R E S E A R C H
Open Access
Split feasibility problems for total
quasi-asymptotically nonexpansive
mappings
Xiong Rui Wang
1, Shih-sen Chang
2*, Lin Wang
2and Yun-he Zhao
2*Correspondence:
2College of Statistics and
Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
Full list of author information is available at the end of the article
Abstract
The purpose of this paper is to propose an algorithm for solving thesplit feasibility problemsfortotal quasi-asymptotically nonexpansive mappingsin infinite-dimensional Hilbert spaces. The results presented in the paper not only improve and extend some recent results of Moudafi [Nonlinear Anal. 74:4083-4087, 2011; Inverse Problem 26:055007, 2010], but also improve and extend some recent results of Xu [Inverse Problems 26:105018, 2010; 22:2021-2034, 2006], Censor and Segal [J. Convex Anal. 16:587-600, 2009], Censoret al.[Inverse Problems 21:2071-2084, 2005], Masad and Reich [J. Nonlinear Convex Anal. 8:367-371, 2007], Censoret al.[J. Math. Anal. Appl. 327:1244-1256, 2007], Yang [Inverse Problem 20:1261-1266, 2004] and others.
MSC: 47J05; 47H09; 49J25
Keywords: split feasibility problem; convex feasibility problem; total
quasi-asymptotically nonexpansive mappings; demi-closeness; Opial condition
1 Introduction
Throughout this paper, we always assume thatH,H are real Hilbert spaces, ‘→’, ‘’ denote strong and weak convergence, respectively, andF(T) is a fixed point set of a map-pingT.
Thesplit feasibility problem(SFP) in finite-dimensional 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 [–]. Thesplit feasibility problemin an infinite-dimensional real Hilbert space can be found in [, , –].
The purpose of this paper is to introduce and study the followingsplit feasibility prob-lemfortotal quasi-asymptotically nonexpansive mappingsin the framework of infinite-dimensional real Hilbert spaces:
findx*∈Csuch thatAx*∈Q, (.)
whereA:H→His a bounded linear operator,S:H→HandT:H→Hare map-pings;C:=F(S) andQ:=F(T). In the sequel, we use to denote the set of solutions of
(SFP)-(.),i.e.,
={x∈C,Ax∈Q}. (.)
2 Preliminaries
We first recall some definitions, notations and conclusions which will be needed in proving our main results.
LetEbe a Banach space. A mappingT:E→Eis said to bedemi-closed at originif for any sequence{xn} ⊂Ewithxnx*and(I–T)xn →,x*=Tx*.
A Banach spaceEis said to havethe Opial property, if for any sequence{xn}withxnx*,
lim inf
n→∞xn–x
*<lim inf
n→∞ xn–y, ∀y∈Ewithy=x
*.
Remark . It is well known that each Hilbert space possesses the Opial property.
Definition . LetHbe a real Hilbert space.
() A mappingG:H→His said to be a ({νn},{μn},ζ)-total quasi-asymptotically non-expansive mappingifF(G)=∅; and there exist nonnegative real sequences{νn},{μn}with νn→ and μn→ and a strictly increasing continuous functionζ :R+ →R+ with ζ() = such that for eachn≥,
p–Gnx≤ p–x+νnζ
p–x+μn, ∀p∈F(G),x∈H. (.)
Now, we give an example of total quasi-asymptotically nonexpansive mapping.
LetCbe a unit ball in a real Hilbert spacel, and letT:C→Cbe a mapping defined by
T: (x,x, . . . , )→,x,ax,ax, . . .
, (x,x, . . . , )∈l,
where{ai}is a sequence in (, ) such that∞i=ai=. It is proved in Goebal and Kirk [] that
(i) Tx–Ty ≤x–y,∀x,y∈C; (ii) Tnx–Tny ≤n
j=ajx–y,∀x,y∈C,∀n≥. Denote byk
= ,k
n = nj=aj,n≥, then
lim
n→∞kn=nlim→∞
n
j=
aj
= .
Lettingνn= (kn– ),∀n≥,ζ(t) =t,∀t≥ and{μn}be a nonnegative real sequence with μn→, from (i) and (ii),∀x,y∈C,n≥, we have
Tnx–Tny≤ x–y+νnζ
x–y+μn. (.)
Again, since ∈Cand ∈F(T), this implies thatF(T)=∅. From (.), we have
p–Tny≤ p–y+νnζ
This shows that the mappingTdefined as above is a total quasi-asymptotically nonexpan-sive mapping.
