ H and C = H , and
let the sequence {xn } be deﬁned as follows:
⎧
zn = xn + λ(T – I)xn ,
⎪
⎪
⎪
⎨ y = α z + ( – α )T z ,
n
n n
n  n
⎪ Cn+ = {v ∈ Cn : yn – v ≤ zn – v ≤ xn – v},
⎪
⎪
⎩
xn+ = PCn+ (x ), n ≥ ,
(.)
Zhang et al. Journal of Inequalities and Applications 2015, 2015:1
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where λ ∈ (, ) and {αn } ⊂ (, η] ⊂ (, ) satisﬁes lim infn→∞ αn ( – αn ) > . If C ∩ Q =
∅, then the sequence {xn } converges strongly to a solution of the hierarchical variational
inequality problem (.).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the ﬁnal manuscript.
Author details
1
College of Statistics and Mathematics, Yunnan University of Finance and Economics, Long Quan Road, Kunming, China.
2
School of Information Engineering, The College of Arts and Sciences, Yunnan Normal University, Long Quan Road,
Kunming, 650222, China. 3 Department of Mathematics, Kunming University, Pu Xin Road No. 2, Kunming Economic and
Technological Development Zone, Kunming, 650214, China.
Acknowledgements
The authors would like to express their thanks to the reviewers and editors for their helpful suggestions and advice. This
work was supported by the National Natural Science Foundation of China (Grant No. 11361070) and the Scientiﬁc
Research Foundation of Postgraduate of Yunnan University of Finance and Economics.
Received: 12 April 2014 Accepted: 11 December 2014 Published: 07 Jan 2015
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Cite this article as: Zhang et al.: The strong convergence theorems for split common ﬁxed point problem of
asymptotically nonexpansive mappings in Hilbert spaces. Journal of Inequalities and Applications 2015, 2015:1
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