# Multiple-Set Split Feasibility Problems for Asymptotically Strict Pseudocontractions

Authors: Shih-Sen Chang; Yeol Je Cho; J. K. Kim; W. B. Zhang; L. Yang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

In this paper, we introduce an iterative method for solving the multiple-set split feasibility problems for asymptotically strict pseudocontractions in infinite-dimensional Hilbert spaces, and, by using the proposed iterative method, we improve and extend some recent results given by some authors.


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## Body

## 1. Introduction

The split feasibility problem (SFP) in finite dimensional spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. 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 [3–5].The split feasibility problem in an infinite dimensional Hilbert space can be found in [2, 4, 6–8].Throughout this paper, we always assume thatH1,H2 are real Hilbert spaces, “→”, “⇀” are denoted by strong and weak convergence, respectively.The purpose of this paper is to introduce and study the followingmultiple-set split feasibility problem for asymptotically strict pseudocontraction (MSSFP) in the framework of infinite-dimensional Hilbert spaces. Find x*∈C such that(1.1)Ax*∈Q,
where A:H1→H2 is a bounded linear operator, {Si} and {Ti}, i=1,2,…,M, are the families of mappings Si:H1→H1 and Ti:H2→H2, respectively, C∶=⋂i=1MF(Si) and Q∶=⋂i=1MF(Ti), where F(Si)={xi∈H1:Sixi=xi} and F(Ti)={yi∈H2:Tiyi=yi} denote the sets of fixed points of Si and Ti, respectively. In the sequel, we use Γ to denote the set of solutions of the problem (MSSFP), that is,(1.2)Γ={x∈C:Ax∈Q}.


## 2. Preliminaries

We first recall some definitions, notations, and conclusions which will be needed in proving our main results.LetE be a Banach space. A mapping T:E→E is said to be demiclosed at origin if, for any sequence {xn}⊂E with xn⇀x* and ∥(I-T)xn∥→0, we have x*=Tx*. A Banach space E is said to have Opial’s property if, for any sequence {xn} with xn⇀x*, we have(2.1)lim⁡inf⁡n→∞‖xn-x*‖<lim⁡inf⁡n→∞‖xn-y‖
for all y∈E with y≠x*.Remark 2.1.
It is well known that each Hilbert space possesses Opial’s property.Definition 2.2.
LetH be a real Hilbert space.
(1)
A mappingG:H→H is called a(γ,{kn})-asymptotically strict pseudocontraction if there exists a constant γ∈[0,1) and a sequence {kn}⊂[1,∞) with kn→1 such that
(2.2)‖Gnx-Gny‖2≤kn‖x-y‖2+γ‖(I-Gn)x-(I-Gn)y‖2,∀x,y∈H.
Especially, if kn=1 for each n≥1 in (2.2) and there exists γ∈[0,1) such that
(2.3)‖Gx-Gy‖2≤‖x-y‖2+γ‖(I-G)x-(I-G)y‖2,∀x,y∈H,
then G:H→H is called a γ-strict pseudocontraction.(2)
A mappingG:H→H is said to be uniformlyL-Lipschitzian if there exists a constant L>0 such that
(2.4)‖Gnx-Gny‖≤L‖x-y‖,∀x,y∈H,n≥1.(3)
A mappingG:H→H is said to be semicompact if, for any bounded sequence {xn}⊂H with limn→∞∥xn-Gxn∥=0, there exists a subsequence {xni}⊂{xn} such that xni converges strongly to a point x*∈H.Now, we give one example of the (γ,{kn})-asymptotically strict pseudocontraction mapping.Example 2.3.
LetB be the unit ball in a Hilbert space l2, and define a mapping T:B→B by
(2.5)T=(x1,x2,…)=(0,x12,a2x2,a3x3,…),
where {ai} is a sequence in (0,1) such that Πi=2∞ai=1/2. It is proved in Goebel and Kirk [9] that(a)
∥Tx=Ty∥≤2∥x-y∥ for all x,y∈B,(b)
∥Tnx-Tny∥≤2Πj=2naj for all n≥2 and x,y∈B.Denote by k11/2=2, kn1/2=2Πj=2naj(n≥2) and γ∈[0,1). Then, we have
(2.6)lim⁡n→∞kn=lim⁡n→∞(2∏j=2naj)2=1,‖Tnx-Tny‖2≤kn‖x-y‖2+γ‖x-y-(Tnx-Tny)‖2,∀n≥1,x,y∈B,
and so the mapping T is a (γ,{kn})-asymptotically strict pseudocontraction.Remark 2.4.
(1) If we putγ=0 in (2.2), then the mapping G:H→H is asymptotically nonexpansive.
(2) If we putγ=0 in (2.3), then the mapping G:H→H is nonexpansive.
(3) Each (γ,{kn})-asymptotically strict pseudocontraction and each γ-strictly pseudocontraction both are demiclosed at origin [10].Proposition 2.5.
LetG:H→H be a (γ,{kn})- asymptotically strict pseudocontraction. If F(G)≠∅, then, for any q∈F(G) and x∈H, the following inequalities hold and they are equivalent:
(2.7)‖Gnx-q‖2≤kn‖x-q‖2+γ‖x-Gnx‖2,(2.8)〈x-Gnx,x-q〉≥1-γ2‖x-Gnx‖2-kn-12‖x-q‖2,(2.9)〈x-Gnx,q-Gnx〉≤1+γ2‖x-Gnx‖2+kn-12‖x-q‖2.Lemma 2.6 (see [11]).
Let{an},  {bn}, and {δn} be sequences of nonnegative real numbers satisfying
(2.10)an+1≤(1+δn)an+bn,∀n≥1.
If ∑i=1∞δn<∞ and ∑i=1∞bn<∞, then the limit limn→∞an exists.