() A mapping G:H →H is said to be ({kn})-quasi-asymptotically nonexpansive if F(G)=∅; and there exists a sequence{kn} ⊂[,∞) withkn→ such that for alln≥,
p–Gnx≤knp–x, ∀p∈F(G),x∈H. (.)
() A mappingG:H→His said to bequasi-nonexpansiveifF(G)=∅such that
p–Gx ≤ p–x, ∀p∈F(G),x∈H. (.)
Remark . It is easy to see that every quasi-nonexpansive mapping is a ({} )-quasi-asymptotically nonexpansive mapping and every {kn}-quasi-asymptotically
nonexpan-sive mapping is a ({νn},{μn},ζ)-total quasi-asymptotically nonexpansive mapping with
{νn=kn– },{μn= }andζ(t) =t,t≥.
Definition .
() A mappingG:H→His said to be uniformlyL-Lipschitzian if there exists a constant
L> such that
Tnx–Tny≤Lx–y, ∀x,y∈Handn≥.
() A mapping G:H→H is said to be semi-compact if for any bounded sequence {xn} ⊂H withlimn→∞xn–Gxn= , there exists a subsequence{xni} ⊂ {xn}such that
xniconverges strongly to some pointx*∈H.
Proposition . Let G:H→H be a({νn},{μn},ζ)-total quasi-asymptotically nonexpan-sive mapping. Then for each q∈F(G)and for each x∈H, the following inequalities are equivalent: for each n≥
q–Gnx≤ q–x+νnζ
q–x+μn, ∀q∈F(G),x∈H; (.)
x–Gnx,x–q≥x–Gnx–νnζ
q–x–μn; (.)
x–Gnx,q–Gnx≤x–Gnx+νnζ
q–x+μn. (.)
Proof
(I) (.)⇔(.) In fact, since
Gnx–q=Gnx–x+x–q
=Gnx–x+x–q+ Gnx–x,x–q, ∀x∈H,q∈F(G),
from (.) we have that
Gnx–x+x–q+ Gnx–x,x–q
≤ x–q+νnζ
Simplifying it, inequality (.) is obtained.
Conversely, from (.) the inequality (.) can be obtained immediately. (II) (.)⇔(.) In fact, since
x–Gnx,x–q= x–Gnx,x–Gnx+Gnx–q
=x–Gnx+ x–Gnx,Gnx–q
it follows from (.) that
x–Gnx+ x–Gnx,Gnx–q≥x–Gnx–νnζ
q–x–μn.
Simplifying it, the inequality (.) is obtained.
Conversely, from (.) the inequality (.) can be obtained immediately.
This completes the proof of Proposition ..
Lemma .[] Let{an},{bn}and{δn}be sequences of nonnegative real numbers satisfying
an+≤( +δn)an+bn, ∀n≥.
If∞i=δn<∞and
∞
i=bn<∞, then the limitlimn→∞anexists.
3 Split feasibility problem
For solving the split feasibility problem (.), let us assume that the following conditions are satisfied:
. HandHare two real Hilbert spaces,A:H→His a bounded linear operator;
. S:H→HandT:H→Hare two uniformlyL-Lipschitzian and
({νn},{μn},ζ)-total quasi-asymptotically nonexpansive mappings satisfying the
following conditions:
(i) TandSboth are demi-closed at origin; (ii) ∞n=(μn+νn) <∞;
(iii) there exist positive constantsMandM*such thatζ(t)≤ζ(M) +M*t,∀t≥.
We are now in a position to give the following result.
Theorem . Let H, H, A, S, T , L,{μn},{νn},ζ be the same as above. Let{xn}be the sequence generated by:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x∈H chosen arbitrarily,
xn+= ( –αn)un+αnSn(un),
un=xn+γA*(Tn–I)Axn, ∀n≥,
(.)
where{αn}is a sequence in[, ], andγ> is a constant satisfying the following conditions:
(iv) <lim infn→∞αn≤lim supn→∞αn< ; andγ∈(,A),
(I) If=∅(whereis the set of solutions to ((SFP)-(.)), then{xn}converges weakly to a pointx*∈.
The proof of conclusion (I)
() First, we prove that for eachp∈, the following limits exist:
lim
n→∞xn–p=nlim→∞un–p. (.)
In fact, sincep∈, we havep∈C:=F(S) andAp∈Q:=F(T). It follows from (.) and (.) that
xn+–p =un–p–αn
un–Snun
=un–p– αnun–p,un–Snun
+αnun–Snun
≤ un–p–αnun–Snun
–νnζ
un–p–μn
+αnun–Snun by (.)
=un–p–αn( –αn)un–Snun+αn
νnζ
un–p+μn
. (.)
