R E S E A R C H
Open Access
Multiple-set split feasibility problems for total
asymptotically strict pseudocontractions
mappings
LI Yang
1*, Shih-Sen Chang
2*, Yeol JE Cho
3and Jong KYU Kim
4* Correspondence: [email protected]; [email protected]
1Department of Mathematics,
South West University of Science and Technology, Mianyang Sichuan 621010, China
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 article is to propose and investigate an algorithm for solving the
multiple-set split feasibility problems for total asymptotically strict pseu-docontractions mappings in infinite-dimensional Hilbert spaces. The results presented in this article improve and extend some recent results of A. Moudafi, H. K. Xu, Y. Censor, A. Segal, T. Elfving, N. Kopf, T. Bortfeld, X. A. Motova, Q. Yang, A. Gibali, S. Reich and others. 2000 AMS Subject Classification: 47J05; 47H09; 49J25.
Keywords:multiple-set split feasibility problem, split feasibility problem, demi-close-ness, Opial condition, total asymptotically strict pseudocontraction
1. Introduction and preliminaries
Throughout this article, we always assume thatH1,H2are real Hilbert spaces,“®”,“⇀” are denoted by strong and weak convergence, respectively, andF(T) is the fixed point set of a mappingT.
LetGbe a nonempty closed convex subset ofH1andT :G®Ga mapping.
Tis said to be a contraction if there exists a constantaÎ (0,1) such that
Tx−Ty≤αx−y, ∀x,y∈G. (1:1)
Banach contraction principle guarantees that every contractive mapping defined on complete metric spaces has a unique fixed point.
Tis said to be a weak contraction if
Tx−Ty≤x−y−ψx−y, ∀x,y∈G. (1:2)
whereψ: [0,∞)®[0, ∞) is a continuous and nondecreasing function such thatψis positive on (0,∞), ψ(0) = 0, and limt®∞ ψ(t) =∞. We remark that the class of weak contractions was introduced by Alber and Guerre-Delabriere [1]. In 2001, Rhoades [2] showed that every weak contraction defined on complete metric spaces has a unique fixed point.
Tis said to be nonexpansive if
Tx−Ty≤x−y, ∀x,y∈G. (1:3)
T is said to be asymptotically nonexpansive if there exists a sequence {kn}⊂ [1,∞) withkn®1 asn®∞such that
Tnx−Tny≤κnx−y, ∀n≥1, x,y∈G. (1:4)
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [3] as a generalization of the class of nonexpansive mappings. They proved that if
G is a nonempty closed convex bounded subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive mapping on G, thenT has a fixed point.
T is said to be total asymptotically nonexpansive if
Tnx−Tny≤x−y+μnφx−y+ξn, ∀n≥1, x,y∈G. (1:5)
wherej: [0,∞)®[0,∞) is a continuous and strictly increasing function withj(0) = 0, and {μn} and {ξn} are nonnegative real sequences such that μn® 0 andξn ®0 as n ®∞. The class of mapping was introduced by Alber et al. [4]. From the definition,
we see that the class of total asymptotically nonexpansive mappings includes the class of asymptotically nonexpansive mappings as special cases, see [5,6] for more details.
T is said to be strictly pseudocontractive if there exists a constant Î [0, 1) such that
Tx−Ty2≤x−y2+κ(I−T)x−(I−T)y2, ∀x,y∈G. (1:6)
The class of strict pseudocontractions was introduced by Browder and Petryshyn [7] in a real Hilbert space. In 2007, Marino and Xu [8] obtained a weak convergence theorem for the class of strictly pseudocontractive mappings, see [8] for more details.
T is said to be an asymptotically strict pseudocontraction if there exist a constant
Î [0, 1) and a sequence {kn}⊂[1,∞) withkn®1 asn®∞such that Tnx−Tny2≤κnx−y2+κI−Tn
x−I−Tny2, ∀n≥1, x,y∈G. (1:7)
The class of asymptotically strict pseudocontractions was introduced by Qihou [9] in 1996. Kim and Xu [10] proved that the class of asymptotically strict pseudocontrac-tions is demiclosed at the origin and also obtained a weak convergence theorem for the class of mappings; see [10] for more details.
In this article, we introduce the following mapping.