## 3. Multiple-Set Split Feasibility Problem

For solving the multiple-set split feasibility problem (1.1), let us assume that the following conditions are satisfied:(C1)
H1 and H2 are two real Hilbert spaces, A:H1→H2 is a bounded linear operator;(C2)
Si:H1→H1, i=1,2,…,M, is a uniformly Li-Lipschitzian and (βi,{ki,n})-asymptotically strict pseudocontraction, and Ti:H2→H2, i=1,2,…,M, is a uniformly L̃i-Lipschitzian and (μi,{k̃i,n})-asymptotically strict pseudocontraction satisfying the following conditions:
(a)
C∶=⋂i=1MF(Si)≠∅ and Q∶=⋂i=1MF(Ti)≠∅,(b)
β=max1≤i≤Mβi<1 and μ=max1≤i≤Mμi<1,(c)
L∶=max1≤i≤MLi<∞ and L̃∶=max1≤i≤ML̃i<∞,(d)
kn=max1≤i≤M{ki,n,k̃i,n} and ∑n=1∞(kn-1)<∞.We are now in a position to give the following result.Theorem 3.1.
LetH1,  H2,  A,  {Si},  {Ti},  C,  Q,  β,  μ,  L,  L̃, and  {kn} be the same as above. Let  {xn} be the sequence generated by
(3.1)x1∈H1chosenarbitrarily,xn+1=(1-αn)un+αnSnn(un),un=xn+γA*(Tnn-I)Axn,∀n≥1,
where Snn=Sn(modM)n, Tnn=Tn(modM)n for all n≥1, {αn} is a sequence in [0,1], and γ>0 is a constant satisfying the following conditions.(e)
αn∈(δ,1-β) for all n≥1 and γ∈(0,(1-μ)/∥A∥2), where δ∈(0,1-β) is a positive constant.
(1)
IfΓ≠∅, then the sequence {xn} converges weakly to a point x*∈Γ.(2)
In addition, if there exists a positive integerj such that Sj is semicompact, then the sequences {xn} and {un} both converge strongly to a point x*∈Γ.Proof.
(1) The proof is divided into 5 steps as follows.
Step 1.
We first prove that, for anyp∈Γ, the limit
(3.2)lim⁡n→∞‖xn-p‖
exists. In fact, since p∈Γ, p∈C∶=⋂i=1MF(Si), and Ap∈Q∶=⋂i=1MF(Ti). From (3.1) and (2.8), it follows that
(3.3)‖xn+1-p‖2=‖un-p-αn(un-Snnun)‖2=‖un-p‖2-2αn〈un-p,un-Snnun〉+αn2‖un-Snnun‖2≤‖un-p‖2-αn{(1-β)‖un-Snnun‖2-(kn-1)‖un-p‖2}+αn2‖un-Snnun‖2=(1+αn(kn-1))‖un-p‖2-αn(1-β-αn)‖un-Snnun‖2.
On the other hand, since(3.4)‖un-p‖2=‖xn-p+γA*(Tnn-I)Axn‖2=‖xn-p‖2+γ2‖A*(Tnn-I)Axn‖2+2γ〈xn-p,A*(Tnn-I)Axn〉,(3.5)γ2‖A*(Tnn-I)Axn‖2=γ2〈A*(Tnn-I)Axn,A*(Tnn-I)Axn〉=γ2〈AA*(Tnn-I)Axn,(Tnn-I)Axn〉≤γ2‖A‖2‖TnnAxn-Axn‖2,(3.6)2γ〈xn-p,A*(Tnn-I)Axn〉=2γ〈Axn-Ap,(Tnn-I)Axn〉=2γ〈(Axn-Ap)+(Tnn-I)Axn-(Tnn-I)Axn,(Tnn-I)Axn〉=2γ{〈TnnAxn-Ap,TnnAxn-Axn〉-‖(Tnn-I)Axn‖2}.
Further, letting x=Axn, Gn=Tnn, q=Ap, γ=μ in (2.9) and noting Ap∈F(Tn), it follows that
(3.7)〈TnnAxn-Ap,TnnAxn-Axn〉≤1+μ2‖(Tnn-I)Axn‖2+kn-12‖Axn-Ap‖2≤1+μ2‖(Tnn-I)Axn‖2+(kn-1)‖A‖22‖xn-p‖2.
Substituting (3.7) into (3.6) and simplifying it, we have
(3.8)2γ〈xn-p,A*(Tnn-I)Axn〉≤γ(μ-1)‖(Tnn-I)Axn‖2+(kn-1)γ‖A‖2‖xn-p‖2.
Substituting (3.5) and (3.8) into (3.4) and simplifying it, we have
(3.9)‖un-p‖2≤‖xn-p‖2+γ2‖A‖2‖TnnAxn-Axn‖2+γ(μ-1)‖(Tnn-I)Axn‖2+(kn-1)γ‖A‖2‖xn-p‖2=‖xn-p‖2-γ(1-μ-γ‖A‖2)‖TnnAxn-Axn‖2+(kn-1)γ‖A‖2‖xn-p‖2.