On the other hand, by condition (iii), we have
ζun–p≤ζ(M) +M*un–p. (.)
Substituting (.) into (.) and simplifying, we have
xn+–p≤
+αnνnM*
un–p–αn( –αn)un–Snun
+αn
νnζ(M) +μn
≤ +νnM*un–p–αn( –αn)un–Snun
+νnζ(M) +μn. (.)
On the other hand,
un–p=xn–p+γA*
Tn–IAxn
=xn–p+γA*Tn–IAxn+ γ xn–p,A*Tn–IAxn, (.)
and
γA*Tn–IAxn=γA*Tn–IAxn,A*Tn–IAxn
=γAA*Tn–IAxn,
Tn–IAxn
≤γATnAxn–Axn, (.)
and
γ xn–p,A*
Tn–IAxn
= γ Axn–Ap,
Tn–IAxn
= γ Axn–Ap+Tn–IAxn–Tn–IAxn,Tn–IAxn
In (.), takingx=Axn, Gn=Tn, q=Ap, and noting Ap∈F(T), from (.) and
condi-tion (iii), we have
TnAxn–Ap,TnAxn–Axn
≤ T
n–IAxn +νnζ
Axn–Ap+μn
≤ T
n–IAxn+ν n
ζ(M) +M*Axn–p+μn
. (.)
Substituting (.) into (.) and simplifying it, we have
γ xn–p,A*
Tn–IAxn
≤γνn
ζ(M) +M*Axn–p
+μn–Tn–I
Axn
. (.)
Substituting (.) and (.) into (.) after simplifying, we have
un–p≤ +γ νnM*Axn–p+γνnζ(M) +μn
–γ –γATn–IAxn. (.)
Substituting (.) into (.) and simplifying it, we have
xn+–p≤
+νnM* +γ νnM*Axn–p
+γνnζ(M) +μn
–γ –γATn–IAxn
–αn( –αn)un–Snun
+νnζ(M) +μn
≤( +ξn)xn–p+ηn–γ
–γATn–IAxn
–αn( –αn)un–Snun, (.)
where
ξn=νn
M*+γM*A+γ νnM*A,
ηn=
+νnM*γ+ νnζ(M) +μn
.
By condition (iii), we have
∞
n=
ξn<∞, and
∞
n=
ηn<∞.
By condition (iv), ( –γA) > . Hence, from (.), we have
xn+–p≤( +ξn)xn–p+ηn, ∀n≥.
By Lemma ., the following limit exists:
lim
Now, we rewrite (.) as follows:
γ –γATn–IAxn
+αn( –αn)un–Snun
≤ xn–p–xn+–p
+ξnxn–p+ηn→ (asn→ ∞).
This together with the condition (iv) implies that
lim
n→∞un–S
nu
n= ; (.)
and
lim
n→∞T
n–IAx
n= . (.)
It follows from (.), (.) and (.) that the limitlimn→∞un–pexists and
lim
n→∞un–p=nlim→∞xn–p.
The conclusion (.) is proved. () Next, we prove that
lim
n→∞xn+–xn= and nlim→∞un+–un= . (.)
In fact, it follows from (.) that
xn+–xn=( –αn)un+αnSn(un) –xn
=( –αn)
xn+γA*Tn–IAxn+αnSn(un) –xn
=( –αn)γA*
Tn–IAxn+αn
Sn(un) –xn
=( –αn)γA*
Tn–IAxn+αn
Sn(un) –un
+αn(un–xn)
=( –αn)γA*
Tn–IAxn+αn
Sn(un) –un
+αnγA*
Tn–IAxn
=γA*Tn–IAxn+αn
Sn(un) –un.
In view of (.) and (.), we have that
lim
n→∞xn+–xn= . (.)
Similarly, it follows from (.), (.) and (.) that
un+–un=xn++γA*
Tn+–IAxn+–
xn+γA*Tn–IAxn
≤ xn+–xn+γA*
Tn+–IAxn+
+γA*Tn–IAxn→ (asn→ ∞). (.)
() Next, we prove that
un–Sun → and Axn–TAxn → (asn→ ∞). (.)
In fact, from (.), we have
ζn:=un–Snun→ (asn→ ∞). (.)
SinceSis uniformlyL-Lipschitzian continuous, it follows from (.) and (.) that
un–Sun ≤un–Snun+Snun–Sun
≤ζn+LSn–un–un
≤ζn+LSn–un–Sn–un–
+Sn–un––un
≤ζn+Lun–un–
+LSn–un––un–+un––un
≤ζn+L( +L)un–un–+Lζn–→ (asn→ ∞).