Definition 1.1 LetHbe a real Hilbert space, andGbe a nonempty closed convex subset ofH. A mapping T:G®Gis said to be (, {μn}, {ξn},j)-total asymptotically strict pseudocontractive, if there exists a constantÎ[0, 1) and sequences {μn}⊂[0,∞), {ξn}⊂[0,∞) withμn®0 andξn®0 asn®∞, and a continuous and strictly increasing functionj: [0,∞)®[0,∞) withj(0) = 0 such that
Tnx−Tny2≤
x−y2+κx−y−Tnx−Tny2
+μnφx−y+ξn, ∀n≥1, x,y∈G. (1:8)
Now, we give an example of total asymptotically strict pseudocontractive mapping. Let C be a unit ball in a real Hilbert space l2 and let T : C ® Cbe a mapping defined by
T:(x1,x2, ...,)→
0,x21,a2x2, a3x3, ...
where {ai} is a sequence in (0, 1) such that∞i=2ai=
1 2. It is proven in Goebal and Kirk [3] that
(i)Tx−Ty≤2x−y, ∀x,y∈C;
(ii)Tnx−Tny≤2n
j=2ajx−y, ∀x,y∈C, ∀n≥2..
Denote byκ 1 2 1 = 2, κ
1 2
n = 2
n
j=2aj, n≥2, then
lim
n→∞kn= limn→∞
2n
j=2aj
2 = 1.
Letting μn=(κn−1), ∀n≥1, φ(t) =t2, ∀t≥0, κ = 0and {ξn} be a nonnega-tive real sequence with ξn®0, then∀x,y∈C, n≥1, we have
Tnx−Tny2≤x−y2+μnφx−y+κ|x−y−
Tnx−Tny2+ξn.
Remark 1.2If j(l) =l2andξn= 0, then total asymptotically strict pseudocontrac-tive mapping is asymptotically strict pseudocontraction mapping.
It is easy to see the following proposition holds.
Proposition 1.3 LetT : G® Gbe a (, {μn}, {ξn}, j)-total asymptotically strict pseudocontractive mapping. IfF(T)=∅, then for eachq Î F(T) and for each x ÎG, the following inequalities hold and are equivalent:
x−q, Tnx−q≤ κ+ 1 2k x−q
2
+κ−1 2k T
nx−q2
+μn
2κφx−q+
ξn
2κ; (1:9)
x−Tnx, x−q≥1−κ
2 T
nx−x2−μn
2 φx−q−
ξn
2; (1:10)
x−Tnx, q−Tnx≤ κ+ 1
2 T
nx−x2+μn
2 φx−q+
ξn
2. (1:11)
Thesplit feasibility problem (SFP) in finite-dimensional spaces was first introduced by Censor and Elfving [11] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [12]. 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 [13-15].
The SFPin an infinite-dimensional Hilbert space can be found in [12,14,16-18]. The purpose of this article is to introduce and study the following multiple-set SFP
(MSSFP) for total asymptotically strict pseudocontractionin the framework of infinite-dimensional Hilbert spaces:
find x∗ ∈C such that Ax∗∈Q, (1:12)
where A: H1 ®H2is a bounded linear operator, Si :H1 ®H1 and Ti : H2 ®H2,
i = 1, 2, ..., Nare mappings,C:Ni=1F(Si)andQ:Ni=1F(Ti). In the sequel, we useΓ to denote the set of solutions of (MSSFP)–(1.12), i.e.,
To prove our main results, we first recall some definitions, notations, and conclusions.
Let Ebe a Banach space. A mappingT :E®E is said to bedemi-closed at origin, if for any sequence {xn}⊂E withxn⇀x* and ||(I-T)xn||®0, thenx* =Tx*.
A Banach space E is said to havethe Opial property, if for any sequence {xn} with xn⇀ x*, then
lim inf
n→∞ xn−x
∗<lim inf
n→∞ xn−y, ∀y∈E with y=x ∗.
Remark 1.4It is well known that each Hilbert space possesses the Opial property. Definition 1.5LetHbea real Hilbert space.
(1) A mappingT: H® His said to be uniformlyL-Lipschitzian, if there exists a constantL> 0, such that
Tnx−Tny≤Lx−y, ∀x,y∈H andn≥1.