Again, substituting (3.9) into (3.3) and simplifying it, we have
(3.10)‖xn+1-p‖2≤(1+αn(kn-1))×{‖xn-p‖2-γ(1-μ-γ‖A‖2)‖TnnAxn-Axn‖2+(kn-1)γ‖A‖2‖xn-p‖2}-αn(1-β-αn)‖un-Snnun‖2≤(1+αn(kn-1))‖xn-p‖2-γ(1-μ-γ‖A‖2)‖TnnAxn-Axn‖2+(1+αn(kn-1))(kn-1)γ‖A‖2‖xn-p‖2-αn(1-β-αn)‖un-Snnun‖2.
By the condition (e), we have
(3.11)‖xn+1-p‖2≤(1+αn(kn-1))‖xn-p‖2+(1+αn(kn-1))(kn-1)γ‖A‖2‖xn-p‖2≤(1+K(kn-1))‖xn-p‖2,
where
(3.12)K=sup⁡n≥1(αn+(1+αn(kn-1))γ‖A‖2)<∞.
By the condition (d), ∑n=1(kn-1)<∞; hence, from Lemma 2.6, we know that the following limit exists:
(3.13)lim⁡n→∞‖xn-p‖.
Step 2.
We will now prove that, for eachp∈Γ, the limit
(3.14)lim⁡n→∞‖un-p‖
exists. In fact, from (3.10) and (3.13), it follows that
(3.15)γ(1-μ-γ‖A‖2)‖(Tnn-I)Axn‖2+αn(1-β-αn)‖un-Snnun‖2≤‖xn-p‖2-‖xn+1-p‖2+K(kn-1)‖xn-p‖2⟶0(n⟶∞).
This together with the condition (e) implies that
(3.16)lim⁡n→∞‖un-Snnun‖=0,(3.17)lim⁡n→∞‖(Tnn-I)Axn‖=0.
Therefore, it follows from (3.4), (3.13), and (3.17) that the limit limn→∞∥un-p∥ exists.
Step 3.
Now, we prove that(3.18)lim⁡n→∞‖xn+1-xn‖=0,lim⁡n→∞‖un+1-un‖=0.
In fact, it follows from (3.1) that
(3.19)‖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 (3.16) and (3.17), we have
(3.20)lim⁡n→∞‖xn+1-xn‖=0.
Similarly, it follows from (3.1), (3.17), and (3.20) that
(3.21)‖un+1-un‖=‖xn+1+γA*(Tn+1n+1-I)Axn+1-(xn+γA*(Tnn-I)Axn)‖≤‖xn+1-xn‖+γ‖A*(Tn+1n+1-I)Axn+1‖+γ‖A*(Tnn-I)Axn‖⟶0(n⟶∞).
The conclusion (3.18) is proved.
Step 4.
Next, we prove that, for eachj=1,2,…,M,
(3.22)‖uiM+j-SjuiM+j‖⟶0,‖AxiM+j-TjAxiM+j‖⟶0(i⟶∞).
In fact, from (3.16), it follows that
(3.23)ζiM+j∶=‖uiM+j-SjiM+juiM+j‖⟶0(i⟶∞).
Since Sj is uniformly Lj-Lipschitzian continuous, it follows from (3.18) and (3.23) that
(3.24)‖uiM+j-SjuiM+j‖≤‖uiM+j-SjiM+juiM+j‖+‖SjiM+juiM+j-SjuiM+j‖≤ζiM+j+Lj‖SjiM+j-1uiM+j-uiM+j‖≤ζiM+j+Lj{‖SjiM+j-1uiM+j-SjiM+j-1uiM+j-1‖+‖SjiM+j-1uiM+j-1-uiM+j‖}≤ζiM+j+Lj2‖uiM+j-uiM+j-1‖+Lj‖SjiM+j-1uiM+j-1-uiM+j-1+uiM+j-1-uiM+j‖≤ζiM+j+Lj(1+Lj)‖uiM+j-uiM+j-1‖+LjζiM+j-1⟶0(i⟶∞).
Similarly, for eachj=1,2,…,M, it follows from (3.17) that
(3.25)ξiM+j∶=‖AxiM+j-TjiM+jAxiM+j‖⟶0(i⟶∞).
Since Tj is uniformly L̃j-Lipschitzian continuous, by the same way as above, from (3.18) and (3.25), we can also prove that
(3.26)‖AxiM+j-TjAxiM+j‖⟶0(i⟶∞).
Step 5.
Finally, we prove thatxn⇀x* and un⇀x*, which is a solution of the problem (MSSFP). In fact, since {un} is bounded, there exists a subsequence {uni}⊂{un} such that uni⇀x*∈H1. Hence, for any positive integer j=1,2,…,M, there exists a subsequence {ni(j)}⊂{ni} with ni(j)(modM)=j such that uni(j)⇀x*. Again, from (3.22), it follows that
(3.27)‖uni(j)-Sjuni(j)‖⟶0(ni(j)⟶∞).