Similarly, from (.), we have
Axn–TnAxn→ (asn→ ∞). (.)
SinceTis uniformlyL-Lipschitzian continuous, by the same way as above, from (.) and (.), we can also prove that
Axn–TAxn → (asn→ ∞). (.)
() Finally, we prove thatxnx*andunx*, which is a solution of (SFP)-(.). Since{un}is bounded, there exists a subsequence{uni} ⊂ {un}such thatunix*(some point inH). From (.), we have
uni–Suni → (asni→ ∞). (.)
By the assumption thatSis demi-closed at zero, we get thatx*∈F(S). Moreover, from (.) and (.), we have
xni=uni–γA*
Tni–IAxnix*.
SinceAis a linear bounded operator, we getAxniAx*. In view of (.), we have
Axni–TAxni → (asni→ ∞).
Now, we prove thatxnx*andunx*.
Suppose, to the contrary, that if there exists another subsequence{unj} ⊂ {un}such that unjy*∈withy*=x*, then by virtue of (.) and the Opial property of Hilbert space, we have
lim inf
ni→∞uni–x
*<lim inf
ni→∞uni–y
*= lim
n→∞un–y
*
= lim
nj→∞unj–y
*<lim inf
nj→∞unj–x
*
= lim
n→∞un–x
*=lim inf
ni→∞uni–x *.
This is a contradiction. Therefore,unx*. By using (.) and (.), we have
xn=un–γA*Tnn–IAxnx*.
The proof of conclusion (II) By the assumption thatS is semi-compact, it follows from (.) that there exists a subsequence of{uni}(without loss of generality, we still denote it by{uni}) such thatuni→u*∈H (some point inH). Sinceunix*. This implies that
x*=u*, and souni→x*∈. By virtue of (.), we know thatlim
n→∞un–x*= and limn→∞xn–x*= ,i.e.,{un}and{xn}both converge strongly tox*∈.
This completes the proof of Theorem ..
Theorem . Let H, Hand A be the same as in Theorem .. Let S:H→Hand T :
H→Hbe two({kn})-quasi-asymptotically nonexpansive mappings with{kn} ⊂[,∞),
kn→satisfying the following conditions:
(i) TandSboth are demi-closed at origin;
(ii) ∞n=(kn– ) <∞.
Let{xn}be the sequence generated by
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x∈H chosen arbitrarily,
xn+= ( –αn)un+αnSn(un),
un=xn+γA*(Tn–I)Axn, ∀n≥,
(.)
where{αn}is a sequence in[, ]andγ > is a constant satisfying the condition (iv) in Theorem .. Then the conclusions in Theorem . still hold.
Proof By assumptions,S:H→HandT:H→Hboth are ({kn})-quasi-asymptotically nonexpansive mappings with{kn} ⊂[,∞),kn→; by Remark .,SandTboth are uni-formlyL-Lipschitzian (whereL=supn≥kn) and ({νn},{μn},ζ)-total quasi-asymptotically
nonexpansive mapping with{νn=kn– },{μn= }andζ(t) =t,t≥. Therefore, all
con-ditions in Theorem . are satisfied. The conclusions of Theorem . can be obtained from
Theorem . immediately.
Theorem . Let H, Hand A be the same as in Theorem .. Let S:H→Hand T :
sequence generated by
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x∈H chosen arbitrarily,
xn+= ( –αn)un+αnSn(un),
un=xn+γA*(Tn–I)Axn, ∀n≥,
(.)
where{αn}is a sequence in[, ]andγ > is a constant satisfying the condition (iv) in Theorem .. Then the conclusions in Theorem . still hold.
Proof By the assumptions,S:H→H andT :H→H are quasi-nonexpansive map-pings. By Remark .,SandT both are uniformlyL-Lipschitzian (whereL= ) and
({})-quasi-asymptotically nonexpansive mappings. Therefore, all conditions in Theorem . are satisfied. The conclusions of Theorem . can be obtained from Theorem .
imme-diately.
Remark . Theorems ., . and . not only improve and extend the corresponding
results of Moudafi [, ], but also improve and extend the corresponding results of Cen-soret al.[, ], Yang [], Xu [], Censor and Segal [], Masad and Reich [] and others.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed to this work equal. All authors read and ap- proved the final manuscript.
Author details
1Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China.2College of Statistics and Mathematics,
Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China.
Acknowledgements
This work was supported by the Scientific Research Fund of Science Technology Department of Sichuan Province (2011JYZ010) and the Natural Science Foundation of Yunnan Province (Grant No.2011FB074).
Received: 23 April 2012 Accepted: 30 August 2012 Published: 18 September 2012 References
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