(2) A mappingT:H®H issaid to be semi-compact, if for any bounded sequence {xn}⊂Hwith limn®∞||xn- Txn|| = 0, then there exists a subsequence
xni
⊂ {xn}
such thatxniconverges strongly to some pointx*ÎH.
Lemma 1.6 [10] LetHbe a real Hilbert space. If {xn} is a sequence inHweakly con-vergent to z, then
lim sup
n→∞ xn−y
2
= lim sup
n→∞ xn−z
2+z−y2 ∀
y∈H.
Proposition 1.7 Assume thatGis a closed convex subset of a real Hilbert spaceH
and let T: G®Gbe a (, {μn}, {ξn},j)-total asymptotically strict pseudocon-traction mapping and uniformlyL-Lipschitzian. Then the demiclosedness principle holds forI
-T in the sense that if {xn} is a sequence inGsuch thatxn ⇀x*, and lim supm®∞lim supn®∞ ||xn- Tmxn|| = 0 then (I- T)x* = 0. In particular,xn⇀x*, and (I- T)xn® 0 ⇒ (I-T)x* = 0, i.e.,Tis demiclosed at origin.
ProofSince {xn} is bounded, we can define a functionfonHby
f(x) = lim sup
n→∞ xn−x
2, ∀x∈H.
By Lemma 1.6, the weak convergencexn⇀ x* implies that
f(x) =f(x∗) +x−x∗2, ∀x∈H.
In particular, for each m≥1,
On the other hand, since T is a (, {μn}, {ξn})-total asymptotically strict pseudo-contraction mapping, by (1.8), we get
fTmx∗= lim sup
n→∞
xn−Tmx∗2
= lim sup
n→∞
xn−Tmxn+Tmxn−Tmx∗2
= lim sup
n→∞
xn−Tmxn2+ 2 xn−Tmxn,Tmxn−Tmx∗
+Tmx
n−Tmx∗2
≤lim sup
n→∞ xn−T
mx
nxn−Tmxn+ 2Lxn−x∗
+ lim sup
n→∞
xn−x∗2+kxn−Tmxn−
x∗−Tmx∗2
+μmφxn−x∗+ξm
Taking lim supm®∞ on both sides and observing the facts that limm®∞ μm = 0, limm®∞ξm= 0 and lim supm®∞lim supn®∞||xn- Tmxn|| = 0, we derive that
lim sup
m→∞ f
Tmx∗≤lim sup
n→∞ xn−x ∗2
+klim sup
m→∞ x
∗−Tmx∗2
(1:15)
Since lim supm®∞f(Tmx*) =f(x*)+lim supm®∞||Tmx* -x*||2, andf(x*) = lim supn®∞ ||xn- x*||2, it follows from (1.15) that lim supm®∞||x* -Tmx*||2= 0. That is,Tmx*® x*; henceTx* =x*.
Lemma 1.8 [19] Let {an}, {bn} and {δn} be sequences of nonnegative real numbers satisfying
an+1≤(1 +δn)an+bn, ∀n≥1.