Since Sj is demiclosed at zero (see Remark 2.4), it follows that x*∈F(Sj). By the arbitrariness of j=1,2,…,M, we have x*∈C∶=⋂j=1MF(Sj).
Moreover, it follows from (3.1) and (3.17) that
(3.28)xni=uni-γA*(Tnini-I)Axni⇀x*.
Since A is a linear bounded operator, it follows that Axni⇀Ax*. For any positive integer k=1,2,…,M, there exists a subsequence {ni(k)}⊂{ni} with ni(k)(modM)=k such that Axni(k)⇀Ax*. In view of (3.22), we have
(3.29)‖Axni(k)-TkAxni(k)‖⟶0(ni(k)⟶∞).
Since Tk is demiclosed at zero, we have Ax*∈F(Tk). By the arbitrariness of k=1,2,…,M, it follows that Ax*∈Q∶=⋂k=1MF(Tk). This together with x*∈C shows that x*∈Γ, that is, x* is a solution to the problem (MSSFP).
Now, we prove thatxn⇀x* and un⇀x*. In fact, assume that there exists another subsequence {unj}⊂{un} such that unj⇀y*∈Γ with y*≠x*. Consequently, by virtue of (3.2) and Opial’s property of Hilbert space, we have
(3.30)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, un⇀x*. By using (3.1) and (3.17), we have
(3.31)xn=un-γA*(Tnn-I)Axn⇀x*.
Therefore, the conclusion (I) follows.
(2) Without loss of generality, we can assume thatS1 is semicompact. It follows from (3.27) that
(3.32)‖uni(1)-S1uni(1)‖⟶0(ni(1)⟶∞).
Therefore, there exists a subsequence of {uni(1)} (for the sake of convenience, we still denote it by {uni(1)}) such that uni(1)→u*∈H. Since uni(1)⇀x*, x*=u* and so uni(1)→x*∈Γ. By virtue of (3.2), we know that
(3.33)lim⁡n→∞‖un-x*‖=0,lim⁡n→∞‖xn-x*‖=0,
that is, {un} and {xn} both converge strongly to the point x*∈Γ. This completes the proof.If we putγ=0 in Theorem 3.1, we can get the following.Corollary 3.2.
LetH,  C,  L and  {kn} be the same as above and  {Si} a family of asymptotically nonexpansive mappings. Let  {xn} be the sequence generated by
(3.34)x1∈H1chosenarbitrarily,xn+1=(1-αn)xn+αnSnn(xn),∀n≥1,
where Snn=Sn(modM)n for all n≥1 and {αn} is a sequence in [0,1] satisfying the following conditions.(e)
αn∈(δ,1-β) for all n≥1, where δ∈(0,1-β) is a positive constant.
(1)
IfΓ≠∅, then the sequence {xn} converges weakly to a point x*∈Γ.(2)
In addition, if there exists a positive integerj such that Sj is semicompact, then the sequence {xn} converges strongly to a point x*∈Γ.The following theorem can be obtained from Theorem3.1 immediately.Theorem 3.3.
LetH1 and  H2 be two real Hilbert spaces,  A:H1→H2 a bounded linear operator,  Si:H1→H1, i=1,2,…,M, a uniformly  Li-Lipschitzian and  βi-strict pseudocontraction, and  Ti:H2→H2,i=1,2…,M, a uniformly  L̃i-Lipschitzian and  μi-strict pseudocontraction satisfying the following conditions: (a)
C∶=⋂i=1MF(Si)≠∅ and Q∶=⋂i=1MF(Ti)≠∅,(b)