If∞i=1δn<∞and
∞
i=1bn<∞, then the limit limn®∞anexists. 2. Multiple-set split feasibility problem
For solving the multiple-set split feasibility problem (1.12), let us assume that the fol-lowing conditions are satisfied:
1. H1 and H2 are two real Hilbert spaces, A : H1 ® H2 is a bounded linear operator;
2. LetG,G˜ be a nonempty closed convex subset ofH1 andH2 respectively, Si: G ® G, i = 1, 2,...,N, is a uniformly Li-Lipschitzian and (bi, {μi,n}, {ξi,n}, ji)-total asymptotically strictly pseudocontractive mapping andTi:G˜ → ˜G, i= 1, 2, ...,N,
is a uniformlyLi˜-Lipschitzian and
kiμ˜i,n
,
˜
ξi,n
,φ˜i
-total asymptotically strictly
pseudocontractive mapping which satisfy the following conditions:
(i)C:N
i=1F(Si)=∅, Q:=
N
i=1F(Ti)=∅; (ii)β= max1≤i≤N βi<1, κ = max1≤i≤N κi<1;;
(iii)L:= max1≤i≤NLi<∞, L˜:= max1≤i≤N Li˜ <∞;
(iv)μn= max1≤i≤N
μi,n, μ˜i,n
, ξn= max1≤i≤N
ξi,n, ξ˜i,n
and
∞
i=1μn<∞,
∞
i=1ξn<∞.. (v)φ= max1≤i≤N
φi,
˜
We are now in a position to give the following result:
Theorem 2.1 Let H1, H2, G, G˜, A, {Si}, {Ti}, C, Q, β, κ, L, L˜, {μn}, {ξn} and j
be the same as above. In addition, there exist positive constantsM andM* such that j (l)≤M*l2 for alll≥M. Let {xn} be the sequence generated by:
⎧ ⎨ ⎩
x1∈G chosen arbitrarily
xn+1= (1−αn)un+αnSnn(un), un=xn+γA∗(Tnn−I)Axn, ∀n≥1,
(2:1)
whereSn
n=Snn(modN), Tnn=Tnn(modN) ∀n≥1, {αn}is a sequence in [0, 1] and g > 0 is a constant satisfying the following conditions:
(vi) αn∈(δ, 1−β), ∀n≥1andγ ∈
0, 1−κ A2
, whereδ Î (0, 1 - b) is a
positive constant.
(I) If=∅(whereΓ is the set of solutions to (MSSFP)–(1.12)), then {xn} con-verges weakly to a point x*ÎΓ.
(II) In addition, if there exists a positive integer jsuch thatSjis semi-compact, then {xn} and {un} both converge strongly tox*ÎΓ.
The proof of conclusion (I)
(1) First we prove that for eachpÎΓ, the following limits exist
lim
n→∞xn−p and nlim→∞un−p. (2:2)
In fact, since jis an increasing function, it results thatj(l)≤ j(M), ifl≤ M and j(l)≤M*l2, ifl≥M. In either case, we can obtain that
φ(λ)≤φ(M) +M∗λ2, ∀λ≥0. (2:3) Since pÎ Γ, thenp∈C:=Ni=1F(Si)and Ap∈Q:=Ni=1F(Ti). From (2.1) and (1.10) we have
xn+1−p 2
=un−p−αn
un−Snnun2
=un−p2−2αn un−p, un−Snnun
+αn2un−Snnun2
≤un−p2−αn(1−β)un−Snnun
2
+αnμnφun−p+αnξn+αn2un−Snnun
2
(by(1.10)) ≤un−p2−αn(1−β−αn)un−Snnun
2
+αnμn
φ(M) +M∗un−p2
+αnξn
=1 +αnμnM∗ un−p
2
−αn(1−β−αn)un−Snnun
2
+αnμnφ(M) +αnξn
(2:4)
On the other hand, since
un−p
2
=xn−p+γA∗
Tnn−I
Axn
2
=xn−p
2
+γ2A∗Tnn−I
Axn
2
+ 2γ xn−p, A∗
Tnn−I
Axn
and
A∗Tnn−IAxn2= A∗Tnn−IAxn, A∗Tnn−IAxn
= AA∗Tnn−IAxn, Tnn−IAxn
≤ A2TnnAxn−Axn
2 ,
(2:6)
It follows from (1.11) we have
xn−p, A∗Tnn−IAxn= Axn−Ap, Tnn−IAxn
= Axn−Ap+Tnn−IAxn−Tnn−IAxn, Tnn−IAxn
= TnnAxn−Ap, TnnAxn−Axn
−Tnn−I
Axn2.
≤ 1 +κ 2 T
n n−I
Axn2+μn
2 φAxn−Ap+
ξn
2 −T
n n−I
Axn2.
≤ κ−1 2 T
n n−I
Axn2+μn 2
φ(M) +M∗Axn−Ap2+ξn 2. ≤ κ−1
2 T
n n−I
Axn2+μn 2 M
∗Axn−Ap2 + μn
2 φ(M) +
ξn
2.