β=max1≤i≤Mβi<1 and μ=sup1≤i≤Mμi<1.Let {xn} be the sequence generated by
(3.35)x1∈H1chosenarbitrarily,xn+1=(1-αn)un+αnSn(un),un=xn+γA*(Tn-I)Axn,∀n≥1,
where Sn=Sn(modM), Tn=Tn(modM), {αn} is a sequence in [0,1], and 0<γ<1 is a constant. If Γ≠∅ and the following condition is satisfied:(c)
αn∈(δ,1-β) for all n≥1 and γ∈(0,(1-μ)/∥A∥2), where δ∈(0,1-β) is a constant,then the sequence {xn} converges weakly to a point x*∈Γ. In addition, if there exists a positive integer j such that Sj is semicompact, then the sequences {xn} and {un} both converge strongly to the point x*.Proof.
By the same way as given in the proof of Theorem3.1 and using the case of strict pseudocontraction with the sequence {kn=1}, we can prove that, for each p∈Γ, the limits limn→∞∥xn-p∥ and limn→∞∥un-p∥ exist,
(3.36)‖un-Snun‖→0,‖Axn-TnAxn‖→0,‖un-un+1‖→0,‖xn-xn+1‖→0,xn⇀x*,un⇀x*∈Γ.
In addition, if there exists a positive integerj such that Sj is semicompact, we can also prove that {xn} and {un} both converge strongly to the point x*. This completes the proof.If you putSi=Ti or Ti=I (: the identity mapping) for each i=1,2…,M in Theorem 3.3, then we have the following.Corollary 3.4.
LetH be a real Hilbert space and  Si:H→H,  i=1,2,…,M, a uniformly  Li-Lipschitzian and  βi-strict pseudocontraction satisfying the following conditions: (a)
C∶=⋂i=1MF(Si)≠∅,(b)
β=max1≤i≤Mβi<1. Let {xn} be the sequence generated by
(3.37)x1∈H1chosenarbitrarily,xn+1=(1-αn)xn+αnSn(xn),∀n≥1,
where Sn=Sn(modM) and {αn} is a sequence in [0,1]. If Γ≠∅ and the following condition is satisfied:(c)
αn∈(δ,1-β) for all n≥1, where δ∈(0,1-β) is a constant,then the sequence {xn} converges weakly to a point x*∈Γ. In addition, if there exists a positive integer j such that Sj is semicompact, then the sequences {xn} converges strongly to the point x*.Remark 3.5.
Theorems3.1 and 3.3 improve and extend the corresponding results of Censor et al. [1, 4, 5], Byrne [2], Yang [7], Moudafi [12], Xu [13], Censor and Segal [14], Masad and Reich [15], and others in the following aspects:(1)
for the framework of spaces, we extend the space from finite dimension Hilbert space to infinite dimension Hilbert space;(2)
for the mappings, we extend the mappings from nonexpansive mappings, quasi-nonexpansive mapping or demicontractive mappings to finite families of asymptotically strictly pseudocontractions;(3)
for the algorithms, we propose some new hybrid iterative algorithms which are different from ones given in [1, 2, 4, 5, 7, 14, 15]. And, under suitable conditions, some weak and strong convergences for the algorithms are proved.



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