(2:7)
Substituting (2.6) and (2.7) into (2.5) and simplifying it, we have
un−p2≤xn−p2+γ2A2TnnAxn−Axn2+γ (κ−1)Tnn−IAxn2
+γ μnM∗Axn−Ap
2
+γ μnφ(M) +γ ξn
=xn−p2−γ1−κ−γA2 TnnAxn−Axn2
+γ μnM∗Axn−Ap2+γ μnφ(M) +γ ξn
≤1 +γ μnM∗A2 xn−p2−γ1−κ−γA2 TnnAxn−Axn2
+γ μnφ(M) +γ ξn
(2:8)
Substituting (2.8) into (2.4) and after simplifying we have
xn+1−p 2
≤1 +αnμnM∗ 1 +γ μnM∗A2 xn−p
2
−γ1−κ−γA2 TnnAxn−Axn2+γ μnφ(M) +γ ξn
−αn(1−β−αn)un−Snnun
2
+αnμnφ(M) +αnξn
≤(1 +δn)xn−p
2
−γ1−κ−γA2 TnnAxn−Axn2
−αn(1−β−αn)un−Snnun
2 +bn
(2:9)
where
δn=αnμnM∗+γ μnM∗A2+γA2αnμ2n(M∗)2 bn=1 +αnμnM∗
γ +αn
μnφ(M) +
1 +αnμnM∗
γ +αn
ξn
By condition (vi) we have
xn+1−p 2
≤(1 +δn)xn−p
2 +bn
By condition (iv),∞n=1δn<∞and∞n=1bn<∞. Hence, from Lemma 1.8 we know that the following limit exists
lim
Consequently, from (2.9) and (2.10) we have that
γ1−κ−γA2 Tnn−IAxn2+αn(1−β−αn)un−Snnun
2
≤xn−p2−xn+1−p2+δnxn−p2+bn→0 (as n→ ∞).
This together with the condition (vi) implies that
lim
n→∞un−S n
nun= 0; (2:11)
and
lim
n→∞T n n−I
Axn= 0. (2:12)
It follows from (2.5), (2.10) and (2.12) that the limit ||un-p|| exists. The conclusion (1) is proved.
(2) Next we prove that
lim
n→∞xn+1−xn= 0 and limn→∞un+1−un= 0. (2:13)
In fact, it follows from (2.1) that
xn+1−xn=(1−αn)un+αnSnn(un)−xn
=(1−αn)
xn+γA∗
Tnn−I
Axn
+αnSnn(un)−xn
=(1−αn) γA∗
Tnn−I
Axn+αn
Snn(un)−xn
=(1−αn) γA∗
Tnn−I
Axn+αn
Snn(un)−un
+αn(un−xn)
=(1−αn) γA∗
Tnn−I
Axn+αn
Snn(un)−un
+αnγA∗
Tnn−I
Axn
=γA∗Tnn−I
Axn+αn
Snn(un)−un.
In view of (2.11) and (2.12) we have that
lim
n→∞xn+1−xn= 0. (2:14)
Similarly, it follows from (2.1), (2.12), and (2.14) that
un+1−un=xn+1+γA∗
Tnn+1+1−IAxn+1−
xn+γA∗Tnn−IAxn
≤ xn+1−xn+γA∗
Tnn+1+1−IAxn+1 +γA∗Tnn−IAxn→0(asn→ ∞).
(2:15)
The conclusion (2.13) is proved.
(3) Next we prove that for eachj= 1, 2,...,N- 1,
uiN+j−SjuiN+j→0 and AxiN+j−TjAxiN+j→0 (as i→ ∞), (2:16)
In fact, from (2.11) we have
Since Sj is uniformly Lj-Lipschitzian continuous, it follows from (2.13) and (2.17) that
uiN+j−SjuiN+j=uiN+j−SiNj +juiN+j+SjiN+juiN+j−SjuiN+j
≤ηiN+j+LjSjiN+j−1uiN+j−uiN+j
≤ηiN+j+LjSiNj +j−1uiN+j−SiNj +j−1uiN+j−1 +LjSiNj +j−1uiN+j−1−uiN+j
≤ηiN+j+L2j uiN+j−uiN+j−1
+LjSiNj +j−1uiN+j−1−uiN+j−1+uiN+j−1−uiN+j
≤ηiN+j+Lj
1 +Lj uiN+j−uiN+j−1+LjηiN+j−1→0 (as i→ ∞) Similarly, for each j= 1, 2,..., N- 1, from (2.13) we have
ςiN+j:=AxiN+j−TjiN+jAxiN+j→0 (asi→ ∞). (2:18)
Since Tjis uniformly Lj˜-Lipschitzian continuous, by the same way as above, from (2.13) and (2.18), we can also prove that
AxiN+j−TjAxiN+j→0 (as i→ ∞). (2:19)
(4) Finally we prove that xn⇀ x* andun⇀ x* which is a solution of (MSSFP)– (1.12).
Since {un} is bounded. There exists a subsequence
uni
⊂ {un}such thatuni x∗ (some point in H1). Hence, for any positive integerj= 1, 2,...,N, there exists a subse-quence {ni(j)}⊂ {ni} withni(j)(modN) =j such thatuni(j)x∗. Again from (2.16) we have
uni(j)−Sjuni(j)→ 0 (asni(j)→ ∞) (2:20)
Since Sj is demiclosed at zero (see Proposition 1.7), it gets thatx* Î F(Sj). By the
arbitrariness of j= 1, 2,...,N, we have x∗∈C:=Nj=1FSj.
Moreover, from (2.1) and (2.12) we have
xni =uni−γA∗
Tni
ni−I
Axni x∗.
SinceAis a linear bounded operator, it getsAxni Ax∗. For any positive integerk= 1, 2,..., N, there exists a subsequence {ni(k)} ⊂{ni} with ni(k)(modN) = ksuch that
Axni(k)Ax∗. In view of (2.16) we have
Axni(k)−TkAxni(k)→0 (asni(k)→ ∞).
Since Tkis demiclosed at zero, we have Ax* Î F(Tk). By the arbitrariness of k= 1, 2,..., N, it yieldsAx∗∈Q:=Nk=1F(Tk). This together withx* ÎCshows thatx*Î Γ, i.
In fact, if there exists another subsequenceuni
⊂ {un}such thatuni(j)y∗∈with
y* ≠x*. Consequently, by virtue of (2.2) and the Opial property of Hilbert space, we have
lim inf
ni→∞
uni−x∗<lim inf
ni→∞
uni−y∗= lim
n→∞un−y ∗
= lim inf
ni→∞
uni−y
∗< lim nj→∞
unj−x
∗
= lim inf
n→∞ un−x∗= limni→∞
uni−x∗.
This is a contradiction. Therefore,un⇀x*. By using (2.1) and (2.12), we have
xn=un−λA∗Tnn−IAxnx∗. The proof of conclusion (II).
Without loss of generality, we can assume that S1 is semi-compact. It follows from (2.20) that
uni(1)−S1uni(1)→0 (asni(1)→ ∞) (2:21)
Therefore, there exists a subsequence ofuni(1)
(for the sake of convenience we still
denote it byuni(1)
such that uni(1)→u∗∈H1(some point inH1). Sinceuni(1)x∗. This implies that x* = u*, and souni(1)→x∗∈. By virtue of (2.2) we know that limn®∞ ||un - x*|| = 0 and limn®∞ ||xn - x*|| = 0, i.e., {un} and {xn} both converge strongly to x*ÎΓ.
This completes the proof of Theorem 2.1.
Remark 2.2Since the class of total asymptotically strict pseudocontractive mappings includes the class of asymptotically strict pseudocontractions mappings and the class of strict pseudocontractions mappings as special cases, Theorem 2.1 improves and extend the corresponding results of Censor et al. [14,15], Yang [17], Moudafi [20], Xu [21], Censor and Segal [22], Censor et al. [23] and others.
Acknowledgements
The authors would like to thank the referees for useful comments and suggestions. This study was supported by the Natural Science Foundation of Sichuan Province (No. 08ZA008).
Author details 1
Department of Mathematics, South West University of Science and Technology, Mianyang Sichuan 621010, China
2College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China 3
Department of Mathematics Education, Rins Gyeongsang National University, Jinju 660-701, Korea4Department of Mathematics Education, Kyungnam University Masan, Kyungnam 631-701, Korea
Authors’contributions
All authors have read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 17 July 2011 Accepted: 7 November 2011 Published: 7 November 2011
References
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doi:10.1186/1687-1812-2011-77
Cite this article as:Yanget al.:Multiple-set split feasibility problems for total asymptotically strict pseudocontractions mappings.Fixed Point Theory and Applications20112011:77.
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